Time-changed Feller’s Brownian motions are birth-death processes

Liping Li Fudan University, Shanghai, China. [email protected]

Abstract.

A Feller’s Brownian motion refers to a Feller process on the interval $[0,\infty)$ that is equivalent to the absorbing Brownian motion before reaching $0$ . It is fully determined by four parameters $(p_{1},p_{2},p_{3},p_{4})$ , reflecting its killing, reflecting, sojourn, and jumping behaviors at the boundary $0$ . On the other hand, a birth-death process is a continuous-time Markov chain on $\mathbb{N}$ with a given transition density matrix $Q$ , and it is characterized by three parameters $(\gamma,\beta,\nu)$ that describe its killing, reflecting, and jumping behaviors at the boundary $\infty$ . The primary objective of this paper is to establish a connection between Feller’s Brownian motion and birth-death process. We will demonstrate that any Feller’s Brownian motion can be transformed into a specific birth-death process through a unique time change transformation, and conversely, any birth-death process can be derived from Feller’s Brownian motion via time change. Specifically, the birth-death process generated by the Feller’s Brownian motion, determined by the parameters $(p_{1},p_{2},p_{3},p_{4})$ , through time change, has the parameters:

\gamma=p_{1},\quad\beta=2p_{2},\quad\nu_{n}=\mathfrak{p}_{n},\;n\in\mathbb{N},

where $\{\mathfrak{p}_{n}:n\in\mathbb{N}\}$ is a sequence derived by allocating weights to the measure $p_{4}$ in a specific manner. Utilizing the pathwise representation of Feller’s Brownian motion, our results provide a pathwise construction scheme for birth-death processes, addressing a gap in the existing literature.

Key words and phrases:

Feller’s Brownian motions, Birth-death processes, Continuous-time Markov chains, Time change, Boundary conditions, Local times, Dirichlet forms, Approximation.

2020 Mathematics Subject Classification:

Primary 60J27, 60J40, 60J46, 60J50, 60J74.

The author is a member of LMNS, Fudan University. He is partially supported by NSFC (No. 11931004 and 12371144).

1. Introduction

To study the boundary behavior of Markov processes, historically, two classical probabilistic models have been developed that seem different but are actually similar. The first model is based on Brownian motion on the half-line $(0,\infty)$ , allowing all possible behaviors of the process at the boundary $0$ while ensuring the strong Markov property. This line of work began with Feller’s research on the theory of transition semigroup [5] and was later extended by Itô and McKean, who completed the pathwise construction of all such processes in [10]. These processes are referred to as Feller’s Brownian motion by Itô and McKean, with a detailed definition provided in [10, §5] (see also Definition 3.1). From an analytical perspective, the key to determining the domain of the generator of Feller’s Brownian motion is the boundary condition at $0$ :

p_{1}f(0)-p_{2}f^{\prime}(0)+\frac{p_{3}}{2}f^{\prime\prime}(0)+\int_{(0,\infty)}[f(0)-f(x)]p_{4}(dx)=0,

(1.1)

where $p_{1},p_{2},p_{3}\geq 0$ are constants, and $p_{4}$ is a positive measure on $(0,\infty)$ satisfying $\int_{(0,\infty)}1\wedge x\,p_{4}(dx)<\infty$ . From a probabilistic perspective, these four parameters $(p_{1},p_{2},p_{3},p_{4})$ represent the four possible types of boundary behavior of Feller’s Brownian motion at $0$ : killing, reflecting, sojourn, and jumping. The work of Itô and McKean [10] deeply explores the mechanisms underlying these four types of boundary behavior, providing a clear and intuitive pathwise representation of Feller’s Brownian motion; see §3.2 for a brief summarization.

The other type of model is a fundamental one in the theory of continuous-time Markov chains, known as the birth-death process. This process is induced by a standard transition matrix, whose derivative at time $0$ is the density matrix $Q$ given by (2.1), on the discrete space $\mathbb{N}$ . Its defining characteristic is that from any given point, the process can only jump to its immediate left or right neighbors, a feature akin to the continuity of Brownian motion trajectories. Since the birth-death process may reach $\infty$ in finite time, describing its behavior at $\infty$ and after reaching $\infty$ is a fundamental problem in the theory of birth-death processes. This line of inquiry also began with Feller, who in [6] observed that the boundary behavior of the birth-death process at $\infty$ bears some intrinsic similarity to the boundary behavior of the aforementioned Feller’s Brownian motion at $0$ . Feller’s approach was analytical; although he only examined partial birth-death processes, he derived the boundary condition at $\infty$ for the resolvent function $F$ (i.e., the function in the domain of the generator) as follows:

\gamma F(\infty)+\frac{\beta}{2}F^{+}(\infty)+\sum_{k\in\mathbb{N}}(F(\infty)-F(k))\nu_{k}=0,

(1.2)

where $F^{+}$ denotes the discrete gradient (see §2.3), $\gamma,\beta\geq 0$ are constants, and $\nu=(\nu_{k})_{k\in\mathbb{N}}$ is a positive measure on $\mathbb{N}$ . Shortly thereafter, building on Feller’s work, Yang in 1965 (see [26, Chapter 7]) completed the analytical construction of all birth-death processes using the resolvent method, thereby finalizing the construction theory from an analytical perspective. In our recent article [18], we proved that all (non-Doob) birth-death processes satisfy the boundary condition (1.2) observed by Feller. Conversely, Wang, in his 1958 doctoral thesis, provided a probabilistic construction of all birth-death processes (see [26, Chapter 6]). However, it should be noted that this probabilistic construction differs from the pathwise construction of Feller’s Brownian motion by Itô and McKean, as it instead uses a series of Doob processes to approximate the original birth-death process. In fact, understanding the probabilistic intuition behind the parameters $(\gamma,\beta,\nu)$ through this construction is challenging, although, by comparing with (1.1), it becomes evident that these parameters should correspond to the killing, reflecting, and jumping behaviors of the birth-death process at $\infty$ , respectively.

Apart from two minor differences, the structural consistency of the boundary conditions (1.1) and (1.2) is quite evident: First, the reflecting term in (1.1) is negative, while in (1.2) it is positive; this is because $0$ and $\infty$ are at opposite ends of their respective state spaces, thus causing the direction of the gradient to be exactly opposite. Second, (1.2) lacks a sojourn term; this is because in the definition of the birth-death process, the index set of the transition matrix is $\mathbb{N}$ , which causes that the birth-death process can not visit $\infty$ for a positive duration (correspondingly, the process that allows sojourn at $\infty$ is called a generalized birth-death process, and its index set is $\mathbb{\mathbb{N}}\cup\{\infty\}$ ; see [6]). Naturally, we hope to find a deeper connection between the two beyond methodology.

For the symmetric case (from an analytical perspective, that is, when the transition semigroup satisfies both the Kolmogorov backward and forward equations; see [6]), there is no jumping behavior at the boundary for either process, and the boundary conditions reduce to the classic Dirichlet, Neumann, or Robin boundaries (see [18, §2.3]). At this point, Feller’s Brownian motion and birth-death process are both special cases of a more general class of processes known as quasidiffusions. These processes are also referred to as generalized diffusions or gap diffusions in some literature, such as [15, 14]. The study of quasidiffusions originated from Kac and Krein’s research on the spectral theory of a class of generalized second-order differential operators [12]. Soon after, these self-adjoint operators were realized as strong Markov processes on some closed subset of $[-\infty,\infty]$ with trajectories that satisfy the skip-free property; see [13]. The skip-free property is a combination of the continuity of Brownian motion trajectories and the characteristic of birth-death process trajectories: the trajectory is continuous where the space is continuous and can only jump to adjacent positions where there is a gap in space. In addition to studying quasidiffusions within the classic Feller framework (see [24]), with the aid of symmetry, we can also use the Dirichlet form theory established by Fukushima (see [3, 8]) to study them, as demonstrated in several recent articles [17, 19, 16]. In general, the connection between symmetric Feller’s Brownian motions and symmetric birth-death processes is very clear: they can not only be unified under the more general quasidiffusion but also, the symmetric birth-death process is a time-changed process of a certain symmetric Feller’s Brownian motion. For example, the minimal birth-death process corresponds to the time-changed process of absorbing Brownian motion, and the $(Q,1)$ -process corresponds to that of reflecting Brownian motion; see [18, §3].

However, for the non-symmetric case (especially the so-called ‘pathological’ case by Feller, where $p_{4}$ or $\nu$ is an infinite measure), the connection between Feller’s Brownian motion and birth-death process becomes very vague. Indeed, both theories have been extensively studied and developed within their respective fields. The former has given rise to a mature theory of diffusion processes, see, e.g., [11, 21], which is one of the most important classes of models in the general theory of Markov processes; the latter, originating from the discrete space, has also evolved a series of theories, such as continuous-time Markov chains (see, e.g., [4]) and Markov jump processes (see, e.g., [2]). However, during their parallel development, the connection between the two is more reflected in the mutual borrowing of research methods, and few people discuss the fit as reflected in the boundary conditions (1.1) and (1.2).

The goal of our paper is to construct the missing bridge between Feller’s Brownian motion and birth-death process. In summary, our main result is that any Feller’s Brownian motion can be transformed into a specific birth-death process through a unique time change transformation; conversely, any birth-death process can be derived through the time change transformation of some Feller’s Brownian motion.

To facilitate the explanation of our results, we first perform a spatial transformation on the birth-death process as described in §4.1, aligning it with the natural scale of Feller’s Brownian motion and reflecting the boundary point $\infty$ so that it moves to $0$ , thereby matching the boundary point of Feller’s Brownian motion. The transformed space is denoted by (see (4.1))

\overline{E}=\{\hat{c}_{n}:n\in\mathbb{N}\}\cup\{0\},

where the subscript $n$ of $\hat{c}_{n}$ represents the point before the transformation, and $\lim_{n\to\infty}\hat{c}_{n}=0$ is topologically consistent with the one-point compactification $\mathbb{N}\cup\{\infty\}$ of $\mathbb{N}$ . Under this spatial transformation, aside from the change in the symbol of the state points, the transformed process, which we denote by $\hat{X}$ , remains indistinguishable from the original birth-death process. For convenience, we refer to $\hat{X}$ as a birth-death process on $\overline{E}$ . Additionally, due to the change in the direction of the gradient at the boundary point, the corresponding boundary condition for $\hat{X}$ is modified to:

\gamma\hat{F}(0)-\frac{\beta}{2}\hat{F}^{+}(0)+\sum_{k\in\mathbb{N}}(\hat{F}(0)-\hat{F}(\hat{c}_{k}))\nu_{k}=0,

(1.3)

where $\hat{F}$ is the transformed function of $F$ in (1.2). After this modification, the aforementioned boundary condition closely resembles (1.1).

Our main results, namely Theorems 4.2 and 5.1, can be stated as follows: Given the birth-death density matrix $Q$ , for the Feller’s Brownian motion $Y$ corresponding to the boundary condition (1.1), it can always (and uniquely, see Theorem 4.4) be transformed into a birth-death process on $\overline{E}$ through a time change transformation induced by a positive continuous additive functional $A=(A_{t})_{t\geq 0}$ . This positive continuous additive functional $A$ given by (4.6) is the integral of the local times of $Y$ with respect to the speed measure $\mu$ determined by $Q$ . Moreover, the parameters $(\gamma,\beta,\nu)$ exhibited in the boundary condition (1.3) of the birth-death process obtained by time change can be derived from the parameters $(p_{1},p_{2},p_{3},p_{4})$ of $Y$ as follows:

\gamma=p_{1},\quad\beta=2p_{2},\quad\nu_{n}=\mathfrak{p}_{n},\;n\in\mathbb{N},

(1.4)

where $\{\mathfrak{p}_{n}:n\in\mathbb{N}\}$ is the sequence obtained by allocating weights to the measure $p_{4}$ in the manner of (5.1) and (5.2). Particularly, $p_{4}=\sum_{n\in\mathbb{N}}\mathfrak{p}_{n}\delta_{\hat{c}_{n}}$ whenever $p_{4}$ is supported on $\{\hat{c}_{n}:n\in\mathbb{N}\}$ . It should be noted that the sojourn parameter $p_{3}$ of $Y$ does not play any role in the expression (1.4). This is because any positive duration of stay at $0$ by $Y$ will be eliminated by the time change.

From the relationship between the two sets of parameters in (1.4), it is not difficult to further deduce that every birth-death process on $\overline{E}$ can be obtained by time change from a certain Feller’s Brownian motion; see Corollary 5.2. This conclusion also provides a pathwise construction for all birth-death processes: the trajectory of $\hat{X}$ is the trace of the corresponding Feller’s Brownian motion on $\overline{E}$ ; see §5.2.

Let us introduce some of the tools that will be used in the proof of our main results. For the symmetric case, we primarily employ the theory of Dirichlet forms. The concepts and notation we use are consistent with the foundational references [3, 8], so they will not be further elaborated upon in the text. For the non-symmetric case where $p_{4}$ is a finite measure, it essentially involves the Ikeda-Nagasawa-Watanabe piecing out transformation (see [9]) of the symmetric case. This is very similar to the discussion of birth-death processes found in [18, §5 and §8]. Based on this piecing out transformation, we can directly construct the trajectories of the birth-death process without relying on Feller’s Brownian motion; see §6. The most challenging scenario arises when $p_{4}$ is an infinite measure. In this case, the piecing out transformation becomes ineffective, and the jumps of Feller’s Brownian motion at the boundary become very frequent and complex. Notably, our previous study [18] also regrettably stops short of studying this case for the birth-death process. To address this situation, we draw on Wang’s idea of constructing an approximating sequence for the birth-death process (see [26]), and we similarly construct a sequence of approximating processes for Feller’s Brownian motion (see §7). The boundary behavior of these processes is relatively simple, and they can be linked to the approximating sequence of the corresponding birth-death process, thereby ultimately establishing the parameter relationship (1.4) between the target processes.

At the end of this section, we briefly describe the commonly used notation. The symbols contained in the Markov process $Y=(\Omega,\mathscr{G},\mathscr{G}_{t},Y_{t},\theta_{t},\mathbf{P}_{x})$ are standard, as seen in [1, 25]. For brevity, some elements are sometimes omitted, such as writing $Y=(\Omega,Y_{t},\mathbf{P}_{x})$ . The expectation corresponding to the probability measure $\mathbf{P}_{x}$ is denoted by $\mathbf{E}_{x}$ . In this paper, the cemetery point for Markov processes is uniformly denoted by $\partial$ . Given a topological space $E$ , $\mathcal{B}(E),\mathcal{B}_{+}(E),\mathcal{B}_{b}(E),C(E)$ and $C_{b}(E)$ represent the space of all Borel measurable, non-negative Borel measurable, bounded Borel measurable functions, totality of continuous functions, and bounded continuous functions, respectively. If $E$ is an interval, $C_{c}(E)$ and $C_{0}(E)$ denote the set of all continuous functions with compact support and the set of all continuous functions that equal $0$ at the open endpoints, respectively. The notation for continuous functions with subscripts indicates differentiability, for example, $C^{2}$ indicates twice continuously differentiable, while $C^{\infty}$ indicates infinitely continuously differentiable. The symbol $\delta_{a}$ stands for the Dirac measure at $a\in[0,\infty)$ .

2. Birth-death processes

In this paper, we will strive to use the same terminologies and symbols as in [18] for the birth-death processes. The following restates only some of the important content.

2.1. Elements of $Q$ -processes

We consider a birth-death density matrix as follows:

Q=(q_{ij})_{i,j\in\mathbb{N}}:=\left(\begin{array}[]{ccccc}-q_{0}&b_{0}&0&0&\cdots\\ a_{1}&-q_{1}&b_{1}&0&\cdots\\ 0&a_{2}&-q_{2}&b_{2}&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots\end{array}\right),

(2.1)

where $a_{k}>0$ for $k\geq 1$ and $b_{k}>0,q_{k}=a_{k}+b_{k}$ for $k\geq 0$ . (Set $a_{0}=0$ for convenience.) A continuous-time Markov chain $X=(X_{t})_{t\geq 0}$ is called a birth-death $Q$ -process (or $Q$ -process for short) if its transition matrix $(p_{ij}(t))_{i,j\in\mathbb{N}}$ is standard and its density matrix is $Q$ , i.e., $p^{\prime}_{ij}(0)=q_{ij}$ for $i,j\in\mathbb{N}$ . Readers are referred to [4, 26] for the terminologies concerning continuous-time Markov chains; see also [18]. In our consideration, two $Q$ -processes with the same transition matrix will not be distinguished. For convenience, we will also refer to such $(p_{ij}(t))_{i,j\in\mathbb{N}}$ as a $Q$ -process when there is no risk of confusion.

There always exists at least one $Q$ -process $X^{\text{min}}$ , known as the minimal $Q$ -process. This process is killed at the first time it almost reaches $\infty$ . Similar to a regular diffusion on an interval, the minimal $Q$ -process can be characterized by the scale function (on $\mathbb{N}$ ):

c_{0}=0,\quad c_{1}=\frac{1}{2b_{0}},\quad c_{k}=\frac{1}{2b_{0}}+\sum_{i=2}^{k}\frac{a_{1}a_{2}\cdots a_{i-1}}{2b_{0}b_{1}\cdots b_{i-1}},\;k\geq 2

(2.2)

and the speed measure $\mu$ on $\mathbb{N}$ :

\mu(\{0\}):=\mu_{0}=1,\quad\mu(\{k\}):=\mu_{k}=\frac{b_{0}b_{1}\cdots b_{k-1}}{a_{1}a_{2}\cdots a_{k}},\;k\geq 1.

(2.3)

The transition matrix $(p^{\text{min}}_{ij}(t))$ of $X^{\text{min}}$ is symmetric with respect to $\mu$ in the sense that $\mu_{i}p_{ij}^{\text{min}}(t)=\mu_{j}p_{ji}^{\text{min}}(t)$ for all $i,j\in\mathbb{N}$ and $t\geq 0$ . For more details about this characterization, please refer to [18, §3.1].

2.2. Resolvent representation of $Q$ -processes

When $\infty$ is regular or an exit (for $X^{\text{min}}$ ) in Feller’s sense (see, e.g., [18, Definition 3.3]), there exist other $Q$ -processes besides the minimal one, such as Doob processes (see, e.g., [18, §5]) and the $(Q,1)$ -process (which is only applicable in the regular case; see, e.g., [18, §3.3]). In this case, we have $c_{\infty}:=\lim_{k\rightarrow\infty}c_{k}<\infty$ .

In what follows, let us present a well-known analytic approach to characterize all birth-death processes by solving their resolvents. For $\alpha>0$ and $i\in\mathbb{N}$ , define

u_{\alpha}(i)=\mathbf{E}_{i}^{\text{min}}e^{-\alpha\zeta^{\text{min}}},

where $\mathbf{E}^{\text{min}}_{i}$ stands for the expectation of $X^{\text{min}}$ starting from $i$ and $\zeta^{\text{min}}$ is the lifetime of $X^{\text{min}}$ . Denote by $(R^{\text{min}}_{\alpha})_{\alpha>0}$ the resolvent of the minimal $Q$ -process. Set

\Phi_{ij}(\alpha):=R^{\text{min}}_{\alpha}(i,\{j\}),\quad\alpha>0,i,j\in\mathbb{N}.

Let $\nu$ be a positive measure on $\mathbb{N}$ and let $\gamma,\beta\geq 0$ be two constants. Set $|\nu|:=\sum_{k\geq 1}\nu_{k}$ . When both

\sum_{k\geq 0}\nu_{k}\left(\sum_{j=k}^{\infty}(c_{j+1}-c_{j})\sum_{i=0}^{j}\mu_{i}\right)<\infty,\quad|\nu|+\beta\neq 0,

(2.4)

and

\beta=0,\quad\text{if }\infty\text{ is an exit}

(2.5)

are satisfied, define, for $\alpha>0$ and $i,j\in\mathbb{N}$ ,

\Psi_{ij}(\alpha):=\Phi_{ij}(\alpha)+u_{\alpha}(i)\frac{\sum_{k\geq 0}\nu_{k}\Phi_{kj}(\alpha)+\beta\mu_{j}u_{\alpha}(j)}{\gamma+\sum_{k\geq 0}\nu_{k}(1-u_{\alpha}(k))+\beta\alpha\sum_{k\geq 0}\mu_{k}u_{\alpha}(k)}.

(2.6)

The matrix $(\Psi_{ij}(\alpha))_{i,j\in\mathbb{N}}$ is called the $(Q,\gamma,\beta,\nu)$ -resolvent matrix. For a constant $M>0$ , the $(Q,M\gamma,M\beta,M\nu)$ -resolvent matrix is obviously the same as the $(Q,\gamma,\beta,\nu)$ -resolvent matrix.

The following theorem is attributed to [26, §7.6]. Note that a $Q$ -process is called honest if its transition semigroup $(p_{ij}(t))$ satisfies $\sum_{j\in\mathbb{N}}p_{ij}(t)=1$ for all $i\in\mathbb{N}$ and $t\geq 0$ .

Theorem 2.1.

The transition matrix $(p_{ij}(t))_{i,j\in\mathbb{N}}$ is a $Q$ -process that is not the minimal one, if and only if there exists a unique (up to a multiplicative positive constant) triple $(\nu,\gamma,\beta)\geq 0$ with (2.4) and (2.5) such that the resolvent of $(p_{ij}(t))_{i,j\in\mathbb{N}}$ is given by

R_{\alpha}(i,\{j\})=\Psi_{ij}(\alpha),\quad\alpha>0,i,j\in\mathbb{N},

where $(\Psi_{ij}(\alpha))_{i,j\in\mathbb{N}}$ is the $(Q,\gamma,\beta,\nu)$ -resolvent matrix. Furthermore,

(1)

$(p_{ij}(t))_{i,j\in\mathbb{N}}$ is honest, if and only if $\gamma=0$ .
(2)

$(p_{ij}(t))_{i,j\in\mathbb{N}}$ is a Doob process, if and only if $0<|\nu|<\infty$ and $\beta=0$ .

2.3. Ray-Knight compactification of $Q$ -processes

In a previous study [18], we obtained a càdlàg modification for each $Q$ -process $X$ , denoted by $\bar{X}=(\bar{X}_{t})_{t\geq 0}$ , using the Ray-Knight compactification. The state space of $\bar{X}$ is $\mathbb{N}\cup\{\infty\}$ , which corresponds to the Alexandroff compactification of $\mathbb{N}$ . It is important to note that the Ray-Knight compactifications of Doob processes are not normal, whereas those of other $Q$ -processes are Feller processes on $\mathbb{N}\cup\{\infty\}$ ; see [18, Corollary 5.2]. If no ambiguity arises, we will not distinguish between the $Q$ -process and its Ray-Knight compactification. Additionally, we will refer to a non-minimal and non-Doob $Q$ -process as a Feller $Q$ -process.

Regarding a Feller $Q$ -process $X$ , we can further derive its infinitesimal generator by utilizing the resolvent representation in Theorem 2.1; see [18, Theorem 6.3]. The crucial fact is that every function $F$ in the domain of the infinitesimal generator satisfies the following boundary condition at $\infty$ :

\frac{\beta}{2}F^{+}(\infty)+\sum_{k\in\mathbb{N}}(F(\infty)-F(k))\nu_{k}+\gamma F(\infty)=0,

(2.7)

where $F(\infty):=\lim_{k\rightarrow\infty}F(k)$ (if it exists), $F^{+}(k):=(F(k+1)-F(k))/(c_{k+1}-c_{k})$ for $k\in\mathbb{N}$ and $F^{+}(\infty):=\lim_{k\rightarrow\infty}F^{+}(k)$ (if it exists). (When deriving the boundary condition (2.7) in [18], we mistakenly wrote the parameter $\frac{\beta}{2}$ as $\beta$ . This error occurred because the scale function used in [18], specifically (2.2), is half of the scale function in [6]. However, in the proof of [18, Theorem 6.3], when citing the result from [6], particularly the equation above [18, (6.8)], we forgot to multiply by two accordingly. As a result, the parameter $\frac{\beta}{2}$ in [18, (6.10)] was incorrectly written as $\beta$ . Unfortunately, this mistake was not corrected before the publication of [18].)

3. Feller’s Brownian motions

The terminology of Feller’s Brownian motions is borrowed from the renowned article [10] by Itô and McKean. This class of Markov processes was first discovered by Feller in [5], with its detailed definition found in [10, §5]. Recall that a Markov process on $[0,\infty)$ is called a Feller process if its semigroup $(T_{t})_{t\geq 0}$ satisfies the following conditions: $T_{0}$ is the identify mapping, $T_{t}C_{0}([0,\infty))\subset C_{0}([0,\infty))$ and $T_{t}f\rightarrow f$ in $C_{0}([0,\infty))$ as $t\rightarrow 0$ for any $f\in C_{0}([0,\infty))$ , where $C_{0}([0,\infty))$ is the Banach space consisting of all continuous functions $f$ on $[0,\infty)$ such that $\lim_{x\rightarrow\infty}f(x)=0$ . We provide a definition for Feller’s Brownian motions in a modern manner as follows.

Definition 3.1.

A Feller process $Y:=(\Omega,{\mathscr{G}},{\mathscr{G}}_{t},Y_{t},\theta_{t},(\mathbf{P}_{x})_{x\in[0,\infty)})$ with lifetime $\zeta$ on $[0,\infty)$ is called a Feller’s Brownian motion if its killed process upon hitting $0$ is identical in law to the absorbing Brownian motion on $(0,\infty)$ .

Remark 3.2.

This definition excludes the special cases discussed in [10, §6]. The ‘Brownian motions’ in these special cases do not satisfy the normal property at $0$ , i.e., $\mathbf{P}_{0}(Y_{0}=0)\neq 1$ , or the quasi-left-continuity at the first time reaching $0$ , i.e., for $x>0$ , $\mathbf{P}_{x}(Y_{\lim_{n\rightarrow\infty}\sigma_{n}}=0)<1$ , where $\sigma_{n}:=\inf\{t>0:Y_{t}=\varepsilon_{n}\}$ for any sequence $\varepsilon_{n}\downarrow 0$ .

The case 4a. in [10, §6] warrants special mention. Similar to Doob processes in the context of $Q$ -processes, it corresponds to the piecing out of the absorbing Brownian motion $Y^{0}$ with lifetime $\zeta^{0}$ on $(0,\infty)$ with respect to a certain probability measure $\lambda$ on $(0,\infty)\cup\{\partial\}$ . The process obtained through this piecing out is a right process on $(0,\infty)$ , but it is not a Feller process. It can, however, be extended to a Ray process $\bar{Y}$ on $[0,\infty)$ , where the initial transition function at $0$ is given by $\bar{T}_{0}(0,\cdot):=\bar{\mathbf{P}}_{0}(\bar{Y}_{0}\in\cdot)=\lambda(\cdot)$ . Analogous to [18, Theorem 5.1], the following expression for the resolvent of $\bar{Y}$ can be easily obtained:

\bar{G}_{\alpha}h(x)=G^{0}_{\alpha}h(x)+\mathbf{E}_{x}\left(e^{-\alpha\zeta^{0}}\right)\cdot\frac{\int_{(0,\infty)}G^{0}_{\alpha}h(x)\lambda(dx)}{1-\mathbf{E}_{\lambda}\left(e^{-\alpha\zeta^{0}}\right)},\quad\forall h\in\mathcal{B}_{+}((0,\infty)),

(3.1)

where $G^{0}_{\alpha}$ is the resolvent of $Y^{0}$ and $\mathbf{E}_{\lambda}\left(e^{-\alpha\zeta^{0}}\right):=\int_{(0,\infty)}\mathbf{E}_{x}\left(e^{-\alpha\zeta^{0}}\right)\lambda(dx)$ .

Indeed, the reflecting Brownian motion on $[0,\infty)$ is a classic example of a Feller’s Brownian motion. However, it’s important to note that not all Feller’s Brownian motions exhibit a.s. continuous sample paths. Some Feller’s Brownian motions may experience jumps into $(0,\infty)$ from $0$ or just before reaching $0$ .

3.1. Infinitesimal generators

It is well known that a function $f\in C_{0}([0,\infty))$ is said to belong to the domain $\mathcal{D}(\mathscr{L})$ of the infinitesimal generator of $Y$ if the limit

\mathscr{L}f:=\lim_{t\downarrow 0}\frac{T_{t}f-f}{t}

exists in $C_{0}([0,\infty))$ . The operator $\mathscr{L}:\mathcal{D}(\mathscr{L})\rightarrow C_{0}([0,\infty))$ thus defined is called the infinitesimal generator of the process $Y$ . The following characterization of infinitesimal generators of Feller’s Brownian motions is attributed to Feller [5].

Theorem 3.3.

A Markov process $Y=(Y_{t})_{t\geq 0}$ on $[0,\infty)$ is a Feller’s Brownian motion, if and only if it is a Feller process on $[0,\infty)$ whose infinitesimal generator on $C_{0}([0,\infty))$ is $\mathscr{L}f=\frac{1}{2}f^{\prime\prime}$ with domain

	$\displaystyle\mathcal{D}(\mathscr{L})=$	$\displaystyle\bigg{\{}f\in C_{0}([0,\infty))\cap C^{2}([0,\infty)):f^{\prime\prime}\in C_{0}([0,\infty)),$		(3.2)
		$\displaystyle\qquad p_{1}f(0)-p_{2}f^{\prime}(0)+\frac{p_{3}}{2}f^{\prime\prime}(0)+\int_{(0,\infty)}[f(0)-f(x)]p_{4}(dx)=0\bigg{\}},$		(3.2)

where $p_{1},p_{2},p_{3}$ are non-negative numbers and $p_{4}$ is a positive measure on $(0,\infty)$ such that

p_{1}+p_{2}+p_{3}+\int_{(0,\infty)}(x\wedge 1)p_{4}(dx)=1

(3.3)

and

|p_{4}|:=p_{4}\left((0,\infty)\right)=+\infty\quad\text{if}\quad p_{2}=p_{3}=0.

(3.4)

The parameters $p_{1},p_{2},p_{3},p_{4}$ are uniquely determined for each Feller’s Brownian motion.

Proof.

For the readers’ convenience, we provide necessary details regarding this proof. The uniqueness of $(p_{1},p_{2},p_{3},p_{4})$ is evident.

The necessity can be established by demonstrating that

		$\displaystyle\mathcal{D}(\mathscr{L})\subset\{f\in C_{0}([0,\infty))\cap C^{2}([0,\infty)):f^{\prime\prime}\in C_{0}([0,\infty))\},$		(3.5)
		$\displaystyle\mathscr{L}f=\frac{1}{2}f^{\prime\prime},\quad\forall f\in\mathcal{D}(\mathscr{L}),$		(3.5)

and then applying [21, II§5, Theorem 2]. To achieve this, let $(G_{\alpha})_{\alpha>0}$ denote the resolvent of $Y$ . Fix $\alpha>0$ . By the Hille-Yosida theorem, $\mathcal{D}(\mathscr{L})=G_{\alpha}C_{0}([0,\infty))\subset C_{0}([0,\infty))$ , and for $f=G_{\alpha}h$ with $h\in C_{0}([0,\infty))$ ,

\mathscr{L}f=\alpha G_{\alpha}h-h\in C_{0}([0,\infty)).

(3.6)

Set $\tau_{0}:=\inf\{t>0:Y_{t}=0\}$ . It follows from Definition 3.1 and Dynkin’s formula that

G_{\alpha}h(x)=G_{\alpha}^{0}h(x)+G_{\alpha}h(0)\mathbf{E}_{x}e^{-\alpha\tau_{0}},\quad x>0,

(3.7)

where $G^{0}_{\alpha}$ is the resolvent of the absorbing Brownian motion on $(0,\infty)$ . Note that $\mathbf{E}_{x}e^{-\alpha\tau_{0}}=e^{-\sqrt{2\alpha}x}$ (see, e.g., [11, page 26, 5)]), and (see, e.g., [21, II§3, #7 and #8])

G^{0}_{\alpha}h\in C^{2}((0,\infty))\cap C_{0}((0,\infty)),\quad\frac{1}{2}\left(G^{0}_{\alpha}h\right)^{\prime\prime}=\alpha G^{0}_{\alpha}h-h.

(3.8)

For any $x>0$ , it holds that

	$\displaystyle\left(G_{\alpha}h\right)^{\prime\prime}(x)$	$\displaystyle=\left(G_{\alpha}^{0}h\right)^{\prime\prime}(x)+G_{\alpha}h(0)\cdot 2\alpha\cdot e^{-\sqrt{2\alpha}x}$		(3.9)
		$\displaystyle=2\alpha G^{0}_{\alpha}h(x)-2h(x)+2\alpha G_{\alpha}h(0)e^{-\sqrt{2\alpha}x},$		(3.9)

and thus

\lim_{x\rightarrow 0}\left(G_{\alpha}h\right)^{\prime\prime}(x)=-2h(0)+2\alpha G_{\alpha}h(0),\quad\lim_{x\rightarrow\infty}\left(G_{\alpha}h\right)^{\prime\prime}(x)=0.

(3.10)

Using (3.6), (3.7), (3.9), and (3.10), we conclude that $f=G_{\alpha}h\in C_{0}([0,\infty))\cap C^{2}([0,\infty))$ , $f^{\prime\prime}\in C_{0}([0,\infty)$ , and $\mathscr{L}f=\frac{1}{2}f^{\prime\prime}$ . In other words, (3.5) is established.

Regarding the sufficiency, we first apply [21, II§5, Theorem 3] to $\mathscr{L}$ , showing that $\mathscr{L}$ with domain (3.2) is indeed the infinitesimal generator of a Feller process $Y$ on $[0,\infty)$ . It remains to demonstrate that the killed process

Y^{0}_{t}:=\left\{\begin{aligned} &Y_{t},\quad&t<\tau_{0},\\ &\partial,\quad&t\geq\tau_{0}\end{aligned}\right.

(3.11)

is identical in law to the absorbing Brownian motion on $(0,\infty)$ . In fact, the proof of [21, II§5, Theorem 3] shows that the resolvent kernel of $Y^{0}=(Y^{0}_{t})_{t\geq 0}$ is given by [21, §3, #7] (with $p(x)=2x$ and $m$ being the Lebesgue measure). This is precisely the resolvent kernel of absorbing Brownian motion. ∎

Remark 3.4.

When $p_{2}=p_{3}=|p_{4}|=0$ and $p_{1}=1$ , the condition (3.4) is not satisfied. However, this case corresponds to the absorbing Brownian motion on $(0,\infty)$ .

The triple $(p_{1},p_{2},p_{4})$ plays a role analogous to $(\gamma,\beta,\nu)$ for a $Q$ -process, with some heuristic explanations for the latter triple presented in [18, §2]. The additional parameter $p_{3}$ measures the sojourn of the Feller’s Brownian motion at $0$ . One one hand, this can be observed by examining the paths of the Feller’s Brownian motion as constructed by Itô and McKean, as illustrated in [10, §15]; see also (3.25). On the other hand, we can examine the symmetric case, where the transition semigroup $(T_{t})_{t\geq 0}$ of $Y$ is symmetric with respect to some $\sigma$ -finite measure $m$ in the sense that

\int_{[0,\infty)}T_{t}f(x)g(x)m(dx)=\int_{[0,\infty)}f(x)T_{t}g(x)m(dx),\quad\forall t\geq 0,f,g\in\mathcal{B}_{b}([0,\infty)),

as stated in the following corollary. In this special case, the parameter $p_{3}/(2p_{2})$ represents the mass of the symmetric measure $m$ at $0$ . It is well known that when this symmetric measure, also known as the speed measure of $Y$ , is larger in a specific region, the motion of $Y_{t}$ will be slower when passing through that region.

Corollary 3.5.

Let $Y=(Y_{t})_{t\geq 0}$ be a Feller’s Brownian motion on $[0,\infty)$ with parameters $p_{1},p_{2},p_{3},p_{4}$ as described in Theorem 3.3. The process $Y$ is symmetric if and only if $p_{2}>0,|p_{4}|=0$ . Furthermore, the symmetric measure must be

m(dx)=\frac{p_{3}}{2p_{2}}\delta_{0}(dx)+1_{(0,\infty)}(x)dx

(3.12)

up to a multiplicative constant, where $\delta_{0}$ denotes the Dirac measure at $0$ , and the Dirichlet form associated with this symmetric Feller’s Brownian motion on $L^{2}([0,\infty),m)$ is

	$\displaystyle{\mathscr{F}}$	$\displaystyle=H^{1}([0,\infty)),$		(3.13)
	$\displaystyle{\mathscr{E}}(f,g)$	$\displaystyle=\frac{1}{2}\int_{0}^{\infty}f^{\prime}(x)g^{\prime}(x)dx+\frac{p_{1}}{2p_{2}}f(0)g(0),\quad f,g\in{\mathscr{F}},$		(3.13)

where

H^{1}([0,\infty)):=\{f\in L^{2}([0,\infty)):f\text{ is absolutely continuous and }f^{\prime}\in L^{2}([0,\infty))\}.

Proof.

We continue to denote by $G_{\alpha}$ the resolvent of $Y$ and by $Y^{0}$ the killed process of $Y$ upon hitting $0$ . Let $g^{0}_{\alpha}(x,y)$ represent the resolvent density of $Y^{0}$ , i.e., $G^{0}_{\alpha}(x,dy)=g^{0}_{\alpha}(x,y)dy$ . We have

g_{\alpha}^{0}(x,y)=\left\{\begin{aligned} &W^{-1}u_{-}(x)u_{+}(y),\quad&x\leq y,\\ &W^{-1}u_{-}(y)u_{+}(x),\quad&y\leq x,\end{aligned}\right.

(3.14)

where $u_{-}(x)=\sinh(\sqrt{2\alpha}x),u_{+}(x)=e^{-\sqrt{2\alpha}x}$ , and $W:=W(u_{-},u_{+})=\frac{\sqrt{2\alpha}}{2}$ is the Wronskain of $u_{-},u_{+}$ (with $W^{-1}:=1/W$ ); see, e.g., [21, II§3, #7]. As established in [10, §15, 4.], for any $h\in C_{0}([0,\infty))$ , it holds that

G_{\alpha}h(0)=\frac{2p_{2}\int_{(0,\infty)}e^{-\sqrt{2\alpha}x}h(x)dx+p_{3}h(0)+\int_{(0,\infty)}G^{0}_{\alpha}h(x)p_{4}(dx)}{p_{1}+\sqrt{2\alpha}p_{2}+\alpha p_{3}+\int_{(0,\infty)}[1-e^{-\sqrt{2\alpha}x}]p_{4}(dx)}.

(3.15)

To demonstrate the necessity, let $m$ be a symmetric measure of $Y$ . From [8, Lemma 4.1.3], it follows that $m|_{(0,\infty)}$ is the symmetric measure of $Y^{0}$ . Hence, without loss of generality, we may assume that $m|_{(0,\infty)}$ is the Lebesgue measure on $(0,\infty)$ . For any $h_{1},h_{2}\in C_{0}((0,\infty))$ , substituting (3.7) and (3.15) into

\int_{[0,\infty)}G_{\alpha}h_{1}(x)h_{2}(x)m(dx)=\int_{[0,\infty)}h_{1}(x)G_{\alpha}h_{2}(x)m(dx),

(3.16)

we obtain

		$\displaystyle\iint_{(0,\infty)\times(0,\infty)}h_{1}(x)h_{2}(y)e^{-\sqrt{2\alpha}x}m(dx)\left(p_{4}G^{0}_{\alpha}\right)(dy)$
		$\displaystyle\quad=\iint_{(0,\infty)\times(0,\infty)}h_{1}(y)h_{2}(x)e^{-\sqrt{2\alpha}x}m(dx)\left(p_{4}G^{0}_{\alpha}\right)(dy),\quad\forall h_{1},h_{2}\in C_{0}((0,\infty)),$

where $\left(p_{4}G^{0}_{\alpha}\right)(dy)=\int_{(0,\infty)}p_{4}(dx)G_{\alpha}^{0}(x,dy)=\left(\int_{(0,\infty)}g^{0}_{\alpha}(x,y)p_{4}(dx)\right)dy$ . This implies that

H(y):=e^{\sqrt{2\alpha}y}\int_{(0,\infty)}g^{0}_{\alpha}(x,y)p_{4}(dx)

is constant. Substituting (3.14) into $H(y)$ , we have

W\cdot H(y)=\int_{(0,\infty)}H_{y}(x)p_{4}(dx),

where

H_{y}(x)=\left\{\begin{aligned} &\sinh(\sqrt{2\alpha}x),\quad&x\leq y,\\ &\sinh(\sqrt{2\alpha}y)e^{-\sqrt{2\alpha}(x-y)},&y\leq x.\end{aligned}\right.

It is straightforward to verify that $H_{y}(x)$ is increasing in $y$ . Letting $y\downarrow 0$ , we conclude that $H(y)\equiv 0$ . Since $H_{y}$ is strictly positive for $y>0$ , it must hold that $|p_{4}|=0$ . Substituting (3.7) and (3.15) into (3.16) again, but now taking $h_{1},h_{2}\in C_{0}([0,\infty))$ , we further obtain

\left(p_{3}-2p_{2}m(\{0\})\right)\cdot\left(h_{1}(0)\int_{0}^{\infty}h_{2}(x)e^{-\sqrt{2\alpha}x}dx-h_{2}(0)\int_{0}^{\infty}h_{1}(x)e^{-\sqrt{2\alpha}x}dx\right)=0.

This clearly implies

p_{3}=2p_{2}\cdot m(\{0\}).

Particularly, if $p_{2}=0$ , then $p_{3}=0$ . This contradicts the condition (3.4). Therefore, we conclude that $p_{2}>0,|p_{4}|=0$ , and the symmetric measure is precisely (3.12).

Next, we consider the case where $p_{2}>0$ and $|p_{4}|=0$ . Note that (3.13) is, in fact, a regular Dirichlet form on $L^{2}([0,\infty),m)$ ; see, e.g., [7, §5.3]. It suffices to show that

G_{\alpha}h\in{\mathscr{F}},\quad{\mathscr{E}}_{\alpha}(G_{\alpha}h,g)=(h,g)_{m},\quad\forall h,g\in C_{c}^{\infty}([0,\infty)).

Indeed, $G_{\alpha}^{0}h\in H^{1}_{0}((0,\infty))=\{f\in H^{1}([0,\infty)):f(0)=0\}$ , and hence (3.7) implies that $G_{\alpha}h\in{\mathscr{F}}$ . Using (3.7), we further have

$\displaystyle{\mathscr{E}}_{\alpha}(G_{\alpha}h,g)$	$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\left(G^{0}_{\alpha}h\right)^{\prime}(x)g^{\prime}(x)dx+\frac{G_{\alpha}h(0)}{2}\int_{0}^{\infty}\left(e^{-\sqrt{2\alpha}x}\right)^{\prime}g^{\prime}(x)dx$	(3.17)
	$\displaystyle\quad+\frac{p_{1}}{2p_{2}}G_{\alpha}h(0)g(0)+\alpha\int_{0}^{\infty}G_{\alpha}^{0}h(x)g(x)dx$
	$\displaystyle\quad+\alpha G_{\alpha}h(0)\int_{0}^{\infty}e^{-\sqrt{2\alpha}x}g(x)dx+\frac{\alpha p_{3}}{2p_{2}}G_{\alpha}h(0)g(0).$

Note that (see, e.g., [21, II§4, (23)])

W\cdot\left(G^{0}_{\alpha}h\right)^{\prime}(x)=u^{\prime}_{+}(x)\int_{0}^{x}u_{-}(y)h(y)dy+u^{\prime}_{-}(x)\int_{x}^{\infty}u_{+}(y)h(y)dy.

This yields that

\lim_{x\rightarrow\infty}\left(G^{0}_{\alpha}h\right)^{\prime}(x)=0,\quad\left(G^{0}_{\alpha}h\right)^{\prime}(0)=2\int_{0}^{\infty}h(y)e^{-\sqrt{2\alpha}y}dy.

(3.18)

Substituting (3.8), (3.15), and (3.18) into (3.17), we can obtain that

{\mathscr{E}}_{\alpha}(G_{\alpha}h,g)=\int_{0}^{\infty}h(x)g(x)dx+\frac{p_{3}}{2p_{2}}h(0)g(0)=(h,g)_{m}.

This completes the proof. ∎

It is worth noting that the $L^{2}$ -generator of symmetric Feller’s Brownian motion, i.e., the $L^{2}$ -generator of the Dirichlet form (3.13), is derived in, e.g., [7, Theorem 5.3].

3.2. Pathwise construction

The sample paths of a Feller’s Brownian motion were constructed by Itô and McKean in [10]. The symmetric cases characterized in Corollary 3.5 correspond to either reflecting Brownian motion (for $p_{1}=0$ ) or elastic Brownian motion (for $p_{1}>0$ ), up to a time change transformation induced by the speed measure (3.12). Their pathwise constructions are quite clear; see also [10, §10]. For the readers’ convenience, we will restate some necessary details of the pathwise construction for non-symmetric cases in this subsection.

3.2.1. $p_{2}=0,p_{3}>0,|p_{4}|<\infty$

This case was examined in [10, §9]. The sample paths of Feller’s Brownian motion can be constructed as follows: Given a reflecting Brownian motion with sample paths $t\mapsto W^{+}_{t}$ starting at a point $x\in[0,\infty)$ , let $Y_{t}:=W^{+}_{t}$ up to the first hitting time $\tau^{+}_{0}:=\inf\{t>0:W^{+}_{t}=0\}$ of $0$ . Then, make $Y$ wait at $0$ for an exponential holding time $\mathfrak{e}$ with the conditional law

\mathbf{P}_{x}(\mathfrak{e}>t|W^{+})=e^{-\frac{p_{1}+|p_{4}|}{p_{3}}t};

(3.19)

at the end of this time (only applicable in the case where $p_{1}+|p_{4}|>0$ ), let it jump to a point in $(0,\infty)\cup\{\partial\}$ according to the distribution $\lambda$ given by

\lambda(dx)|_{(0,\infty)}:=\frac{p_{4}(dx)}{p_{1}+|p_{4}|},\quad\lambda(\{\partial\}):=\frac{p_{1}}{p_{1}+|p_{4}|}.

(3.20)

If the reaching point is in $(0,\infty)$ , let it start afresh; if it jumps to $\partial$ , let $Y_{t}:=\partial$ at all later times.

3.2.2. $p_{2}>0,0<|p_{4}|<\infty$

This case was addressed in [10, §12] using an increasing Lévy process, defined by (3.21). In fact, we can provide an alternative construction via the Ikeda-Nagasawa-Watanabe piecing out procedure (see [9]), similar to the approach in [18, Theorem 8.1] for $Q$ -processes with $|\nu|<+\infty$ ; see also [21, III§4].

To be precise, let us begin with a symmetric Feller’s Brownian motion $Y^{1}=(Y^{1}_{t})_{t\geq 0}$ with parameters $(p_{1}+|p_{4}|,p_{2},p_{3},0)$ . Let $\lambda$ be the probability measure on $(0,\infty)\cup\{\partial\}$ defined as in (3.20). Then the Feller’s Brownian motion $Y=(Y_{t})_{t\geq 0}$ with parameters $(p_{1},p_{2},p_{3},p_{4})$ such that $p_{2}>0,0<|p_{4}|<\infty$ is actually the piecing out of $Y^{1}$ with respect to $\lambda$ , as described in [18, Appendix A]. Intuitively, $Y$ repeatedly splices resurrection paths at the death times of $Y^{1}_{t}$ , with the resurrection points randomly determined by the distribution $\lambda$ .

3.2.3. $|p_{4}|=\infty$

When $|p_{4}|=\infty$ , the pathwise construction becomes significantly more challenging. This was accomplished by Itô and McKean in [10, §12-§15].

We first consider the case where $p_{1}=p_{3}=0$ and $|p_{4}|=+\infty$ ( $p_{2}\geq 0)$ . Let $Z=(Z(t))_{t\geq 0}$ be a subordinator, i.e., an increasing Lévy process on $\mathbb{R}$ with $Z(0)=0$ , whose distribution has the Laplace transform

\mathbf{E}e^{-xZ(t)}=\exp\left\{-t\left[p_{2}x+\int_{(0,\infty)}(1-e^{-xy})p_{4}(dy)\right]\right\}.

(3.21)

In [10], this subordinator is referred to as an increasing differential process. For every $s\geq 0$ , define

Z^{-1}(s):=\inf\{t>0:Z(t)>s\}.

Further, let $W^{+}=(W^{+}_{t})_{t\geq 0}$ be a reflecting Brownian motion on $[0,\infty)$ , independent of $Z$ , with local time $(\ell_{t})_{t\geq 0}$ at $0$ given by

\ell_{t}=\lim_{\varepsilon\downarrow 0}\frac{1}{2\varepsilon}\int_{0}^{t}1_{\{W^{+}_{s}<\varepsilon\}}ds,\quad\forall t\geq 0.

(3.22)

Note that the Revuz measure of $\ell$ with respect to $W^{+}$ is $\frac{\delta_{0}}{2}$ ; see, e.g., [22, X. Proposition 2.4]. It was shown in [10, §13] that

Y^{1}_{t}:=Z(Z^{-1}(\ell_{t}))-\ell_{t}+W^{+}_{t},\quad t\geq 0

(3.23)

is indeed a Feller’s Brownian motion with parameters $(0,p_{2},0,p_{4})$ . For a detailed explanation of the sample paths defined by (3.23), please refer to [10, §12]. Note that this pathwise construction (3.23) also applies to the case $p_{1}=p_{3}=0,p_{2}>0$ , and $|p_{4}|<+\infty$ .

Regarding the general case, we note that

\ell^{Y^{1}}_{t}:=Z^{-1}(\ell_{t}),\quad t\geq 0

(3.24)

is the local time of the process defined by (3.23) at $0$ , as established in [10, §14]. Define $\mathfrak{f}(t):=t+p_{3}\ell^{Y^{1}}_{t}$ and $\mathfrak{f}^{-1}(t):=\inf\{s>0:\mathfrak{f}(s)>t\}$ . Then, the time-changed process of (3.23) with respect to $\mathfrak{f}$ is

Y^{2}_{t}:=Y^{1}_{\mathfrak{f}^{-1}(t)},\quad t\geq 0.

(3.25)

This process $Y^{2}$ is a Feller’s Brownian motion with parameters $(0,p_{2},p_{3},p_{4})$ . The desired Feller’s Brownian motion $Y$ with parameters $(p_{1},p_{2},p_{3},p_{4})$ is obtained as the subprocess of (3.25) perturbed by the multiplicative functional

M_{t}:=e^{-p_{1}\ell^{Y^{1}}_{\mathfrak{f}^{-1}(t)}},\quad t\geq 0.

(3.26)

A rigorous construction of the subprocess can be found in, e.g., [1, III, §3]. Roughly speaking, one can take a random time $\zeta$ such that

\mathbf{P}_{x}(\zeta>t|Y^{1})=e^{-p_{1}\ell^{Y^{1}}_{\mathfrak{f}^{-1}(t)}},\quad\forall t>0,

(3.27)

and then kill the process $Y^{2}$ at time $\zeta$ .

3.3. Local times of Feller’s Brownian motion

In [10, §14], Itô and McKean examined the local time (3.24) of the special Feller’s Brownian motion defined by (3.23) at $0$ . What we focus on here is the local time of a general Feller’s Brownian motion at a given point $a>0$ .

Consider a Feller’s Brownian motion on $[0,\infty)$ :

Y=(\Omega,{\mathscr{G}},{\mathscr{G}}_{t},Y_{t},\theta_{t},(\mathbf{P}_{x})_{x\in[0,\infty)})

with lifetime $\zeta$ , where $({\mathscr{G}}_{t})_{t\geq 0}$ is the augmented natural filtration on $\Omega$ . Let ${\mathscr{G}}_{\infty}:=\bigvee_{t\geq 0}{\mathscr{G}}_{t}$ denote the $\sigma$ -algebra generated by $\bigcup_{t\geq 0}{\mathscr{G}}_{t}$ . We adjoin to $\Omega$ the dead path $[\partial]$ with $Y_{t}([\partial]):=\partial$ for all $t\geq 0$ . A positive continuous additive functional of $Y$ is defined as follows. For detailed discussions, see, e.g., [1, IV§1] and [3, Definition A.3.1].

Definition 3.6.

A family $A=(A_{t})_{t\geq 0}$ of functions from $\Omega$ to $[0,\infty]$ is called a positive continuous additive functional (PCAF for short) of $Y$ if there exists $\Lambda\in{\mathscr{G}}_{\infty}$ such that

\mathbf{P}_{x}(\Lambda)=1\text{ for }x\in[0,\infty)\quad\text{ and }\quad\theta_{t}\Lambda\subset\Lambda\text{ for }t\geq 0,

and the following conditions are satisfied:

(A1)

For each $t\geq 0$ , $A_{t}|_{\Lambda}\in{\mathscr{G}}_{t}|_{\Lambda}$ .
(A2)

For every $\omega\in\Lambda$ , $A_{\cdot}(\omega)$ is continuous on $[0,\infty)$ , $A_{0}(\omega)=0$ , $A_{t}(\omega)<\infty$ for $t<\zeta(\omega)$ , and $A_{t}(\omega)=A_{\zeta(\omega)}(\omega)$ for $t\geq\zeta(\omega)$ .
(A3)

For $\omega\in\Lambda$ and every $t,s\geq 0$ , $A_{t+s}(\omega)=A_{t}(\omega)+A_{s}(\theta_{t}\omega)$ .

The set $\Lambda$ is called a defining set of $A$ . We further make the convention $A_{t}([\partial])=0$ for all $t\geq 0$ .

The fine support of a PCAF $(A_{t})_{t\geq 0}$ is defined as

\text{Supp}(A):=\{x\in[0,\infty):\mathbf{P}_{x}(R=0)=1\},

where $R(\omega):=\inf\{t>0:A_{t}(\omega)>0\}$ . According to [1, V. Theorem 3.13], for a given $a>0$ , there exists a PCAF (unique up to a multiplicative constant) of $Y$ with fine support $\{a\}$ .

Recall that $\tau_{0}=\inf\{t>0:Y_{t}=0\}$ . The killed process $Y^{0}$ can be written as

Y^{0}=\left(\Omega,{\mathscr{G}},{\mathscr{G}}_{t},Y^{0}_{t},\theta^{0}_{t},(\mathbf{P}_{x})_{x\in(0,\infty)}\right),

where $Y^{0}_{t}$ is defined as in (3.11), $\theta^{0}_{t}(\omega):=\theta_{t}(\omega)$ for $t<\tau_{0}(\omega)$ and $\theta^{0}_{t}(\omega):=[\partial]$ for $t\geq\tau_{0}(\omega)$ ; see, e.g., [25, (12.21i)]. The lifetime of $Y^{0}$ is $\zeta^{0}:=\zeta\wedge\tau_{0}(=\tau_{0})$ . We can define the PCAFs for $Y^{0}$ and their fine supports analogously. The following fact is crucial to our discussion.

Lemma 3.7.

Given $a>0$ , let $\ell^{a}=(\ell^{a}_{t})_{t\geq 0}$ be a PCAF of $Y$ with fine support $\{a\}$ . Then

\ell^{a,0}_{t}:=\left\{\begin{aligned} &\ell^{a}_{t},\quad&t<\zeta^{0},\\ &\ell^{a}_{\zeta^{0}},\quad&t\geq\zeta^{0}\end{aligned}\right.

is a PCAF of $Y^{0}$ with fine support $\{a\}$ .

Proof.

Assuming without loss of generality that the defining set of $\ell^{a}$ is $\Omega$ , it is straightforward to verify that $\ell^{a,0}$ satisfies properties (A1) and (A2). To establish property (A3) for $\ell^{a,0}$ , it suffices to consider the case where $t+s\geq\tau_{0}(\omega)$ . If $t\geq\tau_{0}(\omega)$ , then $\ell^{a,0}_{t}(\omega)=\ell^{a}_{\tau_{0}(\omega)}(\omega)$ and $\ell^{a,0}_{s}(\theta^{0}_{t}\omega)=\ell^{a,0}_{s}([\partial])=\ell^{a}_{0}([\partial])=0$ . Hence, we have

\ell^{a,0}_{t+s}(\omega)=\ell^{a,0}_{t}(\omega)+\ell^{a,0}_{s}(\theta^{0}_{t}\omega).

(3.28)

If $t<\tau_{0}(\omega)$ , (3.28) can be verified using $\tau_{0}(\theta_{t}\omega)=\tau_{0}(\omega)-t\leq s$ . Thus, $\ell^{a,0}$ is indeed a PCAF of $Y^{0}$ . The fine support of $\ell^{a,0}$ is $\{a\}$ because $\ell^{a,0}_{t}\leq\ell^{a}_{t}$ and $\mathbf{P}_{a}(\tau_{0}>0)=1$ . ∎

According to Definition 3.1, $Y^{0}$ is identical in law to the absorbing Brownian motion on $(0,\infty)$ . Therefore, we can define the Revuz measure $\nu_{a}$ of $\ell^{a,0}$ with respect to $Y^{0}$ as follows:

	$\displaystyle\int_{(0,\infty)}f(x)\nu_{a}(dx)$	$\displaystyle=\lim_{t\downarrow 0}\frac{1}{t}\int_{(0,\infty)}\left(\mathbf{E}_{x}\left[\int_{0}^{t}f(Y^{0}_{t})d\ell^{a,0}_{t}\right]\right)dx$
		$\displaystyle=\lim_{t\downarrow 0}\frac{1}{t}\int_{(0,\infty)}\left(\mathbf{E}_{x}\left[\int_{0}^{t\wedge\tau_{0}}f(Y_{t})d\ell^{a}_{t}\right]\right)dx.$

Note that $\nu_{a}$ is a constant multiple of $\delta_{a}$ . Unlike the approach of normalizing local times by the values of their potentials as described in [1, V. Theorem 3.13], we opt to use the unique PCAF with fine support $\{a\}$ in the following sense.

Definition 3.8.

Given $a>0$ , a PCAF $L^{a}=(L^{a}_{t})_{t\geq 0}$ with $\text{Supp}(L^{a})=\{a\}$ is called the local time of $Y$ at $a$ if the Revuz measure of $L^{a,0}:=(L^{a}_{t\wedge\zeta^{0}})_{t\geq 0}$ with respect to $Y^{0}$ is $\delta_{a}$ .

In the symmetric case, where $p_{2}>0$ and $|p_{4}|=0$ , we can also define the Revuz measure of the local time $L^{a}$ with respect to $Y$ and the symmetric measure $m$ . According to [3, Proposition 4.1.10], this Revuz measure is also equal to the Dirac measure $\delta_{a}$ . Therefore, the definition of local time provided here is consistent with the definition of local time in the theory of Dirichlet forms.

4. Time-changed Feller’s Brownian motions are birth-death processes

Consider the birth-death density matrix (2.1) and from now on, assume that $\infty$ is regular for the minimal $Q$ -process. Particularly, the scale function $(c_{k})_{k\geq 0}$ given by (2.2) satisfies

c_{\infty}=\lim_{k\rightarrow\infty}c_{k}<\infty,

and the speed measure $\mu$ , as defined in (2.3), is finite; see, e.g., [18, Remark 3.4].

The aim of this section is to demonstrate that any Feller’s Brownian motion can be converted into a $Q$ -process through a time change transformation and a spatial homeomorphism. The special cases of absorbing and reflecting Brownian motions have been analyzed in [18, §3]. In these cases, the transformed $Q$ -processes are the minimal $Q$ -process and the $(Q,1)$ -process, respectively.

4.1. Spatial transformation

The formulation we will present encounters a significant issue because the boundary point $0$ of Feller’s Brownian motion and the boundary point $\infty$ of the $Q$ -process are located at opposite ends of their respective state spaces. Additionally, unlike Feller’s Brownian motion, the $Q$ -process is not on the natural scale. However, both issues can be resolved by applying a straightforward spatial transformation to the $Q$ -process. To address this, define $\hat{c}_{n}:=c_{\infty}-c_{n}$ for $n\in\mathbb{N}$ . Let

E:=\{\hat{c}_{n}:n\in\mathbb{N}\},\quad\overline{E}=E\cup\{0\}

(4.1)

and

\Xi:E\rightarrow\mathbb{N},\quad\hat{c}_{n}\mapsto n.

(4.2)

Clearly, $\Xi$ can be extended to a homeomorphism between $\overline{E}$ and $\mathbb{N}\cup\{\infty\}$ , where $\overline{E}$ is endowed with the relative topology of ${\mathbb{R}}$ .

For each $Q$ -process $X=(X_{t})_{t\geq 0}$ on $\mathbb{N}\cup\{\infty\}$ ,

\hat{X}_{t}:=\Xi^{-1}(X_{t}),\quad t\geq 0

defines a continuous-time Markov chain on $\overline{E}$ . It is a Feller process on $\overline{E}$ whenever $X$ is a Feller $Q$ -process. For convenience, we also refer to $\hat{X}=(\hat{X}_{t})_{t\geq 0}$ as a Doob process or a Feller $Q$ -process (on $E$ or $\overline{E}$ ).

4.2. Time change

We first prepare the ingredient, specifically the PCAF, for the time change transformation on a Feller’s Brownian motion $Y$ . For $\hat{c}_{n}\in E$ , let $L^{\hat{c}_{n}}=(L^{\hat{c}_{n}}_{t})_{t\geq 0}$ denote the local time of $Y$ at $\hat{c}_{n}$ as defined in Definition 3.8. Define

A_{t}=\sum_{n\in\mathbb{N}}\mu_{n}L^{\hat{c}_{n}}_{t},\quad t\geq 0.

(4.3)

We will show that $A=(A_{t})_{t\geq 0}$ is a PCAF of $Y$ .

Lemma 4.1.

The family of functions $A=(A_{t})_{t\geq 0}$ defined as (4.3) is a PCAF of the Feller’s Brownian motion $Y$ . Furthermore,

(1)

If $p_{2}=0,p_{3}>0$ , and $|p_{4}|<\infty$ , then the fine support of $A$ is $E$ ;
(2)

Otherwise, the fine support of $A$ is $\overline{E}$ .

Proof.

We first demonstrate that

\mathbf{E}_{x}\int_{0}^{\infty}e^{-t}dA_{t}=\sum_{n\in\mathbb{N}}\mu_{n}\mathbf{E}_{x}\int_{0}^{\infty}e^{-t}dL^{\hat{c}_{n}}_{t}<\infty,\quad\forall x\in[0,\infty).

(4.4)

The main task is to estimate $\mathbf{E}_{x}\int_{0}^{\infty}e^{-t}dL^{\hat{c}_{n}}_{t}$ for each $n\in\mathbb{N}$ . Let $T_{n}:=\inf\{t>0:Y_{t}=\hat{c}_{n}\}$ . According to [1, V. Theorem 3.13], there exists a positive constant $k_{n}$ such that

\mathbf{E}_{x}\int_{0}^{\infty}e^{-t}dL^{\hat{c}_{n}}_{t}=k_{n}\mathbf{E}_{x}e^{-T_{n}},\quad\forall x\in[0,\infty).

(4.5)

By the strong Markov property of $Y$ and [1, IV, Proposition 1.13], we obtain

\mathbf{E}_{x}\int_{0}^{\infty}e^{-t}dL^{\hat{c}_{n}}_{t}=\mathbf{E}_{x}\int_{0}^{\tau_{0}}e^{-t}dL^{\hat{c}_{n}}_{t}+\mathbf{E}_{x}e^{-\tau_{0}}\cdot\mathbf{E}_{0}\int_{0}^{\infty}e^{-t}dL^{\hat{c}_{n}}_{t}.

Integrating both sides with respect to the Lebesgue measure $\mathfrak{m}$ on $[0,\infty)$ , we have

	$\displaystyle\mathbf{E}_{\mathfrak{m}}\int_{0}^{\infty}e^{-t}dL^{\hat{c}_{n}}_{t}$	$\displaystyle=\mathbf{E}_{\mathfrak{m}}\int_{0}^{\tau_{0}}e^{-t}dL^{\hat{c}_{n}}_{t}+\mathbf{E}_{\mathfrak{m}}e^{-\tau_{0}}\cdot\mathbf{E}_{0}\int_{0}^{\infty}e^{-t}dL^{\hat{c}_{n}}_{t}$
		$\displaystyle=(1-\mathbf{E}_{\hat{c}_{n}}e^{-\tau_{0}})+\mathbf{E}_{\mathfrak{m}}e^{-\tau_{0}}\cdot\mathbf{E}_{0}\int_{0}^{\infty}e^{-t}dL^{\hat{c}_{n}}_{t},$

where the second equality follows from [3, (4.1.3)]. Substituting (4.5) and $\mathbf{E}_{x}e^{-\tau_{0}}=e^{-\sqrt{2}x}$ for $x>0$ into the above equation, we obtain

k_{n}=\frac{\sqrt{2}(1-e^{-\sqrt{2}c})}{\sqrt{2}\mathbf{E}_{\mathfrak{m}}e^{-T_{n}}-\mathbf{E}_{0}e^{-T_{n}}}.

(4.6)

The strong Markov property also implies that for any $0<x<\hat{c}_{n}$ ,

	$\displaystyle\mathbf{E}_{x}e^{-T_{n}}$	$\displaystyle=\mathbf{E}_{x}\left(e^{-T_{n}};T_{n}<\tau_{0}\right)+\mathbf{E}_{x}\left(e^{-T_{n}};T_{n}>\tau_{0}\right)$		(4.7)
		$\displaystyle=\mathbf{E}_{x}\left(e^{-T_{n}};T_{n}<\tau_{0}\right)+\mathbf{E}_{x}\left(e^{-\tau_{0}};T_{n}>\tau_{0}\right)\mathbf{E}_{0}e^{-T_{n}}.$		(4.7)

For $x>\hat{c}_{n}$ , it holds that

\mathbf{E}_{x}e^{-T_{n}}=e^{-\sqrt{2}(x-\hat{c}_{n})}.

(4.8)

Substituting (4.7), (4.8) into (4.6) and using [11, Problem 6 of page 29], we deduce that

k_{n}=\frac{\sqrt{2}(e^{\sqrt{2}\hat{c}_{n}}-e^{-\sqrt{2}\hat{c}_{n}})}{2(e^{\sqrt{2}\hat{c}_{n}}-\mathbf{E}_{0}e^{-T_{n}})}.

(4.9)

Since $\mathbf{E}_{0}e^{-T_{n}}\leq 1$ , it follows that

\mathbf{E}_{x}\int_{0}^{\infty}e^{-t}dL^{\hat{c}_{n}}_{t}\leq k_{n}\leq\frac{\sqrt{2}(e^{\sqrt{2}\hat{c}_{n}}-e^{-\sqrt{2}\hat{c}_{n}})}{2(e^{\sqrt{2}\hat{c}_{n}}-1)}=\frac{\sqrt{2}}{2}(1+e^{-\sqrt{2}\hat{c}_{n}})\leq\sqrt{2}.

Note that $\sum_{n\in\mathbb{N}}\mu_{n}<\infty$ . Therefore, (4.4) can be concluded.

To prove that (4.3) is a PCAF of $Y$ , we start by considering the defining sets $\Lambda_{n}\in{\mathscr{G}}_{\infty}$ of $L^{\hat{c}_{n}}$ . Define

\Lambda:=\left(\bigcap_{n\in\mathbb{N}}\Lambda_{n}\right)\bigcap\left\{\omega\in\Omega:A_{t}(\omega)<\infty,\forall t<\zeta(\omega)\right\}.

Clearly, $A_{t}|_{\bigcap_{n}\Lambda_{n}}\in{\mathscr{G}}_{t}|_{\bigcap_{n}\Lambda_{n}}$ . Note that

\left\{\omega\in\Omega:A_{t}(\omega)<\infty,\forall t<\zeta(\omega)\right\}=\bigcap_{t>0}\left(\{A_{t}<\infty,t<\zeta\}\cup\{\zeta\leq t\}\right).

(4.10)

Thus, it is straightforward to verify that $\Lambda\in{\mathscr{G}}_{\infty}$ and that $\Lambda$ satisfies all the conditions in Definition 3.1 except for

\mathbf{P}_{x}(\Lambda)=1,\quad\forall x\in[0,\infty).

(4.11)

To show (4.11), assume for contradiction that

\mathbf{P}_{x}\left(\left(\bigcap_{t>0}\left(\{A_{t}<\infty,t<\zeta\}\cup\{\zeta\leq t\}\right)\right)^{c}\right)=\mathbf{P}_{x}\left(\bigcup_{t>0}\{A_{t}=\infty,t<\zeta\}\right)>0.

This implies $\mathbf{P}_{x}(\bigcup_{t>0}\{A_{t}=\infty\})>0$ . Since $A_{t}$ is increasing in $t$ , $\{A_{t}=\infty\}$ is also increasing in $t$ . Therefore, there exists $t_{0}>0$ such that $\mathbf{P}_{x}(A_{t_{0}}=\infty)>0$ . We have

	$\displaystyle\mathbf{E}_{x}\int_{0}^{\infty}e^{-t}dA_{t}$	$\displaystyle\geq\mathbf{E}_{x}\left(\int_{0}^{t_{0}}e^{-t}dA_{t};\{A_{t_{0}}=\infty\}\right)$
		$\displaystyle\geq e^{-t_{0}}\mathbf{E}_{x}\left(\int_{0}^{t_{0}}dA_{t};\{A_{t_{0}}=\infty\}\right)$
		$\displaystyle=\infty.$

This contradicts (4.5). Thus, (4.11) holds.

Finally, we examine the fine support of $A$ . Note that $dL^{\hat{c}_{n}}_{t}$ (as a measure in $t$ ) vanishes outside $\{t:Y_{t}=\hat{c}_{n}\}$ . Thus, $dA_{t}$ vanishes on $\{t:Y_{t}\notin\overline{E}\}$ . Consequently, by [1, V. Corollary 3.10], we have $\text{Supp}(A)\subset\overline{E}$ . On the other hand, since $A_{t}\geq\mu_{n}L^{\hat{c}_{n}}_{t}$ , it follows from the definition that $\hat{c}_{n}\in\text{Supp}(A)$ . Therefore, $E\subset\text{Supp}(A)$ .

If $p_{2}=0,p_{3}>0$ , and $|p_{4}|<\infty$ , then $Y_{t}=0$ for all $0\leq t<\mathfrak{e}$ , $\mathbf{P}_{0}$ -a.s., where $\mathfrak{e}$ is given by (3.19). Note that $\mathbf{P}_{0}(\mathfrak{e}>0)=1$ . Hence, $\mathbf{P}_{0}(R\geq\mathfrak{e}>0)=1$ , where $R=\inf\{t>0:A_{t}>0\}$ . By the definition of fine support, it follows that $0\notin\text{Supp}(A)$ . Therefore, $\text{Supp}(A)=E$ .

It remains to show $0\in\text{Supp}(A)$ for the remaining cases. We proceed by contradiction. Suppose $0\notin\text{Supp}(A)$ . Then $\mathbf{P}_{0}(R>0)=1$ . For $\mathbf{P}_{0}$ -a.s. $\omega\in\Omega$ and $0<t<R(\omega)$ , we have $L^{\hat{c}_{n}}_{t}(\omega)=0$ for all $n\in\mathbb{N}$ . According to [1, V Theorem 3.8], $Y_{t}(\omega)\notin E$ for all $0<t<R(\omega)$ . Note that $Y_{t}(\omega)$ is càdlàg in $t$ . From the pathwise representation of $Y$ (see §3.2), we can obtain that if $Y_{t-}(\omega)\neq Y_{t}(\omega)$ , then $Y_{t-}(\omega)=0$ . This fact, combined with $Y_{t}(\omega)\notin E$ for $0<t<R(\omega)$ and $Y_{0}(\omega)=0$ , implies that $Y_{t}(\omega)=0$ for all $0\leq t<R(\omega)$ . Particularly, $Y_{t}(\omega)$ is continuous in $t\in[0,R(\omega))$ . This is impossible when $p_{2}>0,|p_{4}|<\infty$ , because before the first jumping time, the sample paths of $Y$ are those of a symmetric Feller’s Brownian motion with parameters $(p_{1}+|p_{4}|,p_{2},p_{3},0)$ . This process, as a regular diffusion process on $[0,\infty)$ , can not stay at any point for an extended period; see, e.g., [23, V. (47.1)]. For the case where $|p_{4}|=\infty$ , the pathwise representation of $Y$ in §3.2.3 indicates that $Y_{t}(\omega)$ is exactly a Brownian path $W^{+}_{t}(\omega)$ (up to a transformation of time change (3.25)) for $t\in[0,R(\omega))$ . This also contradicts $Y_{t}(\omega)=0$ for $t\in[0,R(\omega))$ . ∎

With the Feller’s Brownian motion $Y$ and its PCAF $A=(A_{t})_{t\geq 0}$ given by (4.3), we can now introduce the time-changed process of $Y$ with respect to $A$ . Define the right-continuous inverse of $A_{t}(\omega)$ for each $\omega\in\Omega$ as

\gamma_{t}(\omega):=\left\{\begin{aligned} &\inf\{s:A_{s}(\omega)>t\}\quad&\text{for }t<A_{\zeta(\omega)-}(\omega),\\ &\infty\quad&\text{for }t\geq A_{\zeta(\omega)-}(\omega).\end{aligned}\right.

Further, let

\check{Y}_{t}(\omega):=Y_{\gamma_{t}(\omega)}(\omega),\quad\check{\zeta}(\omega):=A_{\zeta(\omega)-}(\omega)(=A_{\zeta(\omega)}(\omega)),\quad t\geq 0,\omega\in\Omega.

Note that $Y_{t}(\omega):=\partial$ for $\zeta(\omega)\leq t\leq\infty$ , so $\check{Y}_{t}(\omega)=\partial$ for $t\geq\check{\zeta}(\omega)$ . According to [3, Proposition A.3.8(iv)], we may assume without loss of generality that $\check{Y}_{t}(\omega)\in\text{Supp}(A)\cup\{\partial\}$ for all $t\geq 0$ and all $\omega\in\Omega$ . Set $\check{{\mathscr{G}}}_{t}:={\mathscr{G}}_{\gamma_{t}}$ and $\check{\theta}_{t}:=\theta_{\gamma_{t}}$ . It is well known that the time-changed process

\check{Y}:=\left(\Omega,{\mathscr{G}},\check{{\mathscr{G}}}_{t},\check{Y}_{t},\check{\theta}_{t},\left(\mathbf{P}_{x}\right)_{x\in\text{Supp}(A)}\right)

(4.12)

with lifetime $\check{\zeta}$ is a right process on $\text{Supp}(A)$ ; see, e.g., [3, Theorem A.3.11].

Theorem 4.2.

Let $Y$ be a Feller’s Brownian motion with parameters $(p_{1},p_{2},p_{3},p_{4})$ as specified in Theorem 3.3, and let $\check{Y}$ be the time-changed process (4.12) of $Y$ with respect to the PCAF (4.3). Then $\Xi(\check{Y}):=\left(\Xi(\check{Y}_{t})\right)_{t\geq 0}$ is a $Q$ -process whose birth-death density matrix is (2.1), where $\Xi$ is defined as (4.2). Furthermore,

(1)

If $p_{2}=0,p_{3}>0$ , and $|p_{4}|=0$ , then $\Xi(\check{Y})$ is the minimal $Q$ -process;
(2)

If $p_{2}=0,p_{3}>0$ , and $0<|p_{4}|<\infty$ , then $\Xi(\check{Y})$ is a Doob process;
(3)

Otherwise, $\Xi(\check{Y})$ is a Feller $Q$ -process.

Proof.

Denote the transition semigroup and resolvent of $\check{Y}$ by $(\check{T}_{t})_{t\geq 0}$ and $(\check{G}_{\alpha})_{\alpha>0}$ , respectively. Define

\check{p}_{ij}(t):=\check{T}_{t}1_{\{\hat{c}_{j}\}}(\hat{c}_{i}),\quad t\geq 0,i,j\in\mathbb{N},

and

\check{\Psi}_{ij}(\alpha):=\int_{0}^{\infty}e^{-\alpha t}\check{p}_{ij}(t)dt,\quad\alpha>0,i,j\in\mathbb{N}.

The goal is to prove that $(\check{p}_{ij}(t))_{i,j\in\mathbb{N}}$ is a $Q$ -process.

We first demonstrate that $(\check{p}_{ij}(t))_{i,j\in\mathbb{N}}$ is a standard transition matrix. According to [26, §2.5, Theorem 1], it suffices to verify that $\check{\Psi}_{ij}(\alpha)$ satisfies the following conditions:

		$\displaystyle\check{\Psi}_{ij}(\alpha)\geq 0,\quad\alpha\sum_{j\in\mathbb{N}}\check{\Psi}_{ij}(\alpha)\leq 1,$		(4.13)
		$\displaystyle\check{\Psi}_{ij}(\alpha)-\check{\Psi}_{ij}(\beta)+(\alpha-\beta)\sum_{k\in\mathbb{N}}\check{\Psi}_{ik}(\alpha)\check{\Psi}_{kj}(\beta)=0,\quad\forall\alpha,\beta>0,$
		$\displaystyle\lim_{\alpha\rightarrow\infty}\alpha\check{\Psi}_{ij}(\alpha)=\delta_{ij},$

where $\delta_{ij}$ is the Kronecker delta. To accomplish this, note that

\check{\Psi}_{ij}(\alpha)=\check{G}_{\alpha}1_{\{\hat{c}_{j}\}}(\hat{c}_{i}).

Thus, the first condition in (4.13) is straightforward, and the third condition follows from the right continuity of $\check{Y}$ . To prove the second condition, it is sufficient to demonstrate

\check{G}_{\alpha}1_{\{0\}}(\hat{c}_{i})=0

(4.14)

and then apply the resolvent equation of $\check{G}_{\alpha}$ . In fact, it follows from the definition of $\check{Y}$ , [22, Proposition 4.9 of page 8], and the Fubini theorem that

	$\displaystyle\check{G}_{\alpha}1_{\{0\}}(\hat{c}_{i})$	$\displaystyle=\mathbf{E}_{\hat{c}_{i}}\int_{0}^{\infty}e^{-\alpha A_{\gamma_{t}}}1_{\{0\}}(Y_{\gamma_{t}})dt=\mathbf{E}_{\hat{c}_{i}}\int_{0}^{\infty}e^{-\alpha A_{t}}1_{\{0\}}(Y_{t})dA_{t}$
		$\displaystyle=\sum_{n\in\mathbb{N}}\mu_{n}\mathbf{E}_{\hat{c}_{i}}\int_{0}^{\infty}e^{-\alpha A_{t}}1_{\{0\}}(Y_{t})dL^{\hat{c}_{n}}_{t}=0.$

Thus, (4.14) is established.

Recall that $Y^{0}$ , with lifetime $\zeta^{0}$ , is an absorbing Brownian motion on $(0,\infty)$ . According to Lemma 3.7 and Lemma 4.1, we have that

A^{0}_{t}:=A_{t\wedge\zeta^{0}},\quad t\geq 0

is a PACF of $Y^{0}$ , with Revuz measure $\sum_{n\in\mathbb{N}}\mu_{n}\delta_{\hat{c}_{n}}$ . Define

\hat{Y}^{0}=\left(\Omega,{\mathscr{G}},\hat{{\mathscr{G}}}_{t},\hat{Y}^{0}_{t},\hat{\theta}^{0}_{t},(\mathbf{P}_{x})_{x\in E}\right),

with lifetime $\hat{\zeta}^{0}:=A^{0}_{\zeta^{0}}=A_{\zeta^{0}}$ , as the time-changed process of $Y^{0}$ with respect to the PACF $A^{0}$ . Specifically,

\hat{{\mathscr{G}}}_{t}:={\mathscr{G}}_{\gamma^{0}_{t}},\quad\hat{\theta}^{0}_{t}:=\theta^{0}_{\gamma^{0}_{t}},\quad\hat{Y}^{0}_{t}:=Y^{0}_{\gamma^{0}_{t}},

with

\gamma^{0}_{t}(\omega):=\left\{\begin{aligned} &\inf\{s:A^{0}_{s}(\omega)>t\}\quad&\text{for }t<\hat{\zeta}^{0},\\ &\infty\quad&\text{for }t\geq\hat{\zeta}^{0}.\end{aligned}\right.

It has been established in [18, Lemma 3.1] that $\Xi(\hat{Y}^{0})$ is exactly the minimal $Q$ -process.

Next, let us examine the killed process of $\check{Y}$ upon hitting $0\in\overline{E}$ . More precisely, let $\check{\eta}_{n}:=\inf\{t>0:\check{Y}_{t}=\hat{c}_{n}\}$ for all $n\geq 1$ . From the definition (4.12) of $\check{Y}$ , we find that for $\mathbf{P}_{\hat{c}_{i}}$ -a.s. $\omega\in\Omega$ , $\check{\eta}_{n}(\omega)$ is increasing in $n$ for $n>i$ . Hence,

\check{\eta}_{\infty}(\omega):=\lim_{n\rightarrow\infty}\check{\eta}_{n}(\omega)

(4.15)

is well defined for $\mathbf{P}_{\hat{c}_{i}}$ -a.s. $\omega$ and all $i\in\mathbb{N}$ . We will show that $\check{\eta}_{\infty}=A_{\zeta^{0}}=\hat{\zeta}^{0}$ , and hence the killed process

\check{Y}^{0}_{t}:=\left\{\begin{aligned} &\check{Y}_{t},\quad&t<\check{\zeta}\wedge\check{\eta}_{\infty},\\ &\partial,\quad&t\geq\check{\zeta}\wedge\check{\eta}_{\infty},\end{aligned}\right.

with lifetime $\check{\zeta}^{0}:=\check{\zeta}\wedge\check{\eta}_{\infty}(=\check{\eta}_{\infty})$ , on $E$ is identical to $\hat{Y}^{0}$ . To prove this, fix $\mathbf{P}_{\hat{c}_{i}}$ and let $\eta_{n}:=\inf\{t>0:Y_{t}=\hat{c}_{n}\}$ for $n>i$ . Note that $Y_{\eta_{n}}=\hat{c}_{n}$ and $Y_{t}\neq\hat{c}_{n}$ for all $t<\eta_{n}$ , $\mathbf{P}_{\hat{c}_{i}}$ -a.s. It follows from [1, V. Theorem 3.8] that $A_{\eta_{n}}=A_{\eta_{n}-\varepsilon_{0}}$ for some $\varepsilon_{0}>0$ and $A_{\eta_{n}+\varepsilon}>A_{\eta_{n}}$ for all $\varepsilon>0$ . Consequently,

\gamma_{A_{\eta_{n}}}=\inf\{s>0:A_{s}>A_{\eta_{n}}\}=\eta_{n},

and

\gamma_{t}\leq\eta_{n}-\varepsilon_{0},\quad\forall t<A_{\eta_{n}}.

These yield

\check{Y}_{A_{\eta_{n}}}=Y_{\gamma_{A_{\eta_{n}}}}=Y_{\eta_{n}}=\hat{c}_{n}

and

\check{Y}_{t}=Y_{\gamma_{t}}\neq\hat{c}_{n},\quad\forall t<A_{\eta_{n}}.

In other words, $\check{\eta}_{n}=A_{\eta_{n}}$ , $\mathbf{P}_{\hat{c}_{i}}$ -a.s. Noting that $\lim_{n\rightarrow\infty}\eta_{n}=\zeta^{0}$ , we obtain that $\check{\eta}_{\infty}=\lim_{n\rightarrow\infty}\check{\eta}_{n}=\lim_{n\rightarrow\infty}A_{\eta_{n}}=A_{\zeta^{0}}$ .

According to the argument in the previous two paragraphs, we can conclude that $\check{p}^{0}_{ij}(t):=\mathbf{P}_{\hat{c}_{i}}(\check{Y}_{t}=\hat{c}_{j},t<\check{\eta}_{\infty})$ is the transition matrix of the minimal $Q$ -process. This implies that

\lim_{t\rightarrow 0}\frac{\check{p}^{0}_{ij}(t)-\delta_{ij}}{t}=q_{ij}.

(4.16)

Mimicking the proof of [18, Proposition 3.6], we can also obtain that

\lim_{t\rightarrow 0}\frac{\mathbf{P}_{\hat{c}_{i}}(\check{\eta}_{\infty}\leq t)}{t}=0.

(4.17)

Combining (4.16) and (4.17), we obtain

\lim_{t\rightarrow 0}\frac{\check{p}_{ij}(t)-\delta_{ij}}{t}=q_{ij}.

In other words, $(\check{p}_{ij}(t))_{i,j\in\mathbb{N}}$ is a $Q$ -process.

Finally, let us classify the $Q$ -process $\Xi(\check{Y})$ for different cases. For the first case, where $p_{2}=0,p_{3}>0$ and $|p_{4}|=0$ , we observe that $t\mapsto A_{t}$ does not increase after time $\zeta^{0}$ (according to the sample path representation in §3.2.1). Consequently, $\gamma_{t}=\infty$ for all $t\geq A_{\zeta^{0}}$ , which implies that $\hat{Y}_{t}=\partial$ for all $t\geq A_{\zeta^{0}}=\check{\eta}_{\infty}$ . Thus, $\Xi(\check{Y})$ aligns with the minimal $Q$ -process. In the second case, where $p_{2}=0,p_{3}>0$ and $0<|p_{4}|<\infty$ , $t\mapsto A_{t}$ may continue to increase after time $\zeta^{0}$ . A similar argument shows that $\Xi(\check{Y})$ is not the minimal $Q$ -process. However, according to Lemma 4.1, $\check{Y}$ is a right process with state space $\text{Supp}(A)=E$ . Therefore, $\check{Y}_{\check{\eta}_{\infty}}\in E\cup\{\partial\}$ , $\mathbf{P}_{\hat{c}_{i}}$ -a.s. for all $i\in\mathbb{N}$ . Based on [18, Corollary 5.2], it follows that $\Xi(\check{Y})$ is a Doob process.

For the remaining cases, it suffices to show that $\check{Y}_{\check{\eta}_{\infty}}=0$ , $\mathbf{P}_{\hat{c}_{i}}$ -a.s. for all $i\in\mathbb{N}$ . To demonstrate this, consider

\gamma_{\check{\eta}_{\infty}}=\inf\{t>0:A_{t}>\check{\eta}_{\infty}\}=\inf\{t>0:A_{t}>A_{\zeta^{0}}\}.

If it were false that $\gamma_{\check{\eta}_{\infty}}=\zeta^{0}$ , then there would exist some $\varepsilon_{0}>0$ such that $A_{\zeta^{0}+\varepsilon_{0}}=A_{\zeta^{0}}$ . This implies that $Y_{t}\notin E$ for all $\zeta^{0}<t<\zeta^{0}+\varepsilon_{0}$ by [1, V. Theorem 3.8]. Since $Y_{\zeta^{0}}=0$ and $Y$ is right continuous, it follows that $Y_{t}=0$ for all $\zeta^{0}\leq t<\zeta^{0}+\varepsilon_{0}$ . However, this contradicts the previous argument that $Y$ cannot remain at $0$ for any extended period, as noted in the last paragraph of the proof of Lemma 4.1. Therefore, we have $\check{Y}_{\check{\eta}_{\infty}}=Y_{\zeta^{0}}=0$ . ∎

In the context of general Markov process, the Feller’s Brownian motion $Y$ with parameters $(p_{1},p_{2},p_{3},p_{4})$ is termed conservative if $\mathbf{P}_{x}(\zeta<\infty)=0$ for any $x\in[0,\infty)$ , which is equivalent to $p_{1}=0$ (see [10, §15]). In terms of continuous-time Markov chains, a conservative $Q$ -process is also referred to as an honest $Q$ -process (see §2.2).

Corollary 4.3.

If the Feller’s Brownian motion $Y$ is conservative, then for any $a>0$ and $x\in[0,\infty)$ ,

\mathbf{P}_{x}(L^{a}_{\infty}=\infty)=1.

(4.18)

Particularly, the $Q$ -process $\left(\Xi(\check{Y}_{t})\right)_{t\geq 0}$ , obtained in Theorem 4.2, is also conservative.

Proof.

Write $T:=T_{a}:=\inf\{t>0:Y_{t}=a\}$ . We first show that

\mathbf{P}_{x}(T<\infty)=1.

(4.19)

To demonstrate this, let $\sigma:=\{t>0:Y_{t}\geq a\}\leq T$ . Observe that

\{\sigma=\infty\}=\{Y_{t}<a,\forall t>0\}.

The sample path of $Y$ is described by (3.23) up to a time change transformation (see (3.25)). Given that $p_{1}=0$ , it follows that

\{\sigma=\infty\}\subset\{W^{+}_{t}<a,\forall t>0\}.

Therefore, $\mathbf{P}_{x}(\sigma=\infty)\leq\mathbf{P}_{x}(W^{+}_{t}<a,\forall t>0)=0$ . On the other hand, by the strong Markov property, we have

\mathbf{P}_{x}(T=\infty,\sigma<\infty)=\mathbf{E}_{x}\left(\mathbf{P}_{Y_{\sigma}}(T=\infty);\{\sigma<\infty\}\right).

Since $Y_{\sigma}\geq a$ , $\mathbf{P}_{x}$ -a.s. on $\{\sigma<\infty\}$ , it follows that $\mathbf{P}_{Y_{\sigma}}(T=\infty)=0$ , $\mathbf{P}_{x}$ -a.s. on $\{\sigma<\infty\}$ . Therefore, $\mathbf{P}_{x}(T=\infty,\sigma<\infty)=0$ . Combining this with $\mathbf{P}_{x}(\sigma=\infty)=0$ , we can eventually derive (4.19).

Next, by mimicking the computation in (4.9), we can show that for $\alpha>0$ ,

\mathbf{E}_{a}\int_{0}^{\infty}e^{-\alpha t}dL^{a}_{t}=\frac{1}{\sqrt{2\alpha}}\frac{e^{\sqrt{2\alpha}a}-e^{-\sqrt{2\alpha}a}}{e^{\sqrt{2\alpha}a}-\mathbf{E}_{0}e^{-\alpha T}}.

It follows from (4.19) (with $x=0$ ) that

\lim_{\alpha\rightarrow 0}\mathbf{E}_{a}\int_{0}^{\infty}e^{-\alpha t}dL^{a}_{t}=\infty.

(4.20)

Finally, by using (4.19) and (4.20), we can apply [1, V. Theorem 3.17] to conclude (4.18). ∎

4.3. Uniqueness of PCAF for time change

We continue to examine the time change transformation in Theorem 4.2. The goal is to demonstrate that (4.3) is indeed the unique PCAF of $Y$ for which the corresponding time-changed process is a $Q$ -process with the given density matrix (2.3).

Theorem 4.4.

Let $Y$ be a Feller’s Brownian motion and $A^{1}=(A^{1}_{t})_{t\geq 0}$ be a PCAF of $Y$ such that $\text{Supp}(A^{1})\subset\overline{E}$ . Denote by $\check{Y}^{1}$ the time-changed process of $Y$ with respect to the PCAF $A^{1}$ . If $X^{1}:=\Xi(\check{Y}^{1})$ is a $Q$ -process with the given density matrix (2.3), then

A^{1}_{t}=A_{t},\quad\forall t\geq 0,

where $A=(A_{t})_{t\geq 0}$ is defined as (4.3).

Proof.

Let $\ell=(\ell_{t})_{t\geq 0}$ denote the local time of $Y$ at $0$ , as discussed in [10] or [1, V§3]. (We will soon see that the normalization of this local time is not necessary for our proof.) Define

\bar{A}_{t}:=A_{t}+\ell_{t},\quad t\geq 0.

Clearly, $\bar{A}=(\bar{A}_{t})_{t\geq 0}$ is a PCAF of $Y$ . To verify the condition in [1, V§3, Proposition 3.11] for $A^{1}$ and $\bar{A}$ , consider $g\in\mathcal{B}_{+}([0,\infty))$ such that

\mathbf{E}_{x}\int_{0}^{\infty}g(Y_{t})d\bar{A}_{t}=0,\quad\forall x\in[0,\infty).

Then $g|_{\overline{E}}\equiv 0$ . Since $\text{Supp}(A^{1})\subset\overline{E}$ , we have

\mathbf{E}_{x}\int_{0}^{\infty}g(Y_{t})dA^{1}_{t}=0,\quad\forall x\in[0,\infty).

Applying [1, V§3, Proposition 3.11] to $A^{1}$ and $\bar{A}$ , we obtain that

A^{1}_{t}=\sum_{n\in\mathbb{N}}\tilde{\mu}_{n}L^{\hat{c}_{n}}_{t}+\tilde{\mu}_{\infty}\ell_{t},

for some $\tilde{\mu}_{n}\geq 0$ and $\tilde{\mu}_{\infty}\geq 0$ .

The fact that $X^{1}$ is a $Q$ -process implies that

\mathbf{E}_{x}\int_{0}^{\infty}e^{-\alpha t}1_{\{0\}}(\check{Y}^{1}_{t})dt=0

(4.21)

for $\alpha>0$ and $x\in\text{Supp}(A^{1})$ . However, the left-hand side of (4.21) can be expressed as

\mathbf{E}_{x}\int_{0}^{\infty}e^{-\alpha A^{1}_{t}}1_{\{0\}}(Y_{t})dA^{1}_{t}=\tilde{\mu}_{\infty}\cdot\mathbf{E}_{x}\int_{0}^{\infty}e^{-\alpha A^{1}_{t}}d\ell_{t}.

Hence, we conclude that $\tilde{\mu}_{\infty}=0$ .

On the other hand, it is straightforward (or follows from the proof of Theorem 4.2) that the time-changed process of the absorbing Brownian motion $Y^{0}$ , with lifetime $\zeta^{0}$ , with respect to the PCAF $(A^{1}_{t\wedge\zeta^{0}})_{t\geq 0}$ corresponds precisely to the minimal $Q$ -process. Particularly, the Revuz measure of

A^{1}_{t\wedge\zeta^{0}}=\sum_{n\in\mathbb{N}}\tilde{\mu}_{n}L^{\hat{c}_{n}}_{t\wedge\zeta^{0}}

with respect to $Y^{0}$ is actually the speed measure $\mu$ . Therefore, we can conclude that $\tilde{\mu}_{n}=\mu_{n}$ for all $n\in\mathbb{N}$ . ∎

5. Parameters of birth-death processes obtained by time change

According to Theorem 2.1, the $Q$ -process $\Xi(\check{Y})$ obtained in Theorem 4.2 (excluding the first case, which yields the minimal $Q$ -process) admits a resolvent representation given by a triple $(\gamma,\beta,\nu)$ , which is unique up to a multiplicative constant. It is certainly interesting to explore how the parameters $(p_{1},p_{2},p_{3},p_{4})$ of Feller’s Brownian motion determine $(\gamma,\beta,\nu)$ .

5.1. Main result

Recall that the sequence $\{\hat{c}_{n}:n\in\mathbb{N}\}$ is given in (4.1). Based on the measure $p_{4}$ on $(0,\infty)$ , we define a sequence $\{\mathfrak{p}_{n}:n\in\mathbb{N}\}$ as follows:

\mathfrak{p}_{0}:=\int_{(\hat{c}_{1},\hat{c}_{0}]}\frac{x-\hat{c}_{1}}{\hat{c}_{0}-\hat{c}_{1}}p_{4}(dx)+p_{4}\left((\hat{c}_{0},\infty)\right)

(5.1)

and

\mathfrak{p}_{n}:=\int_{(\hat{c}_{n+1},\hat{c}_{n}]}\frac{x-\hat{c}_{n+1}}{\hat{c}_{n}-\hat{c}_{n+1}}p_{4}(dx)+\int_{(\hat{c}_{n},\hat{c}_{n-1})}\frac{\hat{c}_{n-1}-x}{\hat{c}_{n-1}-\hat{c}_{n}}p_{4}(dx),\quad n\geq 1.

(5.2)

Note that if $|p_{4}|<\infty$ , then $|p_{4}|=\sum_{n\in\mathbb{N}}\mathfrak{p}_{n}$ . Our main result is as follows.

Theorem 5.1.

Let $Y$ be a Feller’s Brownian motion with parameters $(p_{1},p_{2},p_{3},p_{4})$ as described in Theorem 3.3 (excluding the first case, i.e., $p_{2}=0,p_{3}>0$ , and $|p_{4}|=0$ , in Theorem 4.2). Then the parameters $(\gamma,\beta,\nu)$ that determine the resolvent matrix of the $Q$ -process $\Xi(\check{Y})$ , obtained in Theorem 4.2, are given by

\gamma=p_{1},\quad\beta=2p_{2},\quad\nu_{n}=\mathfrak{p}_{n},\;n\in\mathbb{N}

up to a multiplicative constant, where $\{\mathfrak{p}_{n}\}$ is the sequence defined by (5.1) and (5.2).

The proof of Theorem 5.1 will be completed in the the following three sections. Here, we present a consequence: not only is every time-changed Feller’s Brownian motion a $Q$ -process, as shown in Theorem 4.2, but also every $Q$ -process can be derived from a Feller’s Brownian motion through time change.

Corollary 5.2.

For every $Q$ -process $X$ , there exists a Feller’s Brownian motion $Y$ such that the $Q$ -process obtained from $Y$ through time change, as described in Theorem 4.2, is identical in law to $X$ .

Proof.

Let $(\gamma,\beta,\nu)$ be the triple determining the resolvent matrix of $X$ . When $X$ is the minimal $Q$ -process, we can choose $Y$ with $p_{2}=0,p_{3}>0$ and $|p_{4}|=0$ . In this case, the first case of Theorem 4.2 applies. When $X$ is a Doob process, i.e., $\beta=0,|\nu|<\infty$ , we can choose

p_{1}=\gamma,\quad p_{2}=0,\quad p_{3}>0,\quad p_{4}=\sum_{n\in\mathbb{N}}\nu_{n}\delta_{\hat{c}_{n}}.

Then, we apply the second case of Theorem 4.2 and Theorem 5.1. When $X$ is non-minimal and non-Doob, we can choose

p_{1}=\gamma,\quad p_{2}=\frac{\beta}{2},\quad p_{3}=0,\quad p_{4}=\sum_{n\in\mathbb{N}}\nu_{n}\delta_{\hat{c}_{n}}.

(To satisfy (3.3), it may be necessary to multiply all parameters by a positive constant.) ∎

5.2. Pathwise construction of $Q$ -processes

In this subsection, we discuss the pathwise construction of a Feller $Q$ -process $X$ based on Theorem 5.1. For simplicity, we focus on the honest case where $\gamma=0$ , and consider the Feller process $\hat{X}=\Xi^{-1}(X)$ on $\overline{E}$ with parameters $(\nu,\beta,0)$ .

In the context of $Q$ -processes, the $(Q,1)$ -process plays a role analogous to that of the reflecting Brownian motion in the context of Feller’s Brownian motion. We denote its corresponding $Q$ -process on $\overline{E}$ by $\hat{W}^{+}$ . This process is the time-changed process of the reflecting Brownian motion $W^{+}$ with respect to $A^{+}_{t}:=\sum_{n\in\mathbb{N}}\mu_{n}L^{+,\hat{c}_{n}}_{t}$ , where $L^{+,\hat{c}_{n}}$ denotes the local time of $W^{+}$ at $\hat{c}_{n}$ as defined in Definition 3.1. Let $\gamma^{+}$ be the right-continuous inverse of $A^{+}$ . The local time of $\hat{W}^{+}$ at $0$ , denoted by $\hat{\ell}$ , can be derived from the Brownian local time $\ell$ through the corresponding time change transformation, specifically, $\hat{\ell}_{t}=\ell_{\gamma^{+}_{t}}$ .

We initially considered replacing $W^{+}$ with $\hat{W}^{+}$ and constructing the general $Q$ -process similarly to (3.23), by defining

\hat{X}^{1}_{t}:=Z(Z^{-1}(\hat{\ell}_{t}))-\hat{\ell}_{t}+\hat{W}^{+}_{t},\quad t\geq 0,

(5.3)

where $Z$ is the subordinator (3.21) with $p_{2}=\frac{\beta}{2}$ and $p_{4}=\sum_{n\in\mathbb{N}}\nu_{n}\delta_{\hat{c}_{n}}$ , which is independent of $\hat{W}^{+}$ . (A similar idea appears in [10, §20].) However, at a time $t$ when $\hat{\ell}_{t}$ increases to a discontinuous point $x$ ( $=\hat{\ell}_{t}$ ) of $Z(Z^{-1})$ , where $h(x):=Z(Z^{-1}(x))-x=l$ for some $0<l\in E$ and $h(x-)=0$ , the term $h(\hat{\ell}_{t})$ in (5.3) will continuously decrease from $l$ for a short period thereafter (see [10, §12]). Meanwhile, $\hat{W}^{+}$ ‘diffuses’ within the space $\overline{E}$ near $0$ . This discrepancy causes the process defined by (5.3) to move outside of $\overline{E}$ .

Note that (5.3) can be rewritten as

\hat{X}^{1}_{t}=Z(Z^{-1}(\ell_{\gamma^{+}_{t}}))-\ell_{\gamma^{+}_{t}}+W^{+}_{\gamma^{+}_{t}},\quad t\geq 0.

However, according to (3.23) and Theorem 5.1, the correct pathwise representation of $\hat{X}$ should be

\hat{X}_{t}=Z(Z^{-1}(\ell_{\gamma_{t}}))-\ell_{\gamma_{t}}+W^{+}_{\gamma_{t}},\quad t\geq 0,

where $\gamma$ is the right-continuous inverse of (4.3). In other words, (5.3) incorrectly reverses the order of adding jumps using $Z$ and the transformation of time change.

6. Proof of Theorem 5.1 for $|p_{4}|<\infty$

In this section, we will prove Theorem 5.1 for various cases where $|p_{4}|<\infty$ . Simultaneously, we will provide a more comprehensive characterization of the corresponding $Q$ -processes. From now on, we will denote $X_{t}:=\Xi(\check{Y}_{t})$ for $t\geq 0$ and use $X:=(X_{t})_{t\geq 0}$ for convenience.

6.1. Doob processes

According to Theorem 4.2, $X$ is a Doob process (but not the minimal $Q$ -process) if and only if $p_{2}=0,p_{3}>0$ , and $0<|p_{4}|<\infty$ .

Theorem 6.1.

If $p_{2}=0,p_{3}>0$ and $0<|p_{4}|<\infty$ , then the parameters $(\gamma,\beta,\nu)$ that determine the resolvent matrix of the Doob process $X=\Xi(\check{Y})$ are given by

\gamma=p_{1},\quad\beta=0,\quad\nu_{n}=\mathfrak{p}_{n},\quad n\in\mathbb{N}.

Particularly, the instantaneous distribution of $X$ is

\pi(\{\partial\})=\frac{p_{1}}{p_{1}+|p_{4}|},\quad\pi(\{n\})=\frac{\mathfrak{p}_{n}}{p_{1}+|p_{4}|},\;n\in\mathbb{N}.

Proof.

The fact that $\beta=0$ follows directly from Theorems 2.1 and 4.2. Fix $\mathbf{P}_{\hat{c}_{i}}$ for some $i\in\mathbb{N}$ , and let $\check{\eta}_{\infty}$ be defined as (4.15), representing the first flying time (see [18, Corollary 4.7]) of $\check{Y}$ . According to [18, Theorem 5.1], it suffices to demonstrate that

\mathbf{P}_{\hat{c}_{i}}(\check{Y}_{\check{\eta}_{\infty}}=\partial)=\frac{p_{1}}{p_{1}+|p_{4}|},\quad\mathbf{P}_{\hat{c}_{i}}(\check{Y}_{\check{\eta}_{\infty}}=\hat{c}_{n})=\frac{\mathfrak{p}_{n}}{p_{1}+|p_{4}|},\quad n\in\mathbb{N}.

(6.1)

We will use the same symbols as those in the proof of Theorem 4.2. Define

\sigma_{1}:=\inf\{t>\zeta^{0}:Y_{t}\neq 0\},\quad\sigma_{2}:=\inf\{t>\sigma_{1}:Y_{t}\in E\},

and

\sigma:=\inf\{t>0:Y_{t}\in E\}.

Note that $\check{\eta}_{\infty}=A_{\zeta^{0}}$ . From the pathwise representation of $Y$ in §3.2.1, it follows that

\gamma_{\check{\eta}_{\infty}}=\inf\{t>0:A_{t}>A_{\zeta^{0}}\}=\sigma_{2}.

It is straightforward to verify that $Y_{\sigma_{2}}=Y_{\sigma}\circ\theta_{\sigma_{1}}$ and that $\{Y_{\sigma_{2}}=\hat{c}_{n}\}\subset\{Y_{\sigma_{1}}\in(\hat{c}_{n+1},\hat{c}_{n-1})\}$ (with the convention $\hat{c}_{-1}:=\infty$ ). By the strong Markov property of $Y$ , we have

	$\displaystyle\mathbf{P}_{\hat{c}_{i}}(\check{Y}_{\check{\eta}_{\infty}}=\hat{c}_{n})$	$\displaystyle=\mathbf{P}_{\hat{c}_{i}}(Y_{\sigma_{2}}=\hat{c}_{n})=\mathbf{P}_{\hat{c}_{i}}\left(Y_{\sigma}\circ\theta_{\sigma_{1}}=\hat{c}_{n},Y_{\sigma_{1}}\in(\hat{c}_{n+1},\hat{c}_{n-1})\right)$
		$\displaystyle=\mathbf{P}_{\hat{c}_{i}}\left(\mathbf{P}_{Y_{\sigma_{1}}}(Y_{\sigma}=\hat{c}_{n});Y_{\sigma_{1}}\in(\hat{c}_{n+1},\hat{c}_{n-1})\right).$

Note that $\mathbf{P}_{\hat{c}_{i}}\left(Y_{\sigma_{1}}\in dx\right)=\lambda(dx)$ , where $\lambda$ is given by (3.20). For $x\in(\hat{c}_{n+1},\hat{c}_{n-1})$ , we have

\mathbf{P}_{x}(Y_{\sigma}=\hat{c}_{n})=\left\{\begin{aligned} &\frac{x-\hat{c}_{n+1}}{\hat{c}_{n}-\hat{c}_{n+1}},\quad&x\in(\hat{c}_{n+1},\hat{c}_{n}],\\ &\frac{\hat{c}_{n-1}-x}{\hat{c}_{n-1}-\hat{c}_{n}},\quad&x\in(\hat{c}_{n},\hat{c}_{n-1}),\end{aligned}\right.

with the convention $\frac{\infty}{\infty}:=1$ . Thus, a straightforward computation yields

\mathbf{P}_{\hat{c}_{i}}(\check{Y}_{\check{\eta}_{\infty}}=\hat{c}_{n})=\int_{(\hat{c}_{n+1},\hat{c}_{n-1})}\mathbf{P}_{x}(Y_{\sigma}=\hat{c}_{n})\lambda(dx)=\frac{\mathfrak{p}_{n}}{p_{1}+|p_{4}|}.

Finally, $\mathbf{P}_{\hat{c}_{i}}(\check{Y}_{\check{\eta}_{\infty}}=\partial)=1-\sum_{n\in\mathbb{N}}\mathbf{P}_{\hat{c}_{i}}(\check{Y}_{\check{\eta}_{\infty}}=\hat{c}_{n})=\frac{p_{1}}{p_{1}+|p_{4}|}$ . Therefore, (6.1) is established. This completes the proof. ∎

6.2. Symmetric case

The symmetric case, where $p_{2}>0$ and $|p_{4}|=0$ , can be analyzed using Dirichlet form theory. In this framework, the time change of a Markov process corresponds to a trace Dirichlet form (see [3, Chapter 5]).

To state our results, we first define the following quadratic form for a function $f$ on $\mathbb{N}$ :

{\mathscr{A}}(f,f):=\sum_{k\in\mathbb{N}}\frac{(f(k+1)-f(k))^{2}}{c_{k+1}-c_{k}}.

If ${\mathscr{A}}(f,f)<\infty$ , it follows from $c_{\infty}=\lim_{k\rightarrow\infty}c_{k}<\infty$ and the Cauchy-Schwarz inequality that $f(\infty):=\lim_{k\rightarrow\infty}f(k)$ exists.

Theorem 6.2.

If $p_{2}>0$ and $|p_{4}|=0$ , then the parameters $(\gamma,\beta,\nu)$ determining the resolvent matrix of the $Q$ -process $X=\Xi(\check{Y})$ are

\gamma=p_{1},\quad\beta=2p_{2},\quad\nu_{n}=0,\;n\in\mathbb{N}.

(6.2)

Furthermore, $X$ is symmetric with respect to the speed measure $\mu$ , and the associated Dirichlet form on $L^{2}(\mathbb{N}\cup\{\infty\},\mu)$ is given by

		$\displaystyle\check{{\mathscr{F}}}=\{f\in L^{2}(\mathbb{N}\cup\{\infty\},\mu):{\mathscr{A}}(f,f)<\infty\}$
		$\displaystyle\check{{\mathscr{E}}}(f,g)=\frac{1}{2}\sum_{k\in\mathbb{N}}\frac{(f(k+1)-f(k))(g(k+1)-g(k))}{c_{k+1}-c_{k}}+\frac{p_{1}}{2p_{2}}f(\infty)g(\infty),\quad f,g\in\check{{\mathscr{F}}}.$

Proof.

In this case, the Revuz measure of the PCAF (4.5) with respect to $Y$ is exactly $\tilde{\mu}:=\sum_{k\in\mathbb{N}}\mu_{k}\delta_{\hat{c}_{k}}$ , by Definition 3.8 and [3, Proposition 4.1.10]. According to [3, Theorem 5.2.2], the time-changed process $\check{Y}$ is symmetric with respect to $\tilde{\mu}$ , and its associated Dirichlet form $(\tilde{\mathscr{E}},\tilde{\mathscr{F}})$ on $L^{2}(\overline{E},\tilde{\mu})$ is derived in [3, (5.2.4)].

In what follows, we will compute $(\tilde{{\mathscr{E}}},\tilde{{\mathscr{F}}})$ . Regarding $\tilde{{\mathscr{F}}}$ , we note that the extended Dirichlet space of (3.13) is

H^{1}_{e}([0,\infty)):=\{f\text{ is absolutely continuous on }[0,\infty)\text{ and }f^{\prime}\in L^{2}([0,\infty))\},

as detailed in [3, Theorem 2.2.11]. According to [3, (5.2.4)], it is straightforward to compute that

\tilde{{\mathscr{F}}}=H^{1}_{e}([0,\infty))|_{\overline{E}}\cap L^{2}(\overline{E},\tilde{\mu})=\{\Xi(f):f\in\check{{\mathscr{F}}}\},

(6.3)

where $\Xi(f)(\hat{c}_{n}):=f(n)$ for all $n\in\mathbb{N}$ . The quadratic form $\tilde{{\mathscr{E}}}$ can be formulated using [3, Theorem 5.5.9]. The crucial step is to compute the Feller measure $U$ on $\overline{E}\times\overline{E}$ and the supplementary Feller measure $V$ on $\overline{E}$ , as defined in [3, (5.5.7)]. In fact, by mimicking the proof of [20, Theorem 2.1], we can show that the strongly local part of $\tilde{{\mathscr{E}}}$ vanishes, and $U$ is supported on $\{(\hat{c}_{n},\hat{c}_{n+1}),(\hat{c}_{n+1},\hat{c}_{n}):n\in\mathbb{N}\}\subset\overline{E}\times\overline{E}$ (with the notation $(\cdot,\cdot)$ indicating a pair of points, not an interval) with

U(\{(\hat{c}_{n},\hat{c}_{n+1})\})=U(\{(\hat{c}_{n+1},\hat{c}_{n})\})=\frac{1}{2|\hat{c}_{n}-\hat{c}_{n+1}|}=\frac{1}{2|c_{n+1}-c_{n}|}.

Regarding the measure $V$ , we note that $\mathbf{P}_{x}(\tau\geq\zeta)=0$ for any $x\in[0,\infty)\setminus\overline{E}$ , where $\tau:=\inf\{t\in[0,\zeta]:Y_{t}\notin[0,\infty)\setminus\overline{E}\}$ . Thus, by the definition [3, (5.5.7)] of $V$ , we have $V\equiv 0$ . Therefore, applying [3, Theorem 5.5.9], we obtain that for $u\in\tilde{{\mathscr{F}}}$ ,

\tilde{\mathscr{E}}(u,u)=\frac{1}{2}\sum_{k\in\mathbb{N}}\frac{(u(\hat{c}_{k+1})-u(\hat{c}_{k}))^{2}}{c_{k+1}-c_{k}}+\frac{p_{1}}{2p_{2}}u(0)^{2}.

(6.4)

Applying the spatial transformation $\Xi$ to (6.3) and (6.4) yields the expression for $(\check{{\mathscr{E}}},\check{{\mathscr{F}}})$ .

Finally, the parameters $(\gamma,\beta,\nu)$ are given by (6.2), as discussed in [18, Lemma 7.1 and Theorem 8.1]. ∎

6.3. Non-symmetric case with finite jumping measure

We turn to examine the non-symmetric case where $p_{2}>0$ and $0<|p_{4}|<\infty$ . The sample path representation of Feller’s Brownian motion for this case is detailed in §3.2.2. Let $\pi$ be a probability measure on $\mathbb{N}\cup\{\partial\}$ given by

\pi(\{\partial\}):=\frac{p_{1}}{p_{1}+|p_{4}|},\quad\pi(\{n\})=\pi_{n}:=\frac{\mathfrak{p}_{n}}{p_{1}+|p_{4}|},\quad n\in\mathbb{N},

(6.5)

where $\{\mathfrak{p}_{n}\}$ is specified by (5.1) and (5.2).

Theorem 6.3.

If $p_{2}>0$ and $0<|p_{4}|<\infty$ , then the parameters $(\gamma,\beta,\nu)$ determining the resolvent matrix of the $Q$ -process $X=\Xi(\check{Y})$ are

\gamma=p_{1},\quad\beta=2p_{2},\quad\nu_{n}=\mathfrak{p}_{n},\quad n\in\mathbb{N}.

(6.6)

Furthermore, $X$ can be obtained by piecing out $X^{1}$ with respect to the probability measure $\pi$ on $\mathbb{N}\cup\{\partial\}$ , where $X^{1}$ is the symmetric $Q$ -process corresponding to the parameters $(p_{1}+|p_{4}|,2p_{2},0)$ and $\pi$ is defined as (6.5).

Proof.

Let $Y^{1}$ , with lifetime $\zeta^{1}$ , be the symmetric Feller’s Brownian motion with parameters $(p_{1}+|p_{4}|,p_{2},p_{3},0)$ , as discussed in §3.2.2. It represents the killed process of $Y$ at $\zeta^{1}$ , and $Y$ can be obtained by piecing out $Y^{1}$ with respect to the probability measure $\lambda$ given by (3.20). Particularly, we have

\mathbf{P}_{x}(Y_{\zeta^{1}}\in\cdot)=\lambda(\cdot),\quad\forall x\in[0,\infty).

By following the steps used to obtain $\hat{Y}^{0}=\check{Y}^{0}$ in the proof of Theorem 4.2, we can also show that the killed process $\check{Y}^{1}$ of $\check{Y}$ at time $\check{\zeta}^{1}:=A_{\zeta^{1}}$ is identical in law to the time-changed process of $Y^{1}$ with respect to the PCAF $A^{1}_{t}:=A_{t\wedge\zeta^{1}},t\geq 0$ . Note that the Revuz measure of $A^{1}$ with respect to $Y^{1}$ is exactly $\sum_{n\in\mathbb{N}}\mu_{n}\delta_{\hat{c}_{n}}$ . Hence, according to Theorem 6.2, $\Xi(\check{Y}^{1})$ is identical in law to $X^{1}$ .

Denote by $\tilde{R}_{\alpha}^{1}$ and $\tilde{R}_{\alpha}$ the resolvents of $\check{Y}^{1}$ and $\check{Y}$ , respectively. For a bounded function $f$ on $\overline{E}$ , Dynkin’s formula gives us:

\tilde{R}_{\alpha}f(\hat{c})=\mathbf{E}_{\hat{c}}\int_{0}^{\infty}e^{-\alpha t}f(\check{Y}_{t})dt=\tilde{R}^{1}_{\alpha}f(\hat{c})+\mathbf{E}_{\hat{c}}\left(e^{-\alpha\check{\zeta}^{1}}\tilde{R}_{\alpha}f(\check{Y}_{\check{\zeta}^{1}})\right),\quad\forall\hat{c}\in\overline{E}.

(6.7)

Let $\sigma:=\inf\{t>0:Y_{t}\in E\}$ and $\sigma_{1}:=\inf\{t>\zeta^{1}:Y_{t}\in E\}$ . Then,

\gamma_{\check{\zeta}^{1}}=\inf\{t>0:A_{t}>A_{\zeta^{1}}\}=\sigma_{1}.

Note that $Y_{\sigma_{1}}=Y_{\sigma}\circ\theta_{\zeta^{1}}$ . According to the construction procedures of piecing out as stated in [18, Appendix A], it is straightforward to see that $Y_{\zeta^{1}}$ (with distribution $\lambda$ ) is independent of $Y^{1}$ . Thus, $Y_{\zeta^{1}}$ is also independent of $\check{\zeta}^{1}=A_{\zeta^{1}}$ . From this independence and the strong Markov property of $Y$ , it follows that

$\displaystyle\mathbf{E}_{\hat{c}}\left(e^{-\alpha\check{\zeta}^{1}}\tilde{R}_{\alpha}f(\check{Y}_{\check{\zeta}^{1}})\right)$	$\displaystyle=\mathbf{E}_{\hat{c}}\left(e^{-\alpha\check{\zeta}^{1}}\tilde{R}_{\alpha}f(Y_{\sigma}\circ\theta_{\zeta^{1}})\right)$	(6.8)
	$\displaystyle=\mathbf{E}_{\hat{c}}\left(e^{-\alpha\check{\zeta}^{1}}\mathbf{E}_{Y_{\zeta^{1}}}\left(\tilde{R}_{\alpha}f(Y_{\sigma})\right)\right)$
	$\displaystyle=\mathbf{E}_{\hat{c}}\left(e^{-\alpha\check{\zeta}^{1}}\right)\cdot\mathbf{E}_{\lambda}\left(\tilde{R}_{\alpha}f(Y_{\sigma})\right).$

Similar to (6.4), we can also deduce that $\mathbf{P}_{\lambda}(Y_{\sigma}\in\cdot)=\tilde{\pi}(\cdot)$ , where $\tilde{\pi}(\{\hat{c}_{n}\}):=\pi_{n}$ for all $n\in\mathbb{N}$ and $\tilde{\pi}(\{\partial\}):=\pi(\{\partial\})$ . Substituting (6.8) into (6.7) yields

\tilde{R}_{\alpha}f(\hat{c})=\tilde{R}^{1}_{\alpha}f(\hat{c})+\mathbf{E}_{\hat{c}}\left(e^{-\alpha\check{\zeta}^{1}}\right)\cdot\tilde{\pi}(\tilde{R}_{\alpha}f).

Integrating both sides with respect to $\tilde{\pi}$ , we get

\tilde{\pi}(\tilde{R}_{\alpha}f)=\frac{\tilde{\pi}(\tilde{R}^{1}_{\alpha}f)}{1-\mathbf{E}_{\tilde{\pi}}\left(e^{-\alpha\check{\zeta}^{1}}\right)}.

It follows that

\tilde{R}_{\alpha}f(\hat{c})=\tilde{R}^{1}_{\alpha}f(\hat{c})+\frac{\tilde{\pi}(\tilde{R}^{1}_{\alpha}f)}{1-\mathbf{E}_{\tilde{\pi}}\left(e^{-\alpha\check{\zeta}^{1}}\right)}\cdot\mathbf{E}_{\hat{c}}\left(e^{-\alpha\check{\zeta}^{1}}\right),\quad\forall\hat{c}\in\overline{E}.

Finally, repeating the argument in the proof of [18, Theorem 8.1] (the steps after (8.5)) and applying the spatial transformation to $\check{Y}$ , we can conclude that $X$ is the piecing out of $X^{1}$ with respect to $\pi$ . Particularly, the parameters $(\gamma,\beta,\nu)$ for $X$ are given by (6.6). ∎

7. Approximation of Feller’s Brownian motion

We now turn to the ‘pathological’ case where $|p_{4}|=\infty$ . Our primary method involves constructing a sequence of Markov processes that converge to Feller’s Brownian motion, using the strategy outlined in the construction of $Q$ -processes in [26, §6.1-§6.6]. Each constituent process is the piecing out of the absorbing Brownian motion with respect to a specific probability measure $\lambda^{(n)}$ on $(0,\infty)\cup\{\partial\}$ , whose resolvent is expressed as (3.1) with $\lambda=\lambda^{(n)}$ . We refer to these processes as Doob’s Brownian motions, with $\lambda^{(n)}$ termed the instantaneous distribution (of the piecing out transformation).

7.1. Approximating sequence of Doob’s Brownian motions

We begin by introducing an important transformation on the sample paths. Let $x(t)$ be a right-continuous function on $[0,\infty)\cup\{\partial\}$ . Consider two sequences of positive constants $(\alpha_{m})$ and $(\beta_{m})$ such that

0(=:\beta_{0})<\alpha_{1}\leq\beta_{1}<\alpha_{2}\leq\beta_{2}<\cdots.

(These sequences may consist of finite numbers.) We say that the function $y(t)$ is obtained from $x(t)$ by the $C(\alpha_{m},\beta_{m})$ -transformation if

		$\displaystyle y(t)=x(t),$	$\displaystyle\quad 0\leq t<\alpha_{1},$
		$\displaystyle y(d_{m}+t)=x(\beta_{m}+t),$	$\displaystyle\quad 0\leq t<\alpha_{m+1}-\beta_{m},$

where $d_{1}:=\alpha_{1}$ and $d_{m+1}:=d_{m}+(\alpha_{m+1}-\beta_{m})$ . Intuitively speaking, the $C(\alpha_{m},\beta_{m})$ -transformation discards the path of $x(t)$ corresponding to the interval $[\alpha_{m},\beta_{m})$ , keeps the segment $[0,\alpha_{1})$ unchanged, and shifts the remaining parts to the left, connecting them in the original order without intersection, thereby obtaining a new right-continuous path $y(t)$ .

Fix $n\in\mathbb{N}$ . Define $\eta:=\inf\{t>0:Y_{t-}=0\}$ and $\sigma^{(n)}:=\inf\{t>0:Y_{t}\geq\hat{c}_{n}\}\wedge\zeta$ ( $\inf\emptyset:=\infty$ ). We then define a sequence of stopping times as follows:

\eta^{(n)}_{1}:=\eta,\quad\sigma^{(n)}_{1}:=\inf\{t\geq\eta_{1}^{(n)}:Y(t)\geq\hat{c}_{n}\}\wedge\zeta,

and if $\eta^{(n)}_{m-1},\sigma^{(n)}_{m-1}$ are already defined, we set

\eta^{(n)}_{m}:=\inf\{t\geq\sigma^{(n)}_{m-1}:Y_{t-}=0\}\wedge\zeta

and

\sigma^{(n)}_{m}:=\inf\{t\geq\eta^{(n)}_{m}:Y(t)\geq\hat{c}_{n}\}\wedge\zeta.

Note that if $\eta^{(n)}_{m}<\zeta$ , then $Y_{\eta^{(n)}_{m}}=0$ due to the quasi-left-continuity of $Y$ . Particularly, $\eta=\tau_{0}=\inf\{t>0:Y_{t}=0\}$ . We define $\eta$ using the left limit of $Y$ instead of directly using $\tau_{0}$ because, although our focus is on Feller’s Brownian motion, the discussion in this section also applies to Doob’s Brownian motion. For Doob’s Brownian motion, defining the ‘return’ time to $0$ requires the use of the left limit.

For $n\in\mathbb{N}$ and every $\omega\in\Omega$ , by performing the $C(\eta^{(n)}_{m}(\omega),\sigma^{(n)}_{m}(\omega))$ -transformation on $Y_{t}(\omega)$ , we obtain a new path, denoted by $Y^{(n)}_{t}(\omega)$ .

Lemma 7.1.

The process

Y^{(n)}:=\left(\Omega,{\mathscr{G}},Y^{(n)}_{t},(\mathbf{P}_{x})_{x\in(0,\infty)}\right)

is a Doob’s Brownian motion with instantaneous distribution

\lambda^{(n)}(\cdot)=\mathbf{P}_{0}(Y_{\sigma^{(n)}}\in\cdot)

supported on $[\hat{c}_{n},\infty)\cup\{\partial\}$ .

Proof.

Fix $x\in(0,\infty)$ and $\alpha>0$ . We aim to compute the resolvent of $Y^{(n)}$ for a positive and bounded Borel measurable function $f$ on $(0,\infty)$ :

G^{(n)}_{\alpha}f(x)=\mathbf{E}_{x}\int_{0}^{\infty}e^{-\alpha t}f(Y^{(n)}_{t})dt.

Let $\zeta_{1}:=\eta^{(n)}_{1}=\eta$ and $\zeta_{m}:=\eta^{(n)}_{m}-\sigma^{(n)}_{m-1}$ for $m\geq 2$ . By the definition of $Y^{(n)}_{t}$ , we have

G^{(n)}_{\alpha}f(x)=\mathbf{E}_{x}\int_{0}^{\zeta_{1}}e^{-\alpha t}f(Y_{t})dt+\sum_{m\geq 1}\mathbf{E}_{x}\int_{\sum_{i=1}^{m}\zeta_{i}}^{\sum_{i=1}^{m+1}\zeta_{i}}e^{-\alpha t}f(Y_{t-\sum_{i=1}^{m}\zeta_{i}+\sigma^{(n)}_{m}})dt.

(7.1)

Set

	$\displaystyle J_{m}$	$\displaystyle:=\mathbf{E}_{x}\int_{\sum_{i=1}^{m}\zeta_{i}}^{\sum_{i=1}^{m+1}\zeta_{i}}e^{-\alpha t}f(Y_{t-\sum_{i=1}^{m}\zeta_{i}+\sigma^{(n)}_{m}})dt$
		$\displaystyle=\mathbf{E}_{x}e^{-\alpha\sum_{i=1}^{m}\zeta_{i}}\int_{0}^{\zeta_{m+1}}e^{-\alpha t}f(Y_{t+\sigma^{(n)}_{m}})dt.$

Note that $\zeta_{m+1}=\eta\circ\theta_{\sigma^{(n)}_{m}}$ . It follows from the strong Markov property that

	$\displaystyle J_{m}$	$\displaystyle=\mathbf{E}_{x}\left(e^{-\alpha\sum_{i=1}^{m}\zeta_{i}}\mathbf{E}_{Y_{\sigma^{(n)}_{m}}}\int_{0}^{\eta}e^{-\alpha t}f(Y_{t})dt\right)$
		$\displaystyle=\mathbf{E}_{x}\left(e^{-\alpha\sum_{i=1}^{m}\zeta_{i}}G^{0}_{\alpha}f(Y_{\sigma^{(n)}_{m}})\right),$

where $G^{0}_{\alpha}$ is the resolvent of the absorbing Brownian motion.

Let us prove that

J_{m}=\mathbf{E}_{x}e^{-\alpha\eta}\cdot\left(\mathbf{E}_{\lambda^{(n)}}e^{-\alpha\eta}\right)^{m-1}\cdot\int_{0}^{\infty}G^{0}_{\alpha}f(x)\lambda^{(n)}(dx),\quad m\geq 1.

(7.2)

For $m=1$ , we have $Y_{\sigma^{(n)}_{1}}=Y_{\sigma^{(n)}}\circ\theta_{\eta}$ , and thus, by the strong Markov property of $Y$ ,

	$\displaystyle J_{1}$	$\displaystyle=\mathbf{E}_{x}\left(e^{-\alpha\eta}G^{0}_{\alpha}(Y_{\sigma^{(n)}}\circ\theta_{\eta})\right)=\mathbf{E}_{x}\left(e^{-\alpha\eta}\mathbf{E}_{Y_{\eta}}\left(G^{0}_{\alpha}\left(Y_{\sigma^{(n)}}\right)\right)\right)$
		$\displaystyle=\mathbf{E}_{x}e^{-\alpha\eta}\cdot\mathbf{E}_{0}\left(G^{0}_{\alpha}\left(Y_{\sigma^{(n)}}\right)\right)=\mathbf{E}_{x}e^{-\alpha\eta}\cdot\int_{0}^{\infty}G^{0}_{\alpha}f(x)\lambda^{(n)}(dx).$

In general, note that $Y_{\sigma^{(n)}_{i}}=Y_{\sigma^{(n)}}\circ\theta_{\eta^{(n)}_{i}}$ and $\zeta_{i}=\eta^{(n)}_{i}-\sigma^{(n)}_{i-1}=\eta\circ\theta_{\sigma^{(n)}_{i-1}}$ for all $i\geq 2$ . Then the case $m=2$ can be deduced by the strong Markov property as follows:

	$\displaystyle J_{2}$	$\displaystyle=\mathbf{E}_{x}\left(e^{-\alpha(\zeta_{1}+\zeta_{2})}\mathbf{E}_{Y_{\eta^{(n)}_{2}}}\left(G^{0}_{\alpha}\left(Y_{\sigma^{(n)}}\right)\right)\right)$
		$\displaystyle=\mathbf{E}_{x}e^{-\alpha(\eta+\eta\circ\theta_{\sigma^{(n)}_{1}})}\cdot\int_{0}^{\infty}G^{0}_{\alpha}f(x)\lambda^{(n)}(dx)$
		$\displaystyle=\mathbf{E}_{x}\left(e^{-\alpha\eta}\mathbf{E}_{Y_{\sigma^{(n)}_{1}}}\left(e^{-\alpha\eta}\right)\right)\cdot\int_{0}^{\infty}G^{0}_{\alpha}f(x)\lambda^{(n)}(dx).$

Using $Y_{\sigma^{(n)}_{1}}=Y_{\sigma^{(n)}}\circ\theta_{\eta}$ and $Y_{\eta}=0$ , we obtain that

	$\displaystyle J_{2}$	$\displaystyle=\mathbf{E}_{x}\left(e^{-\alpha\eta}\mathbf{E}_{Y_{\sigma^{(n)}}\circ\theta_{\eta}}\left(e^{-\alpha\eta}\right)\right)\cdot\int_{0}^{\infty}G^{0}_{\alpha}f(x)\lambda^{(n)}(dx)$
		$\displaystyle=\mathbf{E}_{x}e^{-\alpha\eta}\cdot\mathbf{E}_{0}\left(\mathbf{E}_{Y_{\sigma^{(n)}}}e^{-\alpha\eta}\right)\cdot\int_{0}^{\infty}G^{0}_{\alpha}f(x)\lambda^{(n)}(dx)$
		$\displaystyle=\mathbf{E}_{x}e^{-\alpha\eta}\cdot\mathbf{E}_{\lambda^{(n)}}\left(e^{-\alpha\eta}\right)\cdot\int_{0}^{\infty}G^{0}_{\alpha}f(x)\lambda^{(n)}(dx).$

The general case can be formulated similarly.

Finally, by substituting (7.2) into (7.1), we conclude that

G^{(n)}_{\alpha}f(x)=G_{\alpha}^{0}f(x)+\mathbf{E}_{x}e^{-\alpha\eta}\frac{\int_{0}^{\infty}G^{0}_{\alpha}f(x)\lambda^{(n)}(dx)}{1-\mathbf{E}_{\lambda^{(n)}}\left(e^{-\alpha\eta}\right)}.

(7.3)

This result corresponds precisely to the resolvent of the Doob’s Brownian motion with instantaneous distribution $\lambda^{(n)}$ . ∎

The sequence of Doob’s Brownian motions $Y^{(n)}$ approximates $Y$ in the case where $p_{3}=0$ in the following sense.

Theorem 7.2.

Assume that $p_{3}=0$ . Then, for any $x\in(0,\infty)$ ,

\mathbf{P}_{x}(\lim_{n\rightarrow\infty}Y^{(n)}_{t}=Y_{t})=1,\quad\forall t\geq 0.

(7.4)

Proof.

A crucial consequence of the assumption $p_{3}=0$ is that

\left|\{t\geq 0:Y_{t}=0\}\right|=0,\quad\mathbf{P}_{x}\text{-a.s.},

where $|\cdot|$ denotes the Lebesgue measure of the time set; see, e.g., [10, §14]. This can also be observed from the pathwise representation (3.23), as $Y_{t}=0$ always indicates $W^{+}_{t}=0$ . Hence, in principle, we may repeat the proof of Theorem A.1 step by step to complete the proof. Below, we provide a concise proof using the pathwise representation of $Y_{t}$ .

Fix $t\geq 0$ . Let $\rho^{(n)}_{t}(\omega)$ denote the total duration discarded from the path of $Y_{\cdot}(\omega)$ when constructing $Y^{(n)}_{s}$ for $0\leq s\leq t$ . Specifically, we have $Y^{(n)}_{t}(\omega)=Y_{{t+\rho^{(n)}_{t}}(\omega)}(\omega)$ . It suffices to show

\mathbf{P}_{x}(\lim_{n\rightarrow\infty}\rho^{(n)}_{t}=0)=1.

(7.5)

Let us first consider the special case where $Y$ is the reflecting Brownian motion $W^{+}$ . To differentiate this specific case from the general one, we denote $\eta,\sigma^{(n)},\eta^{(n)}_{m},\sigma^{(n)}_{m},\rho^{(n)}_{t}$ by $\tilde{\eta},\tilde{\sigma}^{(n)},\tilde{\eta}^{(n)}_{m},\tilde{\sigma}^{(n)}_{m},\tilde{\rho}^{(n)}_{t}$ , respectively. Clearly, the instantaneous distribution of the approximating Doob’s Brownian motion, denoted by $W^{+,(n)}$ , is $\delta_{\hat{c}_{n}}$ . Since $W^{+,(n)}$ is conservative by the expression of its resolvent in (3.1), we have $\tilde{\rho}^{(n)}_{t}<\infty$ for all $n$ . Note that $\tilde{\rho}^{(n)}_{t}$ is decreasing as $n\rightarrow\infty$ . It follows that

	$\displaystyle\tilde{\rho}^{(n)}_{t}(\omega)$	$\displaystyle\leq\left\|\left\{0\leq s\leq t+\tilde{\rho}^{(n)}_{t}(\omega):W^{+}_{s}<\hat{c}_{n}\right\}\right\|$
		$\displaystyle\leq\left\|\left\{0\leq s\leq t+\tilde{\rho}^{(0)}_{t}(\omega):W^{+}_{s}<\hat{c}_{n}\right\}\right\|,$

which implies that

	$\displaystyle\lim_{n\rightarrow\infty}\tilde{\rho}^{(n)}_{t}(\omega)$	$\displaystyle\leq\lim_{n\rightarrow\infty}\left\|\left\{0\leq s\leq t+\tilde{\rho}^{(0)}_{t}(\omega):W^{+}_{s}<\hat{c}_{n}\right\}\right\|$		(7.6)
		$\displaystyle=\left\|\left\{0\leq s\leq t+\tilde{\rho}^{(0)}_{t}(\omega):W^{+}_{s}=0\right\}\right\|=0.$		(7.6)

Now, consider a general Feller’s Brownian motion $Y$ . According to its pathwise representation (3.23) (with lifetime $\zeta$ given in (3.27)), $Y_{t}=0$ or $Y_{t-}=0$ always implies $W^{+}_{t}=0$ . Since $Y_{t}\geq W^{+}_{t}$ , it follows that $\tilde{\sigma}^{(n)}\geq\sigma^{(n)}$ . (Recall that the notation with tilde is defined for $W^{+}$ .) Consequently, each discarding interval $[\eta^{(n)}_{m},\sigma^{(n)}_{m})$ for $Y^{(n)}$ must be contained within some discarding interval for $W^{+,(n)}$ (though the converse is not necessarily true). Particularly, $\rho^{(n)}_{t}\leq\tilde{\rho}^{(n)}_{t}$ . Therefore, (7.5) follows from (7.6). ∎

Remark 7.3.

If we do not assume that $p_{3}=0$ , then the sequence of Doob’s Brownian motions $Y^{(n)}$ will converge to a Feller’s Brownian motion $\tilde{Y}$ with parameters $(p_{1},p_{2},0,p_{4})$ in the sense of (7.4). This is because, according to Lemma 8.2, $Y$ can be obtained from $\tilde{Y}$ , with lifetime $\tilde{\zeta}$ , through a time change transformation corresponding to a strictly increasing PCAF. Therefore, they share the same sequence of distributions $\lambda^{(n)}$ , that is, $\mathbf{P}_{0}(Y_{\sigma^{(n)}}\in\cdot)=\mathbf{P}_{0}(\tilde{Y}_{\tilde{\sigma}^{(n)}}\in\cdot)$ , where $\tilde{\sigma}^{(n)}:=\inf\{t>0:\tilde{Y}_{t}\geq\hat{c}_{n}\}\wedge\tilde{\zeta}$ .

7.2. Representation of instantaneous distributions

The following lemma describes the relationship between the instantaneous distributions.

Lemma 7.4.

For every $m,n\in\mathbb{N}$ with $m>n$ , the following relationship holds:

\lambda^{(n)}=\frac{\frac{\int_{[\hat{c}_{m},\hat{c}_{n})}x\lambda^{(m)}(dx)}{\hat{c}_{n}}\delta_{\hat{c}_{n}}+\lambda^{(m)}|_{[\hat{c}_{n},\infty)\cup\{\partial\}}}{1-\int_{[\hat{c}_{m},\hat{c}_{n})}\left(1-\frac{x}{\hat{c}_{n}}\right)\lambda^{(m)}(dx)}.

(7.7)

Proof.

Take a positive and bounded Borel measurable function $f$ on $(0,\infty)$ . Note that $Y_{\sigma^{(n)}}=Y_{\sigma^{(n)}}\circ\theta_{\sigma^{(m)}}$ , since $\sigma^{(n)}\geq\sigma^{(m)}$ . Using the strong Markov property of $Y$ , we have

\displaystyle\lambda^{(n)}(f)=\mathbf{E}_{0}f(Y_{\sigma^{(n)}}\circ\theta_{\sigma^{(m)}})=\mathbf{E}_{0}\left(\mathbf{E}_{Y_{\sigma^{(m)}}}f(Y_{\sigma^{(n)}})\right).

(7.8)

For $y\geq\hat{c}_{n}$ , it holds $\mathbf{P}_{y}$ -a.s. that $Y_{\sigma^{(n)}}=Y_{0}=y$ . Thus,

	$\displaystyle\mathbf{E}_{0}\left(\mathbf{E}_{Y_{\sigma^{(m)}}}f(Y_{\sigma^{(n)}});Y_{\sigma^{(m)}}\geq\hat{c}_{n}\right)$	$\displaystyle=\mathbf{E}_{0}\left(f(Y_{\sigma^{(m)}});Y_{\sigma^{(m)}}\geq\hat{c}_{n}\right)$		(7.9)
		$\displaystyle=\int_{[\hat{c}_{n},\infty)}f(x)\lambda^{(m)}(dx).$		(7.9)

For $\hat{c}_{m}\leq y<\hat{c}_{n}$ , we have

\mathbf{E}_{y}f(Y_{\sigma^{(n)}})=\mathbf{E}_{y}\left(f(Y_{\sigma^{(n)}});\sigma^{(n)}<\eta\right)+\mathbf{E}_{y}\left(f(Y_{\sigma^{(n)}});\sigma^{(n)}>\eta\right).

(7.10)

Since $Y_{\sigma^{(n)}}=\hat{c}_{n}$ on $\{\sigma^{(n)}<\eta\}$ , it follows that

\mathbf{E}_{y}\left(f(Y_{\sigma^{(n)}});\sigma^{(n)}<\eta\right)=f(\hat{c}_{n})\cdot\frac{y}{\hat{c}_{n}}.

(7.11)

Using $Y_{\sigma^{(n)}}=Y_{\sigma^{(n)}}\circ\theta_{\eta}$ on $\{\sigma^{(n)}>\eta\}$ and applying the strong Markov property of $Y$ , we get

$\displaystyle\mathbf{E}_{y}\left(f(Y_{\sigma^{(n)}});\sigma^{(n)}>\eta\right)$	$\displaystyle=\mathbf{E}_{y}\left(\mathbf{E}_{Y_{\eta}}f(Y_{\sigma^{(n)}});\sigma^{(n)}>\eta\right)$	(7.12)
	$\displaystyle=\mathbf{E}_{0}f(Y_{\sigma^{(n)}})\cdot\mathbf{E}_{y}(\sigma^{(n)}>\eta)$
	$\displaystyle=\lambda^{(n)}(f)\cdot\left(1-\frac{y}{\hat{c}_{n}}\right).$

Substituting (7.11) and (7.12) into (7.10) yields

\mathbf{E}_{y}f(Y_{\sigma^{(n)}})=f(\hat{c}_{n})\cdot\frac{y}{\hat{c}_{n}}+\lambda^{(n)}(f)\cdot\left(1-\frac{y}{\hat{c}_{n}}\right),\quad\forall\hat{c}_{m}\leq y<\hat{c}_{n}.

Therefore,

	$\displaystyle\mathbf{E}_{0}$	$\displaystyle\left(\mathbf{E}_{Y_{\sigma^{(m)}}}f(Y_{\sigma^{(n)}});\hat{c}_{m}\leq Y_{\sigma^{(m)}}<\hat{c}_{n}\right)$
		$\displaystyle=\mathbf{E}_{0}\left(f(\hat{c}_{n})\cdot\frac{Y_{\sigma^{(m)}}}{\hat{c}_{n}}+\lambda^{(n)}(f)\cdot\left(1-\frac{Y_{\sigma^{(m)}}}{\hat{c}_{n}}\right);\hat{c}_{m}\leq Y_{\sigma^{(m)}}<\hat{c}_{n}\right)$
		$\displaystyle=\int_{[\hat{c}_{m},\hat{c}_{n})}\left(f(\hat{c}_{n})\cdot\frac{y}{\hat{c}_{n}}+\lambda^{(n)}(f)\cdot\left(1-\frac{y}{\hat{c}_{n}}\right)\right)\lambda^{(m)}(dy).$

Combining this with (7.8) and (7.9), we can verify (7.7). ∎

Based on this lemma, we can define a measure $\lambda$ on $(0,\infty)\cup\{\partial\}$ as follows:

\lambda|_{(\hat{c}_{n},\infty)\cup\{\partial\}}:=\frac{\lambda^{(n)}|_{(\hat{c}_{n},\infty)\cup\{\partial\}}}{\Lambda_{n}},\quad n\in\mathbb{N},

(7.13)

where $\Lambda_{n}:=1-\int_{[\hat{c}_{n},\hat{c}_{0})}\left(1-\frac{x}{\hat{c}_{0}}\right)\lambda^{(n)}(dx)$ . Clearly, $\lambda$ is a well-defined $\sigma$ -finite measure on $(0,\infty)\cup\{\partial\}$ . The following lemma provides the representation of $\lambda^{(n)}$ in terms of this measure $\lambda$ .

Lemma 7.5.

For each $n\in\mathbb{N}$ , the following holds:

\frac{1}{\Lambda_{n}}=\lambda\left((\hat{c}_{n},\infty)\cup\{\partial\}\right)+\frac{\hat{c}_{0}\left(1-\lambda\left((\hat{c}_{0},\infty)\cup\{\partial\}\right)\right)-\int_{(\hat{c}_{n},\hat{c}_{0}]}x\lambda(dx)}{\hat{c}_{n}},

(7.14)

and

		$\displaystyle\lambda^{(n)}\|_{(\hat{c}_{n},\infty)\cup\{\partial\}}=\Lambda_{n}\cdot\lambda\|_{(\hat{c}_{n},\infty)\cup\{\partial\}}$		(7.15)
		$\displaystyle\lambda^{(n)}(\{\hat{c}_{n}\})=\frac{\Lambda_{n}}{\hat{c}_{n}}\left(\hat{c}_{0}\left(1-\lambda\left((\hat{c}_{0},\infty)\cup\{\partial\}\right)\right)-\int_{(\hat{c}_{n},\hat{c}_{0}]}x\lambda(dx)\right).$		(7.15)

Particularly,

\int_{(0,\infty)}\left(1\wedge x\right)\lambda(dx)<\infty.

(7.16)

Proof.

According to Lemma 7.4, we have

\lambda^{(n)}(\{\hat{c}_{n}\})=\frac{\Lambda_{n}}{\Lambda_{n+1}}\cdot\left(\int_{[\hat{c}_{n+1},\hat{c}_{n})}\frac{x}{\hat{c}_{n}}\lambda^{(n+1)}(dx)+\lambda^{(n+1)}(\{\hat{c}_{n}\})\right).

It follows that

	$\displaystyle\frac{\hat{c}_{n}\lambda^{(n)}(\{\hat{c}_{n}\})}{\Lambda_{n}}$	$\displaystyle=\frac{\hat{c}_{n+1}\lambda^{(n+1)}(\{\hat{c}_{n+1}\})}{\Lambda_{n+1}}+\frac{\int_{(\hat{c}_{n+1},\hat{c}_{n})}x\lambda^{(n+1)}(dx)+\hat{c}_{n}\lambda^{(n+1)}(\{\hat{c}_{n}\})}{\Lambda_{n+1}}$
		$\displaystyle=\frac{\hat{c}_{n+1}\lambda^{(n+1)}(\{\hat{c}_{n+1}\})}{\Lambda_{n+1}}+\int_{(\hat{c}_{n+1},\hat{c}_{n}]}x\lambda(dx).$

Let $h_{n}:=\frac{\hat{c}_{n}\lambda^{(n)}(\{\hat{c}_{n}\})}{\Lambda_{n}}$ . Then $h_{n}$ is decreasing in $n$ , and

h_{0}-\lim_{n\rightarrow\infty}h_{n}=\sum_{n=0}(h_{n}-h_{n+1})=\sum_{n=0}^{\infty}\int_{(\hat{c}_{n+1},\hat{c}_{n}]}x\lambda(dx)=\int_{(0,\hat{c}_{0}]}x\lambda(dx).

Particularly, (7.16) holds true. The induction also yields that

h_{n}=h_{n+m}+\int_{(\hat{c}_{n+m},\hat{c}_{n}]}x\lambda(dx),\quad\forall m\geq 1,

and by letting $m\rightarrow\infty$ , we obtain

h_{n}=\lim_{m\rightarrow\infty}h_{m}+\int_{(0,\hat{c}_{n}]}x\lambda(dx)=h_{0}-\int_{(\hat{c}_{n},\hat{c}_{0}]}x\lambda(dx).

(7.17)

Therefore, the first identity in (7.15) is established. The second identity in (7.15) follows directly from (7.13). Finally, substituting (7.15) into the definition of $\Lambda_{n}$ , we obtain (7.14). ∎

7.3. Parameters of Feller’s Brownian motion

The aim of this subsection is to express the parameters of the original Feller’s Brownian motion in terms of the measure $\lambda$ , which is defined based on the approximating Doob’s Brownian motions.

Theorem 7.6.

Let $Y$ be a Feller’s Brownian motion with parameters $(p_{1},p_{2},0,p_{4})$ , and let $\lambda$ be the measure (7.13) on $(0,\infty)\cup\{\partial\}$ in terms of the approximating Doob’s Brownian motions $Y^{(n)}$ . Then, up to a certain multiplicative constant needed to satisfy (3.3), it holds that

p_{1}=\lambda(\{\partial\}),\quad p_{2}=\hat{c}_{0}\left(1-\lambda\left(\{\partial\}\right)\right)-\int_{(0,\infty)}\left(x\wedge\hat{c}_{0}\right)\lambda(dx),\quad p_{4}=\lambda|_{(0,\infty)}.

(7.18)

Proof.

Define $\tilde{p}^{n}_{2}:=h_{n}=\hat{c}_{0}\lambda^{(0)}(\{\hat{c}_{0}\})-\int_{(\hat{c}_{n},\hat{c}_{0}]}x\lambda(dx)$ for $n\in\mathbb{N}$ (see (7.17)). This sequence is decreasing and converges to

\tilde{p}_{2}:=\hat{c}_{0}\left(1-\lambda\left(\{\partial\}\right)\right)-\int_{(0,\infty)}\left(x\wedge\hat{c}_{0}\right)\lambda(dx).

(7.19)

According to (7.3) and Lemma 7.5, the resolvent of $Y^{(n)}$ can be expressed as

	$\displaystyle G^{(n)}_{\alpha}$	$\displaystyle f(x)$
		$\displaystyle=G^{0}_{\alpha}f(x)+\mathbf{E}_{x}e^{-\alpha\eta}\cdot\frac{\int_{(\hat{c}_{n},\infty)}G^{0}_{\alpha}f(x)\lambda(dx)+\tilde{p}^{n}_{2}\cdot\frac{G^{0}_{\alpha}f(\hat{c}_{n})}{\hat{c}_{n}}}{\frac{1}{\Lambda_{n}}-\int_{(\hat{c}_{n},\infty)}e^{-\sqrt{2\alpha}x}\lambda(dx)-\tilde{p}^{n}_{2}\cdot\frac{e^{-\sqrt{2\alpha}\hat{c}_{n}}}{\hat{c}_{n}}}$
		$\displaystyle=G^{0}_{\alpha}f(x)+\mathbf{E}_{x}e^{-\alpha\eta}\cdot\frac{\int_{(\hat{c}_{n},\infty)}G^{0}_{\alpha}f(x)\lambda(dx)+\tilde{p}^{n}_{2}\cdot\frac{G^{0}_{\alpha}f(\hat{c}_{n})}{\hat{c}_{n}}}{\lambda(\{\partial\})+\int_{(\hat{c}_{n},\infty)}(1-e^{-\sqrt{2\alpha}x})\lambda(dx)+\tilde{p}^{n}_{2}\cdot\frac{1-e^{-\sqrt{2\alpha}\hat{c}_{n}}}{\hat{c}_{n}}}$

for all $f\in C_{c}((0,\infty))$ and $x\in(0,\infty)$ . Note that

\lim_{n\rightarrow\infty}\frac{G^{0}_{\alpha}f(\hat{c}_{n})}{\hat{c}_{n}}=\lim_{x\rightarrow 0}\frac{G^{0}_{\alpha}f(x)}{x}=2\int_{(0,\infty)}e^{-\sqrt{2\alpha}x}f(x)dx

by virtue of the expression of $G^{0}_{\alpha}$ provided in the first paragraph of the proof of Corollary 3.5. Additionally,

\lim_{n\rightarrow\infty}\frac{1-e^{-\sqrt{2\alpha}\hat{c}_{n}}}{\hat{c}_{n}}=\sqrt{2\alpha}.

Given (7.16), we have

	$\displaystyle\lim_{n\rightarrow\infty}$	$\displaystyle G^{(n)}_{\alpha}f(x)$
		$\displaystyle=G^{0}_{\alpha}f(x)+\mathbf{E}_{x}e^{-\alpha\eta}\cdot\frac{\int_{(0,\infty)}G^{0}_{\alpha}f(x)\lambda(dx)+2\tilde{p}_{2}\cdot\int_{(0,\infty)}e^{-\sqrt{2\alpha}x}f(x)dx}{\lambda(\{\partial\})+\int_{(0,\infty)}(1-e^{-\sqrt{2\alpha}x})\lambda(dx)+\tilde{p}_{2}\cdot\sqrt{2\alpha}}.$

By Theorem 7.2, this limit is exactly the resolvent of $Y$ . Comparing this expression with (3.15), we can eventually derive (7.18). ∎

8. Proof of Theorem 5.1 for $|p_{4}|=\infty$

8.1. No sojourn case $p_{3}=0$

In this subsection, we consider the case where $|p_{4}|=\infty$ and further assume that $p_{3}=0$ . According to Theorem 7.2, we can construct a sequence of Doob’s Brownian motions $Y^{(n)}$ that approximates the Feller’s Brownian motion $Y$ . Their instantaneous distributions $\lambda^{(n)}$ are characterized in Lemma 7.5 in terms of the measure $\lambda$ defined by (7.13).

Regarding the time-changed Feller’s Brownian motion $\hat{X}:=\check{Y}$ , we can also construct a sequence of Doob processes $\hat{X}^{(n)}$ with instantaneous distribution $\hat{\lambda}^{(n)}$ , as described in Appendix A, that approximates $\hat{X}$ .

Lemma 8.1.

For all $n\in\mathbb{N}$ , the following holds:

		$\displaystyle\hat{\lambda}^{(n)}(\{\hat{c}_{0}\})=\Lambda_{n}\cdot\left(\int_{(\hat{c}_{1},\hat{c}_{0}]}\frac{x-\hat{c}_{1}}{\hat{c}_{0}-\hat{c}_{1}}\lambda(dx)+\lambda\left((\hat{c}_{0},\infty)\right)\right),$
		$\displaystyle\hat{\lambda}^{(n)}(\{\hat{c}_{i}\})=\Lambda_{n}\cdot\left(\int_{(\hat{c}_{i+1},\hat{c}_{i}]}\frac{x-\hat{c}_{i+1}}{\hat{c}_{i}-\hat{c}_{i+1}}\lambda(dx)+\int_{(\hat{c}_{i},\hat{c}_{i-1})}\frac{\hat{c}_{i-1}-x}{\hat{c}_{i-1}-\hat{c}_{i}}\lambda(dx)\right)$

for $1\leq i\leq n-1$ , and

\hat{\lambda}^{(n)}(\{\hat{c}_{n}\})=\Lambda_{n}\cdot\left(\frac{\tilde{p}^{n}_{2}}{\hat{c}_{n}}+\int_{(\hat{c}_{n},\hat{c}_{n-1})}\frac{\hat{c}_{n-1}-x}{\hat{c}_{n-1}-\hat{c}_{n}}\lambda(dx)\right),

where $\tilde{p}^{n}_{2}=\hat{c}_{0}\lambda^{(0)}(\{\hat{c}_{0}\})-\int_{(\hat{c}_{n},\hat{c}_{0}]}x\lambda(dx)$ .

Proof.

We consider $1\leq i\leq n-1$ and address the other two cases in a similar manner. Note that

\{\hat{X}_{\hat{\sigma}^{(n)}}=\hat{c}_{i}\}=\{Y_{\sigma^{(n)}}\in(\hat{c}_{i+1},\hat{c}_{i-1}),\tau_{\hat{c}_{i}}\circ\theta_{\sigma^{(n)}}<\left(\tau_{\hat{c}_{i+1}}\wedge\tau_{\hat{c}_{i-1}}\right)\circ\theta_{\sigma^{(n)}}\},

where $\tau_{a}:=\inf\{t>0:Y_{t}=a\}$ for $a\in(0,\infty)$ . It follows from the strong Markov property of $Y$ that

	$\displaystyle\hat{\lambda}^{(n)}(\{\hat{c}_{i}\})$	$\displaystyle=\mathbf{P}_{0}\left(\mathbf{P}_{Y_{\sigma^{(n)}}}(\tau_{\hat{c}_{i}}<\tau_{\hat{c}_{i+1}}\wedge\tau_{\hat{c}_{i-1}});Y_{\sigma^{(n)}}\in(\hat{c}_{i+1},\hat{c}_{i-1})\right)$
		$\displaystyle=\int_{(\hat{c}_{i+1},\hat{c}_{i}]}\frac{x-\hat{c}_{i+1}}{\hat{c}_{i}-\hat{c}_{i+1}}\lambda^{(n)}(dx)+\int_{(\hat{c}_{i},\hat{c}_{i-1})}\frac{\hat{c}_{i-1}-x}{\hat{c}_{i-1}-\hat{c}_{i}}\lambda^{(n)}(dx).$

Then applying Lemma 7.5, we can obtain the desired expression of $\lambda^{(n)}(\{\hat{c}_{i}\})$ . ∎

We are now prepared to prove Theorem 5.1 for the case where $p_{3}=0$ and $|p_{4}|=\infty$ .

Proof of Theorem 5.1 for $p_{3}=0,|p_{4}|=\infty$ .

Let

\tilde{\mathfrak{p}}_{0}:=\int_{(\hat{c}_{1},\hat{c}_{0}]}\frac{x-\hat{c}_{1}}{\hat{c}_{0}-\hat{c}_{1}}\lambda(dx)+\lambda\left((\hat{c}_{0},\infty)\right)

and

\tilde{\mathfrak{p}}_{n}:=\int_{(\hat{c}_{n+1},\hat{c}_{n}]}\frac{x-\hat{c}_{n+1}}{\hat{c}_{n}-\hat{c}_{n+1}}\lambda(dx)+\int_{(\hat{c}_{n},\hat{c}_{n-1})}\frac{\hat{c}_{n-1}-x}{\hat{c}_{n-1}-\hat{c}_{n}}\lambda(dx),\quad n\geq 1.

Substituting the expression of $\hat{\lambda}^{(n)}$ from Lemma 8.1 into (2.6), we can formulate the resolvent matrix $\hat{\Psi}^{(n)}_{ij}(\alpha):=\mathbf{E}_{\hat{c}_{i}}\int_{0}^{\infty}e^{-\alpha t}1_{\{\hat{c}_{j}\}}(\hat{X}^{(n)}_{t})dt$ of $\hat{X}^{(n)}$ for $\alpha>0$ and $i,j\in\mathbb{N}$ as follows:

		$\displaystyle\hat{\Psi}^{(n)}_{ij}(\alpha)-\Phi_{ij}(\alpha)$
	$\displaystyle=$	$\displaystyle u_{\alpha}(i)\frac{\sum_{k=0}^{n-1}\tilde{\mathfrak{p}}_{k}\Phi_{kj}(\alpha)+\left(\tilde{p}^{n}_{2}+\hat{c}_{n}\int_{(\hat{c}_{n},\hat{c}_{n-1})}\frac{\hat{c}_{n-1}-x}{\hat{c}_{n-1}-\hat{c}_{n}}\lambda(dx)\right)\frac{\Phi_{nj}(\alpha)}{\hat{c}_{n}}}{\frac{1}{\Lambda_{n}}-\sum_{k=0}^{n-1}\tilde{\mathfrak{p}}_{k}u_{\alpha}(k)-\left(\tilde{p}^{n}_{2}+\hat{c}_{n}\int_{(\hat{c}_{n},\hat{c}_{n-1})}\frac{\hat{c}_{n-1}-x}{\hat{c}_{n-1}-\hat{c}_{n}}\lambda(dx)\right)\frac{u_{\alpha}(n)}{\hat{c}_{n}}}$
	$\displaystyle=$	$\displaystyle u_{\alpha}(i)\frac{\sum_{k=0}^{n-1}\tilde{\mathfrak{p}}_{k}\Phi_{kj}(\alpha)+\left(\tilde{p}^{n}_{2}+\hat{c}_{n}\int_{(\hat{c}_{n},\hat{c}_{n-1})}\frac{\hat{c}_{n-1}-x}{\hat{c}_{n-1}-\hat{c}_{n}}\lambda(dx)\right)\frac{\Phi_{nj}(\alpha)}{\hat{c}_{n}}}{\lambda(\{\partial\})+\sum_{k=0}^{n-1}\tilde{\mathfrak{p}}_{k}(1-u_{\alpha}(k))+\left(\tilde{p}^{n}_{2}+\hat{c}_{n}\int_{(\hat{c}_{n},\hat{c}_{n-1})}\frac{\hat{c}_{n-1}-x}{\hat{c}_{n-1}-\hat{c}_{n}}\lambda(dx)\right)\frac{1-u_{\alpha}(n)}{\hat{c}_{n}}}.$

Note that

\hat{c}_{n}\int_{(\hat{c}_{n},\hat{c}_{n-1})}\frac{\hat{c}_{n-1}-x}{\hat{c}_{n-1}-\hat{c}_{n}}\lambda(dx)\leq\int_{(\hat{c}_{n},\hat{c}_{n-1})}x\lambda(dx)\rightarrow 0

as $n\rightarrow\infty$ , due to (7.16). In addition, (see, e.g., [26, §7.10, (3) and (9)])

\lim_{n\rightarrow\infty}\frac{\Phi_{nj}(\alpha)}{\hat{c}_{n}}=2u_{\alpha}(j)\mu_{j},\quad\lim_{n\rightarrow\infty}\frac{1-u_{\alpha}(n)}{\hat{c}_{n}}=2\alpha\sum_{k\in\mathbb{N}}\mu_{k}u_{\alpha}(k).

Therefore,

	$\displaystyle\lim_{n\rightarrow\infty}\hat{\Psi}^{(n)}_{ij}$	$\displaystyle(\alpha)=\Phi_{ij}(\alpha)$
		$\displaystyle+u_{\alpha}(i)\frac{\sum_{k=0}^{\infty}\tilde{\mathfrak{p}}_{k}\Phi_{kj}(\alpha)+2\tilde{p}_{2}u_{\alpha}(j)\mu_{j}}{\lambda(\{\partial\})+\sum_{k=0}^{\infty}\tilde{\mathfrak{p}}_{k}(1-u_{\alpha}(k))+2\alpha\tilde{p}_{2}\sum_{k\in\mathbb{N}}\mu_{k}u_{\alpha}(k)},$

where $\tilde{p}_{2}$ is defined as (7.19). According to Theorem A.1, this limit is precisely the resolvent matrix of $\hat{X}$ . Hence, the parameters of $\hat{X}$ are (up to a multiplicative constant):

\gamma=\lambda(\{\partial\}),\quad\beta=2\tilde{p}_{2},\quad\nu_{k}=\tilde{\mathfrak{p}}_{k},\;k\in\mathbb{N}.

Using Theorem 7.6, we can eventually obtain the desired conclusion. ∎

8.2. Sojourn case $p_{3}>0$

Let us consider the final case where $p_{3}>0$ and $|p_{4}|=\infty$ . The basic idea is to transform this case into one with $p_{3}=0$ . Recall that $Y^{1}$ , defined by (3.23), is a Feller’s Brownian motion with parameters $(0,p_{2},0,p_{4})$ and $\ell^{Y^{1}}$ , defined by (3.24), is its local time at $0$ . According to §3.2.3, the subprocess $Y^{3}$ of $Y^{1}$ , perturbed by the multiplicative functional

M^{3}_{t}:=e^{-p_{1}\ell^{Y^{1}}_{t}},\quad t\geq 0,

is a Feller’s Brownian motion with parameters $(p_{1},p_{2},0,p_{4})$ . Denote by $\zeta_{3}$ the lifetime of $Y^{3}$ . Mimicking the proof of Lemma 3.7, we can easily show that

\mathfrak{f}_{3}(t):=t\wedge\zeta_{3}+p_{3}\ell^{Y^{1}}_{t\wedge\zeta_{3}},\quad t\geq 0

is a PCAF of $Y^{3}$ with $\text{Supp}(\mathfrak{f}_{3})=[0,\infty)$ .

Lemma 8.2.

The time-changed process of $Y^{3}$ with respect to the PCAF $\mathfrak{f}_{3}$ is a Feller’s Brownian motion with parameters $(p_{1},p_{2},p_{3},p_{4})$ .

Proof.

Given $Y^{1}$ expressed as $Y^{1}=(\Omega^{1},{\mathscr{G}}^{1},Y^{1}_{t},(\mathbf{P}^{1}_{x})_{x\in[0,\infty)})$ , according to [1, III, Theorem 3.3], we can write $Y^{3}=(\Omega^{3},{\mathscr{G}}^{3},Y^{3}_{t},\mathbf{P}^{3}_{x})$ as follows:

		$\displaystyle\Omega^{3}=\Omega^{1}\times[0,\infty],\quad{\mathscr{G}}^{3}:={\mathscr{G}}^{1}\times\mathcal{B}([0,\infty]),$		(8.1)
		$\displaystyle Y^{3}_{t}(\omega,\kappa):=Y^{1}_{t}(\omega)\text{ for }t<\kappa,\text{ and }Y^{3}_{t}(\omega,\kappa):=\partial\text{ for }t\geq\kappa,$		(8.1)

and for $\Gamma\in{\mathscr{G}}^{3}$ with $\Gamma^{\omega}:=\{\kappa\in[0,\infty]:(\omega,\kappa)\in\Gamma\}$ for $\omega\in\Omega^{1}$ ,

\mathbf{P}^{3}_{x}(\Gamma):=\mathbf{E}^{1}_{x}\left(\alpha_{\omega}(\Gamma^{\omega})\right),

where $\alpha_{\omega}$ is a probability measure on $[0,\infty]$ such that $\alpha_{\omega}((t,\infty])=M^{3}_{t}(\omega)$ for all $t\geq 0$ . Note that $\zeta_{3}(\omega,\kappa)=\kappa$ .

Let $\mathfrak{f}^{-1}_{3}(t)$ denote the right-continuous inverse of $\mathfrak{f}_{3}(t)$ . This inverse is continuous and strictly increasing up to $\zeta_{3}$ . The time-changed process $Y^{4}=(\Omega^{4},{\mathscr{G}}^{4},Y^{4}_{t},\mathbf{P}^{4}_{x})$ of $Y^{3}$ with respect to $\mathfrak{f}_{3}$ can be expressed as

\Omega^{4}=\Omega^{3},\quad{\mathscr{G}}^{4}={\mathscr{G}}^{3},\quad\mathbf{P}^{4}_{x}=\mathbf{P}^{3}_{x},

and

Y^{4}_{t}:=Y^{3}_{\mathfrak{f}^{-1}_{3}(t)}\text{ for }t<\zeta_{4}:=\mathfrak{f}_{3}(\zeta_{3}),\text{ and }Y^{4}_{t}:=\partial\text{ for }t\geq\zeta_{4},

where $\zeta_{4}$ is the lifetime of $Y^{4}$ . By substituting the expression (8.1) of $Y^{3}$ into this expression of $Y^{4}$ , we find that for $(\omega,\kappa)\in\Omega^{1}\times[0,\infty]$ ,

Y^{4}_{t}(\omega,\kappa)=Y^{1}_{\mathfrak{f}^{-1}(t)}(\omega),\quad t<\zeta_{4}(\omega,\kappa)=\kappa+p_{3}\ell^{Y^{1}}_{\kappa}(\omega)

where $\mathfrak{f}^{-1}$ is the right-continuous inverse of $\mathfrak{f}(t)=t+p_{3}\ell^{Y^{1}}_{t}(\omega)$ ( $t\geq 0$ ). In other words, $Y^{4}$ is the killed process of $Y^{2}$ , defined in (3.25), at time $\zeta_{4}$ . Note that for all $t\geq 0$ ,

\alpha_{\omega}\left(\{\kappa\in[0,\infty]:\zeta_{4}(\omega,\kappa)\geq t\}\right))=\alpha_{\omega}\left(\{\kappa\in[0,\infty]:\kappa\geq\mathfrak{f}^{-1}(t)\}\right)=e^{-p_{1}\ell^{Y^{1}}_{\mathfrak{f}^{-1}(t)}(\omega)}.

Thus, $Y^{4}$ is the subprocess of $Y^{2}$ perturbed by the multiplicative functional in (3.26). According to the pathwise representation in §3.2.3, we conclude that $Y^{4}$ is a Feller’s Brownian motion with parameters $(p_{1},p_{2},p_{3},p_{4})$ . ∎

Now, we complete the proof of Theorem 5.1 as follows.

Proof of Theorem 5.1 for $p_{3}>0,|p_{4}|=\infty$ .

Let $Y^{3}=(\Omega^{3},Y^{3}_{t},\mathbf{P}^{3}_{x})$ , with lifetime $\zeta_{3}$ , be a Feller’s Brownian motion with parameters $(p_{1},p_{2},0,p_{4})$ . According to Lemma 8.2, the Feller’s Brownian motion $Y=(\Omega,Y_{t},\mathbf{P}_{x})$ with parameters $(p_{1},p_{2},p_{3},p_{4})$ can be represented as the time-changed process of $Y^{3}$ with respect to $\mathfrak{f}_{3}$ . The lifetime of $Y$ is $\zeta=\mathfrak{f}_{3}(\zeta_{3})$ .

Denote by $\check{Y}^{3}$ and $\check{Y}$ the time-changed process obtained in Theorem 4.2 for $Y^{3}$ and $Y$ , respectively. The parameters determining the resolvent matrix of $\check{Y}^{3}$ have been examined in §8.1. It suffices to show that $\check{Y}$ is identical in law to $\check{Y}^{3}$ . Let $L^{3,\hat{c}_{n}}_{t}$ denote the local time of $Y^{3}$ at $\hat{c}_{n}$ as defined in Definition 3.1, and set $A^{3}_{t}:=\sum_{n\in\mathbb{N}}\mu_{n}L^{3,\hat{c}_{n}}_{t}$ for all $t\geq 0$ . It is straightforward to verify that

L^{\hat{c}_{n}}_{t}:=L^{3,\hat{c}_{n}}_{\mathfrak{f}^{-1}_{3}(t)},\quad t\geq 0

is the local time of $Y$ at $\hat{c}_{n}$ in the sense of Definition 3.1. Thus, the PCAF of $Y$ given by (4.3) is $A_{t}=A^{3}_{\mathfrak{f}^{-1}_{3}(t)}$ , and the right-continuous inverse of $A$ is

\gamma_{t}=\inf\{s>0:A_{s}>t\}=\inf\{s>0:A^{3}_{\mathfrak{f}^{-1}_{3}(s)}>t\},\quad\forall t<A_{\zeta}=A^{3}_{\zeta_{3}}.

Since $\mathfrak{f}_{3}$ is strictly increasing up to $\zeta_{3}$ , it follows that

\mathfrak{f}^{-1}_{3}(\gamma_{t})=\inf\{s>0:A^{3}_{s}>t\}=:\gamma^{3}_{t},\quad\forall t<A_{\zeta}=A^{3}_{\zeta_{3}},

where $\gamma^{3}_{t}$ is the right-continuous inverse of $A^{3}$ . Particularly, for $t<A_{\zeta}=A^{3}_{\zeta_{3}}$ ,

\check{Y}_{t}=Y_{\gamma_{t}}=Y^{3}_{\mathfrak{f}^{-1}_{3}(\gamma_{t})}=Y^{3}_{\gamma^{3}_{t}}=\check{Y}^{3}_{t}.

(8.2)

Note that $A_{\zeta}$ is the lifetime of $\check{Y}$ and $A^{3}_{\zeta_{3}}$ is the lifetime of $\check{Y}^{3}$ . Therefore, (8.2) establishes the identification between $\check{Y}$ and $\check{Y}^{3}$ . ∎

Appendix A Approximation of birth-death process

For a Feller $Q$ -process $X$ , we can also construct a sequence of Doob processes $X^{(n)}$ in the same manner as described in Lemma 7.1. This approach is precisely the probabilistic construction method for all $Q$ -processes presented in [26]. For the convenience of readers, we restate some necessary details as follows.

Let us consider the corresponding $Q$ -process $\hat{X}_{t}=\Xi^{-1}(X_{t})$ on $\overline{E}$ , with lifetime denoted by $\hat{\zeta}$ . Define

\hat{\eta}:=\inf\{t>0:\hat{X}_{t-}=0\}

and

\hat{\sigma}^{(n)}:=\inf\left\{t>0:\hat{X}_{t}\in\{\hat{c}_{0},\cdots,\hat{c}_{n}\}\right\}\wedge\hat{\zeta},\quad n\in\mathbb{N}.

Analogous sequences of stopping times $\{\hat{\eta}^{(n)}_{m}:m\geq 1\}$ and $\{\hat{\sigma}^{(n)}_{m}:m\geq 1\}$ can be defined by repeating the procedures used before Lemma 7.1 (see also [26, §6.3]). Applying the $C(\hat{\eta}^{(n)}_{m},\hat{\sigma}^{(n)}_{m})$ -transformation to $\hat{X}$ yields a sequence of Doob processes $\hat{X}^{(n)}$ on $E$ , with instantaneous distribution $\hat{\lambda}^{(n)}(\cdot)=\mathbf{P}_{0}(\hat{X}_{\hat{\sigma}^{(n)}}\in\cdot)$ .

The main result of [26, §6] establishes the convergence of $\hat{X}^{(n)}$ to $\hat{X}$ in the following sense.

Theorem A.1.

Let $X$ be a Feller $Q$ -process and let $X^{(n)}=\Xi(\hat{X}^{(n)})$ denote the approximating sequence of Doob processes. Then, for any $i\in\mathbb{N}$ ,

\mathbf{P}_{i}\left(\lim_{n\rightarrow\infty}X^{(n)}_{t}=X_{t}\right)=1,\quad\forall t\geq 0.

(A.1)

Proof.

The original proof in [26] is quite lengthy and only addresses the honest case. Here, we provide an alternative proof using the right continuity of $X$ . For convenience, we will consider $\hat{X}$ and $\hat{X}^{(n)}$ instead of $X$ and $X^{(n)}$ . Denote by $\hat{\zeta}^{(n)}$ the lifetime of $\hat{X}^{(n)}$ . According to the definition of the $C(\hat{\eta}^{(n)}_{m},\hat{\sigma}^{(n)}_{m})$ -transformation, it is straightforward to observe that

\hat{\zeta}^{(1)}\leq\hat{\zeta}^{(2)}\leq\cdots\leq\hat{\zeta}^{(n)}\leq\cdots\leq\hat{\zeta}.

Let $\hat{\zeta}_{\infty}:=\lim_{n\rightarrow\infty}\hat{\zeta}^{(n)}$ ( $\leq\hat{\zeta}$ ).

Firstly, we show that there exists an integer $N$ such that for any $n\geq N$ ,

\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(n)}_{m}<\infty\right)=1,\quad\forall m\geq 1,i\in\mathbb{N}.

(A.2)

It suffices to prove (A.2) for some integer $n=N$ , due to $\hat{\sigma}^{(n+1)}_{m}\leq\hat{\sigma}^{(n)}_{m}$ for all $n\in\mathbb{N}$ . To do this, note that we can find some $t\geq 0$ and $N\geq 1$ such that

\mathbf{P}_{0}(\hat{X}_{t}\geq\hat{c}_{N}\text{ or }\hat{X}_{t}=\partial)>0,

(A.3)

because otherwise

1=\mathbf{P}_{0}\left(\bigcap_{t\in\mathbb{Q}_{+},n\geq 1}\left\{\hat{X}_{t}<\hat{c}_{n}\right\}\right)=\mathbf{P}_{0}\left(\hat{X}_{t}=0,\forall t\geq 0\right),

where $\mathbb{Q}_{+}$ is the set of all positive rational numbers. To prove (A.2) for $n=N$ with $N$ in (A.3), consider the case $m=1$ . We note that $\hat{\sigma}^{(N)}_{1}=\hat{\sigma}^{(N)}\circ\theta_{\hat{\eta}}$ , and $\hat{\eta}<\infty$ , $\mathbf{P}_{\hat{c}_{i}}$ -a.s. It follows that

\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(N)}_{1}<\infty\right)=\mathbf{P}_{0}\left(\hat{\sigma}^{(N)}<\infty\right)\geq\mathbf{P}_{0}(\hat{X}_{t}\geq\hat{c}_{N}\text{ or }\hat{X}_{t}=\partial)>0.

Thus, there exists a constant $T>0$ such that

\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(N)}_{1}\geq T\right)=\mathbf{P}_{0}\left(\hat{\sigma}^{(N)}\geq T\right)=:\alpha<1.

For $k\geq 2$ , it follows from the strong Markov property that

	$\displaystyle\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(N)}_{1}\geq kT\right)$	$\displaystyle=\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(N)}_{1}\circ\theta_{(k-1)T}\geq T;\hat{\sigma}^{(N)}_{1}\geq(k-1)T\right)$
		$\displaystyle=\mathbf{P}_{\hat{c}_{i}}\left(\mathbf{P}_{\hat{X}_{(k-1)T}}\left(\hat{\sigma}^{(N)}_{1}\geq T\right);\hat{\sigma}^{(N)}_{1}\geq(k-1)T\right)$
		$\displaystyle\leq\alpha\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(N)}_{1}\geq(k-1)T\right).$

By induction, we get $\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(N)}_{1}\geq kT\right)\leq\alpha^{k}$ for all $k\geq 0$ . Hence,

\mathbf{E}_{\hat{c}_{i}}\hat{\sigma}^{(N)}_{1}\leq T\sum_{k\geq 0}\mathbf{P}_{\hat{c}_{i}}(\hat{\sigma}^{(N)}_{1}\geq kT)\leq T\sum_{k\geq 0}\alpha^{k}<\infty.

This indicates (A.2) for the case $m=1$ . For general $m\geq 2$ , (A.2) can also be established by using the strong Markov property. We provide details for the case $m=2$ , with the other cases handled by induction. In fact, we have

	$\displaystyle\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(N)}_{2}<\infty\right)$	$\displaystyle=\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(N)}_{2}<\infty;\hat{X}_{\hat{\sigma}_{1}^{(N)}}\in E\right)+\mathbf{P}_{\hat{c}_{i}}\left(\hat{\sigma}^{(N)}_{2}<\infty;\hat{X}_{\hat{\sigma}_{1}^{(N)}}=\partial\right)$
		$\displaystyle=\mathbf{P}_{\hat{c}_{i}}\left(\mathbf{P}_{\hat{X}_{\hat{\sigma}_{1}^{(N)}}}\left(\hat{\sigma}^{(N)}_{1}<\infty\right);\hat{X}_{\hat{\sigma}_{1}^{(N)}}\in E\right)+\mathbf{P}_{\hat{c}_{i}}\left(\hat{X}_{\hat{\sigma}_{1}^{(N)}}=\partial\right)$
		$\displaystyle=\mathbf{P}_{\hat{c}_{i}}\left(\hat{X}_{\hat{\sigma}_{1}^{(N)}}\in E\right)+\mathbf{P}_{\hat{c}_{i}}\left(\hat{X}_{\hat{\sigma}_{1}^{(N)}}=\partial\right)=1.$

Next, we prove that for all $i\in\mathbb{N}$ ,

\hat{\zeta}_{\infty}=\hat{\zeta},\quad\mathbf{P}_{\hat{c}_{i}}\text{-a.s.}

If $\hat{\zeta}(\omega)<\infty$ , then according to the definition of $\hat{X}^{(n)}$ , we have

\hat{\zeta}^{(n)}(\omega)\geq\hat{\zeta}(\omega)-\left|\{0\leq t<\hat{\zeta}(\omega):\hat{X}_{t}(\omega)<\hat{c}_{n}\}\right|,

where $|\cdot|$ stands for the Lebesgue measure of the time set. Note that

\left|\{0\leq t<\hat{\zeta}(\omega):\hat{X}_{t}(\omega)=0\}\right|=0,\quad\mathbf{P}_{\hat{c}_{i}}\text{-a.s.}

It follows that

	$\displaystyle\hat{\zeta}_{\infty}(\omega)$	$\displaystyle=\lim_{n\rightarrow\infty}\hat{\zeta}^{(n)}(\omega)\geq\hat{\zeta}(\omega)-\lim_{n\rightarrow\infty}\left\|\{0\leq t<\hat{\zeta}(\omega):\hat{X}_{t}(\omega)<\hat{c}_{n}\}\right\|$
		$\displaystyle=\hat{\zeta}(\omega)-\lim_{n\rightarrow\infty}\left\|\{0\leq t<\hat{\zeta}(\omega):\hat{X}_{t}(\omega)=0\}\right\|=\hat{\zeta}(\omega).$

Thus, $\hat{\zeta}_{\infty}(\omega)=\hat{\zeta}(\omega)$ whenever $\hat{\zeta}(\omega)<\infty$ . For the case $\hat{\zeta}(\omega)=\infty$ , we prove that $\hat{\zeta}^{(n)}(\omega)=\infty$ for all $n\geq N$ with $N$ satisfying (A.2). It suffices to consider the case where $\hat{\lambda}^{(n)}(\{\partial\})>0$ , because otherwise $\hat{X}^{(n)}$ is honest by Theorem 2.1. If $\hat{\zeta}^{(n)}<\infty$ , then the total duration discarded from $\hat{X}_{\cdot}(\omega)$ in constructing $\hat{X}^{(n)}_{\cdot}(\omega)$ must be infinite. It follows from (A.2) that

\{\hat{\zeta}^{(n)}<\infty,\hat{\zeta}=\infty\}\subset\{\hat{X}_{\hat{\sigma}^{(n)}_{m}}\in E,\forall m\geq 1\}.

By the strong Markov property, for any integer $M\geq 1$ ,

\mathbf{P}_{\hat{c}_{i}}\left(\hat{X}_{\hat{\sigma}^{(n)}_{m}}\in E,1\leq m\leq M\right)=(1-\hat{\lambda}^{(n)}(\{\partial\}))^{M}.

Therefore,

\mathbf{P}_{\hat{c}_{i}}\left(\hat{\zeta}^{(n)}<\infty,\hat{\zeta}=\infty\right)\leq\lim_{M\rightarrow\infty}(1-\hat{\lambda}^{(n)}(\{\partial\}))^{M}=0.

This implies $\hat{\zeta}_{\infty}(\omega)=\hat{\zeta}(\omega)$ for $\hat{\zeta}(\omega)=\infty$ .

We are now prepared to prove (A.1). Specifically, we need to establish that

\lim_{n\rightarrow\infty}\hat{X}^{(n)}_{t}=\hat{X}_{t},\quad\mathbf{P}_{\hat{c}_{i}}\text{-a.s.}

for a fixed $t\geq 0$ . For $\omega\in\Omega$ , let $\rho^{(n)}_{t}(\omega)$ represent the total duration discarded from the path of $\hat{X}_{\cdot}(\omega)$ in constructing $\hat{X}^{(n)}_{s}(\omega)$ for $0\leq s\leq t$ , namely, $\hat{X}^{(n)}_{t}(\omega)=\hat{X}_{t+\rho^{(n)}_{t}(\omega)}(\omega)$ . Obviously, $\rho^{(n)}_{t}(\omega)$ is decreasing as $n\rightarrow\infty$ . If $t\geq\hat{\zeta}(\omega)=\hat{\zeta}_{\infty}(\omega)$ , then

\hat{X}_{t}(\omega)=\hat{X}^{(n)}_{t}(\omega)=\partial,\quad\forall n\in\mathbb{N}.

If $t<\hat{\zeta}(\omega)=\hat{\zeta}_{\infty}(\omega)$ , then there exists an integer $n_{0}$ (which may depend on $\omega$ ) such that

t<\hat{\zeta}^{(n_{0})}(\omega)\leq\hat{\zeta}^{(n)}(\omega),\quad\forall n\geq n_{0}.

This implies

\rho^{(n)}_{t}(\omega)\leq\rho^{(n_{0})}_{t}(\omega)<\infty.

We have

	$\displaystyle\rho^{(n)}_{t}(\omega)$	$\displaystyle\leq\left\|\{0\leq s\leq t+\rho^{(n)}_{t}(\omega):\hat{X}_{s}(\omega)<\hat{c}_{n}\}\right\|$
		$\displaystyle\leq\left\|\{0\leq s\leq t+\rho^{(n_{0})}_{t}(\omega):\hat{X}_{s}(\omega)<\hat{c}_{n}\}\right\|.$

Thus

	$\displaystyle\lim_{n\rightarrow\infty}\rho^{(n)}_{t}(\omega)$	$\displaystyle\leq\lim_{n\rightarrow\infty}\left\|\{0\leq s\leq t+\rho^{(n_{0})}_{t}(\omega):\hat{X}_{s}(\omega)<\hat{c}_{n}\}\right\|$
		$\displaystyle=\left\|\{0\leq s\leq t+\rho^{(n_{0})}_{t}(\omega):\hat{X}_{s}(\omega)=0\}\right\|=0.$

Therefore, by the right continuity of $\hat{X}$ , we obtain

\lim_{n\rightarrow\infty}\hat{X}^{(n)}_{t}(\omega)=\lim_{n\rightarrow\infty}\hat{X}_{t+\rho^{(n)}_{t}(\omega)}(\omega)=\hat{X}_{t}(\omega),

whenever $t<\hat{\zeta}(\omega)$ . ∎

Acknowledgement

The author wishes to thank Professor Patrick J. Fitzsimmons from the University of California, San Diego, for his invaluable suggestion, which inspired the exploration of this topic.

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Time-changed Feller’s Brownian motions are birth-death processes

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

2. Birth-death processes

2.1. Elements of QQ-processes

2.2. Resolvent representation of QQ-processes

Theorem 2.1.

2.3. Ray-Knight compactification of QQ-processes

3. Feller’s Brownian motions

Definition 3.1.

Remark 3.2.

3.1. Infinitesimal generators

Theorem 3.3.

Proof.

Remark 3.4.

Corollary 3.5.

Proof.

3.2. Pathwise construction

3.2.1. p2=0,p3>0,|p4|<∞p_{2}=0,p_{3}>0,|p_{4}|<\infty

3.2.2. p2>0,0<|p4|<∞p_{2}>0,0<|p_{4}|<\infty

3.2.3. |p4|=∞|p_{4}|=\infty

3.3. Local times of Feller’s Brownian motion

Definition 3.6.

Lemma 3.7.

Proof.

Definition 3.8.

4. Time-changed Feller’s Brownian motions are birth-death processes

4.1. Spatial transformation

4.2. Time change

Lemma 4.1.

Proof.

Theorem 4.2.

Proof.

Corollary 4.3.

Proof.

4.3. Uniqueness of PCAF for time change

Theorem 4.4.

Proof.

5. Parameters of birth-death processes obtained by time change

5.1. Main result

Theorem 5.1.

Corollary 5.2.

Proof.

5.2. Pathwise construction of QQ-processes

6. Proof of Theorem 5.1 for |p4|<∞|p_{4}|<\infty

6.1. Doob processes

Theorem 6.1.

Proof.

6.2. Symmetric case

Theorem 6.2.

Proof.

6.3. Non-symmetric case with finite jumping measure

Theorem 6.3.

Proof.

7. Approximation of Feller’s Brownian motion

7.1. Approximating sequence of Doob’s Brownian motions

Lemma 7.1.

Proof.

Theorem 7.2.

Proof.

Remark 7.3.

7.2. Representation of instantaneous distributions

Lemma 7.4.

Proof.

Lemma 7.5.

Proof.

7.3. Parameters of Feller’s Brownian motion

Theorem 7.6.

Proof.

8. Proof of Theorem 5.1 for |p4|=∞|p_{4}|=\infty

8.1. No sojourn case p3=0p_{3}=0

Lemma 8.1.

Proof.

Proof of Theorem 5.1 for p3=0,|p4|=∞p_{3}=0,|p_{4}|=\infty.

8.2. Sojourn case p3>0p_{3}>0

Lemma 8.2.

Proof.

Proof of Theorem 5.1 for p3>0,|p4|=∞p_{3}>0,|p_{4}|=\infty.

2.1. Elements of $Q$ -processes

2.2. Resolvent representation of $Q$ -processes

2.3. Ray-Knight compactification of $Q$ -processes

3.2.1. $p_{2}=0,p_{3}>0,|p_{4}|<\infty$

3.2.2. $p_{2}>0,0<|p_{4}|<\infty$

3.2.3. $|p_{4}|=\infty$

5.2. Pathwise construction of $Q$ -processes

6. Proof of Theorem 5.1 for $|p_{4}|<\infty$

8. Proof of Theorem 5.1 for $|p_{4}|=\infty$

8.1. No sojourn case $p_{3}=0$

Proof of Theorem 5.1 for $p_{3}=0,|p_{4}|=\infty$ .

8.2. Sojourn case $p_{3}>0$

Proof of Theorem 5.1 for $p_{3}>0,|p_{4}|=\infty$ .