Extremal Invariant Distributions of Infinite Brownian Particle Systems with Rank Dependent Drifts

Consider a collection (finite or infinite) of particles on the real line moving according to mutually independent Brownian motions and such that the $i$-th particle from the left gets a constant drift $g_{i-1}$. The special case when $g_{0}=1$ and $g_{i}=0$ for $i\in{\mathbb{N}}$ is the well studied Atlas model. We refer to the general setting as the ${\boldsymbol{g}}$-Atlas model, where ${\boldsymbol{g}}=(g_{0},g_{1},\ldots)^{\prime}$. Such particle systems were originally introduced in stochastic portfolio theory [13, 7, 14] as models for stock growth evolution in equity markets and have been investigated extensively in recent years in several different directions. In particular, characterizations of such particle systems as uniform scaling limits of jump processes with local interactions on integer lattices, such as the totally asymmetric simple exclusion process, have been studied in [24]. Various types of results for the asymptotic behavior of the empirical measure of the particle states have been studied, such as propagation of chaos, characterization of the associated McKean-Vlasov equation and nonlinear Markov processes [39, 21], large deviation principles [12], characterizing the asymptotic density profile and the trajectory of the leftmost particle via Stefan free-boundary problems [10]. These particle systems also have close connections with Aldous’ “Up the river” stochastic control problem [1], recently solved in [40]. Results on wellposedness of the associated stochastic differential equations (in the weak and strong sense) and on absence of triple collisions (three particles at the same place at the same time) have been studied in [8, 38, 18, 17, 31, 20].

One important direction of investigation has been in the study of the long-time behavior of such particle systems. For finite particle systems, under conditions on the drift vector ${\boldsymbol{g}}$, it follows from results of Harrison and Williams [16, 15] that the multidimensional reflected Brownian motion describing the evolution of the gaps between the ranked particles has a unique stationary (invariant) distribution (see [27]). It is known that convergence of the law at time $t$ to this stationary distribution, as $t\to\infty$, occurs at a geometric rate [9]. Rates of convergence to stationarity, depending explicitly on the drift vector ${\boldsymbol{g}}$ and dimension have been obtained in [19, 5, 4].

In the current work we are interested in infinite particle systems. One basic result on long-time behavior of such particle systems was obtained in [27] which showed that for the infinite Atlas model, i.e. when ${\boldsymbol{g}}={\boldsymbol{g}}^{1}=(1,0,0,\ldots)^{\prime}$, the process describing the gaps between the ranked particles in the system has a simple product form stationary distribution given as $\pi_{0}=\otimes_{n=1}^{\infty}\mbox{Exp}(2)$ (here and later, for $a>0$, $\mbox{Exp}(a)$ denotes the Exponential distribution with mean $1/a$). The paper [27] also conjectured that this is the unique stationary distribution of the (gap sequence in the) infinite Atlas model. However, this was shown to be false in [35] who gave an uncountable collection of product form stationary distributions for the gaps in the infinite Atlas model defined as

As in the finite dimensional settings it is of interest to investigate convergence of the laws at time $t$ to the stationary distributions as $t\to\infty$. Due to the multiplicity of stationary distributions, a meaningful goal is to understand the local stability structure of this infinite dimensional stochastic dynamical system and identify the basins of attraction of the various stationary distributions. Such results, describing (weak and strong) domains of attraction of $\pi_{0}$ have been obtained in [32, 11, 6] and (weak) domains of attraction of $\pi_{a}$ ($a\neq 0$) in [6]. Results analogous to [35, 32] for two-sided infinite Brownian systems have been obtained in [33].

Although the above results give us a good understanding of the local stability structure of the infinite Atlas model, the picture that one has is far from complete. A key obstacle here is that a full characterization of all extremal invariant distributions of the infinite Atlas model is currently an open problem. The goal of this work is to make some progress towards this goal and, moreover, provide some characterization of the structure of the set of invariant distributions. We will in fact consider the more general setting of the ${\boldsymbol{g}}$-Atlas model where the drift vector ${\boldsymbol{g}}\in{\mathcal{D}}$, with

For this setting it is known from the work of [35] that, once more, the process associated with the gap sequence of the ranked particle system has a continuum of stationary distributions given as

where $\bar{g}_{n}\doteq\frac{1}{n}(g_{0}+\cdots+g_{n-1})$. In the special case where ${\boldsymbol{g}}\in{\mathcal{D}}_{1}$, with

$\pi_{a}^{{\boldsymbol{g}}}$ is also an invariant distribution for $a=-2\inf_{n\in{\mathbb{N}}}\bar{g}_{n}=-2\lim_{j\rightarrow\infty}\bar{g}_{N_{j}}$ (see [32, Section 4.2]). Note that ${\boldsymbol{g}}^{1}\in{\mathcal{D}}_{1}$ whereas the zero drift ${\boldsymbol{g}}=(0,0,\ldots)^{\prime}$ is in ${\mathcal{D}}$ but not in ${\mathcal{D}}_{1}$. Roughly speaking, a drift ${\boldsymbol{g}}$ lying in ${\mathcal{D}}_{1}$ produces a ‘stabilizing interaction’ in the subsystem of the lowest $N_{j}$ particles for any $j\geq 1$, due to which the gaps between them stabilize in time owing to stronger upward average drifts of the lower particles in this subsystem in comparison to the average drift of all the $N_{j}$ particles. This intrinsic stabilizing influence from the drift of the particles leads to an additional stationary distribution, namely $\pi_{-2\lim_{j\rightarrow\infty}\bar{g}_{N_{j}}}$ (in comparison to a ${\boldsymbol{g}}\in{\mathcal{D}}\setminus{\mathcal{D}}_{1}$). For ${\boldsymbol{g}}\in{\mathcal{D}}\setminus{\mathcal{D}}_{1}$, where such a mechanism is absent, local stability essentially arises from configurations with a rapidly increasing density of particles as one moves away from the lowest one and hence one only obtains “dense stationary distributions” corresponding to $a>-2\inf_{n\in{\mathbb{N}}}\bar{g}_{n}$.

Using Kakutani’s theorem [22] it is easy to verify that, for different values of $a$, the probability measures $\pi_{a}^{{\boldsymbol{g}}}$ are mutually singular. These distributions are also special in that they have a product form structure. In particular, if the initial distribution of the gap process is chosen to be one of these distributions, then the laws of distinct gaps at any fixed time are independent despite these gaps having a highly correlated temporal evolution mechanism (see (2.3)-(2.4)). We now describe the two main results of this work.

First result: Extremality. In Theorem 3.3 we show that for each ${\boldsymbol{g}}\in{\mathcal{D}}$ and $a>-2\inf_{n\in{\mathbb{N}}}\bar{g}_{n}$, $\pi_{a}^{{\boldsymbol{g}}}$ is an extremal invariant distribution for the gap sequence process of the ${\boldsymbol{g}}$-Atlas model. Further, if ${\boldsymbol{g}}\in{\mathcal{D}}_{1}$, $\pi_{a}^{{\boldsymbol{g}}}$ is also an extremal invariant distribution for $a=-2\inf_{n\in{\mathbb{N}}}\bar{g}_{n}=-2\lim_{j\rightarrow\infty}\bar{g}_{N_{j}}$; in particular for the infinite Atlas model $\pi_{a}=\pi_{a}^{{\boldsymbol{g}}^{1}}$ is extremal for all $a\geq 0$. From equivalence between extremality and ergodicity (cf. Lemma 3.2) it then follows that all these invariant distributions are ergodic as well. This result also identifies non-trivial subsets in the weak domain of attraction of $\pi_{a}^{{\boldsymbol{g}}}$ for each $a>-2\inf_{n\in{\mathbb{N}}}\bar{g}_{n}$ ($a\geq-2\inf_{n\in{\mathbb{N}}}\bar{g}_{n}$ if ${\boldsymbol{g}}\in{\mathcal{D}}_{1}$); see Corollary 3.4.

Questions about extremality and ergodicity of stationary distributions have been addressed previously in the context of interacting particle systems on countably infinite graphs (see [26, 2, 37, 3] and references therein). However, in all these cases, the interactions are Poissonian, namely, the dynamics is given in terms of jumps of particles to neighboring vertices in a countably infinite graph at epochs of Poisson processes associated with edges or vertices. This enables one to use the (explicit) generator of the associated continuous-time jump-Markov processes in an effective way. The interactions in rank based diffusions are very ‘singular’ owing to the local time based dynamics (see (2.3)) and generator based methods seem to be less tractable. Furthermore, unlike previous works, the state space for the gap process (i.e. $\mathbb{R}_{+}^{\infty}$) is not countable and has a non-smooth boundary, and the process has intricate interactions (oblique reflections) with the boundary. Hence, proving extremality requires new techniques. Our proofs are based on constructing appropriate couplings for these infinite dimensional diffusions which then allow us to prove suitable invariance properties (see e.g. (4.13)) and a certain ‘directional strong Feller property’ (see Proposition 4.3). Such coupling techniques, based on ‘mirror’ couplings of driving Brownian motions, are novel in the context of infinite rank based diffusions and provide a new method for establishing semigroup continuity properties for such processes. Moreover, the coupling approach introduced in the paper has the potential to be applicable to broader families of infinite-dimensional diffusion processes for which such directional strong Feller properties may be useful, e.g. in analysis of the ergodic behavior. Although our setting and methods are very different, at a high level, the approach we take, of proving extremality by showing the a.e. constancy of suitable invariant functions, is inspired by the papers [37, 3].

Second result: Characterization of product form stationary distributions. A natural question is whether there are any other product form stationary distributions of the ${\boldsymbol{g}}$-Atlas model than the ones identified in [35]. Our second main result (Theorem 3.5) answers this question in the negative under certain conditions by showing that if ${\boldsymbol{g}}\in{\mathcal{D}}_{1}$ and $\pi$ is a product form stationary distribution of the ${\boldsymbol{g}}$-Atlas model satisfying a mild integrability condition (see (3.2)) then it must be $\pi_{a}^{{\boldsymbol{g}}}$ for some $a\geq-2\lim_{j\rightarrow\infty}\bar{g}_{N_{j}}$. Furthermore, this result gives a novel probabilistic interpretation to $a$ in terms of the resulting force acting on a tagged particle under the combined influence of hardcore interactions (collision local times) and soft potentials (drift terms). See Remark 3.6 for this interpretation and for a conjecture that is suggested by this interpretation.

We now make some comments on proofs. The key step in proving the extremality of $\pi_{a}^{{\boldsymbol{g}}}$ is to establish that any bounded measurable function $\psi$ on ${\mathbb{R}}_{+}^{\infty}$ that is $\pi_{a}^{{\boldsymbol{g}}}$-a.e. invariant, under the action of the semigroup of the Markov process corresponding to the ${\boldsymbol{g}}$-Atlas gap sequence, is constant $\pi_{a}^{{\boldsymbol{g}}}$-a.e. If ${\boldsymbol{g}}={\boldsymbol{g}}^{1}$ and $a=0$, we have that $\pi_{a}^{{\boldsymbol{g}}}=\otimes_{i=1}^{\infty}\mbox{Exp}(2)$, and therefore the coordinate sequence $\{Z_{i}\}_{i=1}^{\infty}$ is iid under $\pi_{a}^{{\boldsymbol{g}}}$. In this case, from the Hewitt-Savage zero-one law it suffices to show that $\psi$ is $\pi_{a}^{{\boldsymbol{g}}}$-a.e. invariant under all finite permutations of the coordinates of ${\mathbb{R}}_{+}^{\infty}$. For this, in turn, it suffices to simply prove the above invariance property for transpositions of the $i$-th and $(i+1)$-th coordinates, for all $i\in{\mathbb{N}}$. For a general $\pi_{a}^{{\boldsymbol{g}}}$, the situation is more involved as the coordinate sequence $\{Z_{i}\}_{i=1}^{\infty}$ is not iid any more. Nevertheless, from the scaling properties of Exponential distributions it follows that, with $c_{n}\doteq 2[n(2\bar{g}_{n}+a)]^{-1}$, the sequence $\{\tilde{Z}_{n}\}_{n\geq 1}$, defined as $\tilde{Z}_{n}=c_{n}^{-1}Z_{n}$, $n\in{\mathbb{N}}$, is iid under $\pi_{a}^{{\boldsymbol{g}}}$. In this case, in order to invoke the Hewitt-Savage zero-one law, one needs to argue that for each $i$ the map $\psi$ is $\pi_{a}^{{\boldsymbol{g}}}$-a.e. invariant under the transformation that takes the $(i,i+1)$ coordinates $(z_{i},z_{i+1})$ to $(\frac{c_{i}}{c_{i+1}}z_{i+1},\frac{c_{i+1}}{c_{i}}z_{i})$ and keeps the remaining coordinates the same. Establishing this property is at the heart of the proof of Theorem 3.3. A key technical idea in the proof is the construction of a mirror coupling of the first $i+1$ Brownian motions, and synchronous coupling of the remaining Brownian motions, in the evolution of the ranked infinite ${\boldsymbol{g}}$-Atlas model corresponding to a pair of nearby initial configurations. Estimates on the probability of a successful coupling, before any of the first $i$-gap processes have hit zero or the lowest $i$-particles have interacted with the higher ranked particles (in a suitable sense), are some of the important ingredients in the proof. We refer the reader to Section 4.1 for additional comments on the proof strategy.

The proof of Theorem 3.5 hinges on establishing a key identity for expectations, under the given product form invariant measure $\pi$, of certain integrals involving the state process of the $i$-th gap and the collision local time for the $(j-1)$-th and $j$-th particle, for $i\neq j$ (see Lemma 5.4). This identity is a consequence of the product form structure of $\pi$ and basic results on local times of continuous semimartingales. One subtlety here is that although the product form structure of an invariant measure $\pi$ implies that the laws of the various gaps at any fixed time $t$ (when the process is initiated at $\pi$) are independent, the laws of the paths of the various gap processes are not, and thus one cannot immediately deduce the independence beween the state of the $i$-th gap at time $t$ and the local time of the $j$-th gap at $0$, at time $t$, for $i\neq j$. By using the form of the dynamics of the ${\boldsymbol{g}}$-Atlas model, the identity in Lemma 5.4 allows us to obtain a recursive system of equations for the moment generating functions of the coordinate projections of $\pi$ that can then be solved explicitly from which it is then readily seen that $\pi$ must be $\pi_{a}^{{\boldsymbol{g}}}$ for a suitable value of $a$. See Section 5.1 for additional comments on the proof idea.

Of course it is immediate to construct non-product form stationary distributions of the ${\boldsymbol{g}}$-Atlas model by considering mixtures of the above product stationary distributions, however one can ask if these mixtures are all the invariant measures of the ${\boldsymbol{g}}$-Atlas model. For the cases where ${\boldsymbol{g}}=(0,0,\ldots)^{\prime}$ this question was answered in the affirmative in [30, Theorem 4.2] under certain integrability constraints on the denseness of particle configurations. For a general ${\boldsymbol{g}}$ providing such a complete characterization is a challenging open problem.

In the context of interacting particle systems on countably infinite graphs, the analogous problem has been solved completely in a few cases such as the simple exclusion process [26] and the zero range process [2] where the extremal probability measures are fully characterized as an explicit collection of certain product form measures. However, in these models the particle density associated with distinct extremal measures are scalar multiples of each other owing to certain homogeneity properties in the dynamics (see, for example, [2, Theorem 1.10]). This, along with the Poissonian nature of the interactions, enables one to prove useful monotonicity properties of the ‘synchronously coupled’ dynamics (see [26, Section 2] and [2, Section 4]) using generator methods that are crucial to the above characterization results.

A key challenge in extending these methods to rank based diffusions of the form considered here is that, in addition to the singular local time interactions, the point process associated with the configuration of particles with gaps distributed as $\pi_{a}^{{\boldsymbol{g}}}$ has an intensity function $\rho(x)$, that grows exponentially as $x\to\infty$ when $a>0$ and, due to a nonlinear dependence on $a$, lacks the scalar multiple property for distinct values of $a$. This is a direct consequence of the inhomogeneity of the topological interactions in our particle systems where the local stability in a certain region of the particle cloud is affected both by the density of particles in the neighborhood and their relative ranks in the full system. Moreover, unlike the above interacting particle systems, in rank based diffusions, when the initial gaps are given by a stationary distribution, the point process of particles is typically not stationary. This phenomenon, where the gaps are stationary while the associated point process is not, referred to as quasi-stationarity in [30], is technically challenging. We note that this latter paper studies one setting where the intensity function grows exponentially and the particle density lacks the scalar multiple property for distinct values of $a$. However their setting, in the context of rank based diffusions, corresponds to the case ${\boldsymbol{g}}=(0,0,\ldots)^{\prime}$, where the unordered particles behave like independent standard Brownian motions, and this fact is crucially exploited in [30].

Rest of the paper is organized as follows. We close this section by summarizing the main notation used in this work. In Section 2 we give the precise formulation of the model. In Section 3, we describe the questions of interest and give our main results. Finally Sections 4 and 5 give the proofs of our main results, namely Theorems 3.3 and 3.5, respectively.

The following notation will be used. Let ${\mathbb{N}}_{0}\doteq\{0,1,2,\dots\}$, ${\mathbb{R}}^{\infty}\doteq\{(x_{0},x_{1},\ldots)^{\prime}:x_{i}\in{\mathbb{R}},\,i\in{\mathbb{N}}_{0}\}$ and ${\mathbb{R}}_{+}^{\infty}\doteq\{(x_{0},x_{1},\ldots)^{\prime}:x_{i}\in{\mathbb{R}}_{+},\,i\in{\mathbb{N}}_{0}\}$. We will equip ${\mathbb{R}}^{\infty}_{+}$ with the partial order ‘$\leq$’ under which ${\boldsymbol{x}}\leq{\boldsymbol{y}}$ for ${\boldsymbol{x}}=(x_{i})_{i\in{\mathbb{N}}_{0}}$, ${\boldsymbol{y}}=(y_{i})_{i\in{\mathbb{N}}_{0}}$, if $x_{i}\leq y_{i}$ for all $i\in{\mathbb{N}}_{0}$. Borel $\sigma$-fields on a metric space $S$ will be denoted as ${\mathcal{B}}(S)$ and the space of probability measures on $(S,{\mathcal{B}}(S))$ will be denoted as ${\mathcal{P}}(S)$ which is equipped with the topology of weak convergence. We will denote by ${\mathbb{Q}}$ the set of rationals. For a Polish space $S$ with a partial order ‘$\leq$’, we say for $\gamma_{1},\gamma_{2}\in{\mathcal{P}}(S)$, $\gamma_{1}\leq_{\mbox{{\tiny st}}}\gamma_{2}$, if there are $S$-valued random variables ${\boldsymbol{X}}_{1},{\boldsymbol{X}}_{2}$ given on a common probability space with ${\boldsymbol{X}}_{i}$ distributed as $\gamma_{i}$, $i=1,2$, and ${\boldsymbol{X}}_{1}\leq{\boldsymbol{X}}_{2}$ a.s. Let ${\mathbb{X}}\doteq{\mathcal{C}}([0,\infty):{\mathbb{R}}_{+}^{\infty})$ which is equipped with the topology of local uniform convergence (with ${\mathbb{R}}_{+}^{\infty}$ equipped with the product topology). For $\gamma\in{\mathcal{P}}({\mathbb{R}}_{+}^{\infty})$, let $L^{2}(\gamma)$ be the collection of all measurable $\psi:{\mathbb{R}}_{+}^{\infty}\to{\mathbb{R}}$ such that $\int_{{\mathbb{R}}_{+}^{\infty}}|\psi({\boldsymbol{z}})|^{2}\gamma(d{\boldsymbol{z}})<\infty$. We denote the inner-product and the norm on $L^{2}(\gamma)$ as $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ respectively.