Doubly stochastic Yule cascades (Part II): The explosion problem in the non-reversible case

Largely motivated by the probabilistic method for the deterministic three-dimensional incompressible Navier-Stokes equations (NSE) introduced by Le Jan and Sznitman [lejan], relaxed in [chaos], the authors introduced a new class of stochastic branching models, referred to as doubly stochastic Yule cascades, in [part1_2021]. This class of branching models may also be viewed in the context of statistically dependent first passage percolation models on tree graphs, or in the context of certain cellular aging models in biology [DA_PS_1988, KB_PP_2010]. Roughly speaking, a doubly stochastic Yule cascade evolves from a single progenitor in which each particle waits for a random length of time depending on the particle’s position on the genealogical tree before giving birth to a new generation of particles, each evolving by the same rules as its parent. If the lengths of times between particle reproductions are sufficiently short then branching may produce infinitely many particles in a finite time, an event referred to as stochastic explosion. In this paper, we establish some sufficient conditions for explosion and non-explosion, and apply them for the doubly stochastic Yule cascades associated with several well-known differential equations and probabilistic models.

We begin by recalling, in the context of binary trees, some of the definitions introduced in [part1_2021] which contains more general definitions. With standard notations, debote by $\mathbb{T}=\{\theta\}\cup\left(\cup_{n=1}^{\infty}\{1,2\}^{n}\right)$ the indexed binary tree with the root $\theta$.

In the above, we used the standard notation $Y_{v}\sim{\rm Exp}(\lambda_{v})$ to denote that $Y_{v}$ is exponentially distributed with intensity $\lambda_{v}$. If $\lambda_{v}\equiv\lambda>0$ is a constant, this is the familiar Yule cascade with intensity $\lambda$. The special case $\lambda=1$ is referred to as the standard Yule cascade.

Equivalently, $\{Y_{v}\}_{v\in\mathbb{T}}$ is a DSY cascade with random intensities $\{\lambda_{v}\}_{v\in\mathbb{T}}$ if and only if $\{T_{v}=\lambda_{v}Y_{v}\}_{v\in\mathbb{T}}$ is a family of i.i.d. mean one exponentially distributed random variables (called holding times or clocks) independent of $\{\lambda_{v}\}_{v\in\mathbb{T}}$. Thus, a DSY cascade is completely specified once the family of random intensities is specified. For this reason, one can write a DSY cascade as $\{\lambda_{v}^{-1}T_{v}\}_{v\in\mathbb{T}}$. The associated counting process defined by

$$N(t)=\left\{\begin{matrix}1&\text{if}&t\leq\frac{{{T}_{\theta}}}{{{\lambda}_{\theta}}},\\ \text{card}\left\{v\in\mathbb{T}:\ \sum\limits_{j=0}^{|v|-1}{\frac{{{T}_{v|j}}}{{{\lambda}_{v|j}}}}<t\leq\sum\limits_{j=0}^{|v|}{\frac{{{T}_{v|j}}}{{{\lambda}_{v|j}}}}\right\}&\text{if}&t>\frac{{{T}_{\theta}}}{{{\lambda}_{\theta}}}\\ \end{matrix}\right.$$

is referred to as the Yule process or doubly stochastic Yule process depending on the respective cascade model being considered. In the definition of $N(t)$, we have used standard notations for the length of a vertex $v=(v_{1},\dots,v_{n})\in\mathbb{T}$ and the truncation of $v$ of length $j$, namely $|v|=n$ and $v|j=(v_{1},\dots,v_{j})$ for $0\leq j\leq|v|$. By convention, $v|0=\theta.$

According to the definition, the explosion event $[\zeta<\infty]$ has a positive probability in an explosive cascade. However, in all applications considered in this paper, explosion is in fact a $0-1$ event (Section 2). The DSY cascades introduced in Section 1 are quite general. To consider the explosion problem, we will further assume a certain branching Markov chain structure underlying the random intensities $\lambda_{v}$.

Using estimates on the spectral radius of an operator defined on the underlying branching Markov chains, the authors obtained in [part1_2021] criteria for non-explosion assuming a time-reversibility condition. These conditions were applied to the Bessel cascade, a particular cascade associated with the Navier-Stokes equations, and to the cascade of the Kolmogorov-Petrovski-Piskunov (KPP) equation in Fourier domain to show that they are both non-explosive. Other more purely probabilistic models were also analyzed there. However, time reversibility is too restrictive and does not apply to several examples of particular importance. These include the cascades induced by the scaling properties of the NSE, referred here as self-similar cascades associated with the NSE. The focus of this paper is to establish explosion and non-explosion criteria for a class of DSY cascades using probabilistic arguments that do not depend on the time-reversibility.

To establish a new non-explosion criterion (Section 2), we rely on the sub-criticality of nonhomogeneous Galton-Watson processes associated with the cascade, which is more intuitive and robust than the method of large deviation estimates used in [part1_2021]. The key idea to obtain non-explosion is to construct a sequence of recurring finite cut-sets through an inspection process. The construction of these cut-sets takes into consideration the uncountable set of paths $s\in\partial{\mathbb{T}}=\{1,2\}^{\infty}$ in a binary tree, which involves a new notion of recurrence that naturally generalizes neighborhood recurrence along each path (Section 2). As an application, Section 2 implies the non-explosion of the self-similar cascade associated with NSE in dimensions $d\geq 12$ and recovers the non-explosion result for the Bessel cascade of the 3-dimensional NSE and the cascade associated with the KPP equation proven in [part1_2021].

We establish a criterion for a.s. explosion (Section 2) based on a “fastest path” constructed by a greedy algorithm: at each time of branching, the fastest path chooses to follow the descendant vertex with the larger intensity $\lambda_{v}$. This criterion allows one to show the explosion of a cascade even if the explosion does not occur along any deterministic path $s\in{\partial{\mathbb{T}}}$, i.e.

$$\sum\limits_{j=0}^{\infty}{\frac{{{T}_{s|j}}}{\lambda_{s|j}}}=\infty\ \ \ {\text{a.s.}}\ \forall s\in\partial\mathbb{T}.$$

As an application, we obtain the a.s. explosion of the self-similar cascade of the 3-dimensional Navier-Stokes equations. As shown in [athreya, alphariccati, smallness], the stochastic explosion of DSY cascades associated with certain evolution equations leads to both nonuniqueness and finite-time blowup of solutions to the initial-value problems in appropriate settings. In particular, the aforementioned explosion of the self-similar DSY cascade corresponding to the Navier-Stokes equations leads to a non-uniqueness result for an equation introduced by Montgomery-Smith [smith] as a model for possible Navier-Stokes finite-time blowup. Remarkably, our DSY framework also provides an alternative way to prove finite-time blowup of the smooth solutions to the Montgomery-Smith equations [smallness]. For the actual Navier-Stokes equations, a possible resolution of the outstanding nonuniqueness and finite-time blowup problems hinges on a better understanding of the connections between geometric structure of the nonlinear term and the branching structure of the associated DSY cascade.

Motivated by the PDE and probabilistic models mentioned above, we are particularly interested in DSY cascades in which the random intensities $\lambda_{v}$ are of the form $\lambda_{v}=\lambda(X_{v})$ where, for each $v\in\mathbb{T}$, $X_{v}$ is a random variable taking values in a measurable space $(S,\mathscr{S})$ and $\lambda:S\to(0,\infty)$ is a measurable function. Throughout the paper, we assume the following two properties:

1. (A)

   For any path $s\in\partial\mathbb{T}$, the sequence $X_{\theta}$, $X_{s|1}$, $X_{s|2}$,…is a time homogeneous Markov chain.

2. (B)

   For any path $s\in\partial\mathbb{T}$, the stationary transition probability $p(x,dy)$ does not depend on $s$.

In [part1_2021], DSY cascades satisfying conditions (A) and (B) are called DSY cascades of type ($\mathscr{M}$). We will refer to the family $\{X_{v}\}_{v\in\mathbb{T}}$ as the underlying branching Markov chain of the DSY cascade.

In the statements of the main results, the following conditions are sometimes required. We use the standard notation $u*v$ for concatenation of vertices $u,v\in\mathbb{T}$.

1. (C)

   For each $v\in\mathbb{T}$, the two subfamilies $\{X_{v*1*w}\}_{w\in\mathbb{T}}$ and $\{X_{v*2*w}\}_{w\in\mathbb{T}}$ are conditionally independent of each other given $X_{v*1}$ and $X_{v*2}$.

2. (D)

   For each $v\in\mathbb{T}$, the conditional joint distribution $(X_{v*1},X_{v*2})$ given $X_{v}$ does not depend on $v$.

The next condition is a generalization of the neighborhood recurrence along a path to Markov branching process. First, for $a\in S,A\in\mathscr{S},$ $n\geq 1$ and $s\in\partial\mathbb{T}$, define

$$I_{n}(a,A)=\mathbb{P}_{a}(X_{s|1}\not\in A,\,X_{s|2}\not\in A,\ldots,\,X_{s|n}\not\in A).$$

(2.1)

Notice that $I_{n}$ does not depend on $s\in\partial\mathbb{T}$ by virtue of Properites (A) and (B).

1. (E)

   (Cut-set recurrence condition) There exists a set $A\in\mathscr{S}$, a function $\psi:S\to\mathbb{R}$ and a number $r>2,$ such that for all $a\in S$, $s\in\partial\mathbb{T}$ and $n\in\mathbb{N}$,

   $$I_{n}(a,A)\leq\psi(a)r^{-n}.$$

While conditions (C) and (D) are intuitive, condition (E) is somewhat technical. It implies a kind of recurrence property for the branching Markov chain; in the language to be made clear later, the set $A$ is cut-set recurrent with respect to $\{X_{v}\}_{v\in\mathbb{T}}$. The intuition is that the Markov chain $\{X_{s|j}\}_{j\geq 0}$ returns to the set $A$ infinitely often in a uniform manner across all the paths $s\in\partial\mathbb{T}$ (cf. Section 3 for the precise definition). Specifically, we have the following result.

The proof will be given in Section 4. This theorem leads to a strategy to show the non-explosion of a DSY cascade by finding an appropriate value $c>0$ such that the set $A=\lambda^{-1}((0,c])$ is cut-set recurrent. Next, we state the main criterion for non-explosion. We say that a function $\phi:(0,\infty)\rightarrow(0,\infty)$ is locally bounded if it maps every bounded set into a bounded set.

The proof of this theorem is in Section 5. In our applications, the case $S=(0,\infty)$ is of particular interest. In this case, we have the following corollary.

We will give in Section 6 two sufficient conditions (relatively simple to check) for condition (E) to be satisfied. An interesting class of DSY cascades is the self-similar DSY cascades. This class includes the cascade of the $\alpha$-Riccati equation, the cascade of the complex Burgers equation, and the cascade of the Navier-Stokes equations with a scale-invariant kernel (Section 12).

In Section 7, we provide a useful characterization of self-similar DSY cascades: these are exactly the DSY cascades in which the branching Markov chain $\{X_{v}\}_{v\in\mathbb{T}}$ satisfies (A), (B), (C) such that along each path $s\in\partial\mathbb{T}$, the sequence $\{X_{s|j}\}_{j\geq 0}$ is a multiplicative random walk on $S$ (i.e. $\ln X_{s|j}$ is an additive random walk on the additive group $\ln S$). Multiplicative random walks on $(0,\infty)$ are natural examples of non-reversible Markov chains, which are of interest in this paper. Self-similar DSY cascades admit a rather simple criterion for non-explosion:

The proof of this result is given in Section 7. The main criterion for a.s. explosion of DSY cascades is as follows.

The proof this theorem is given in Section 8. In Section 12, it is shown that the explosion criterion is satisfied by the self-similar DSY cascades associated with the 3-dimensional Navier-Stokes equations. We also show that for large spatial dimensions ($d\geq 12$), the self-similar DSY cascades are non-explosive. The range $3<d<12$ remains inconclusive to us. Nevertheless, we show that the explosion event in a self-similar DSY cascade is a $0-1$ event. More generally, one has the following.

The proof of this result is given in Section 9. Applications of the non-explosion and explosion criteria are given in the last three sections of the paper. In Section 10, we focus on examples originating from probability, whereas Section 11 gives examples originating from differential equations. The applications to NSE are of particular interest and are mentioned in a separated section (Section 12).

To prepare for the proofs of the main results, we introduce some preliminary notions on a general rooted tree (see also [lyons90, lyons92], [lyons, Sec. 2.5]).

Let $\mathscr{T}\subset\mathbb{V}=\{\theta\}\cup(\cup_{n\geq 1}\mathbb{N}^{n})$ be a connected locally finite random tree rooted at $\theta$. Each vertex $v=(v_{1},\ldots,v_{n})\in\mathscr{T}$ is connected to the root by a unique path, so one can identify a vertex with the path connecting it to the root. For a finite or infinite path $s=(s_{1},s_{2},\ldots)$, we denote by $v*s$ the path obtained by appending $s$ to $v$:

$$v*s=(v_{1},\ldots,v_{n},s_{1},s_{2},\ldots)$$

The length of a path $v=(v_{1},\ldots,v_{n})$, or the genealogical height of vertex $v$, is $|v|=n$. For $1\leq j\leq n$, the truncation of $v$ up to the $j$’th generation is $v|j=(v_{1},\ldots,v_{j})$. We use the convention that $v|0=\theta$ and $|\theta|=0$. The boundary of $\mathscr{T}$ is defined as the set of infinite paths:

$$\partial\mathscr{T}=\{s\in\mathbb{N}^{\infty}:~{}s|j\in\mathscr{T},\ \forall\,j\geq 0\}.$$

Intuitively, a cut-set is a set of vertices that separates the root from the boundary.

For the sake of simplicity, we will write the structure $(\{Y_{v}\}_{v\in\mathbb{V}},\mathscr{T})$ simply as $\{Y_{v}\}_{v\in\mathscr{T}}$.

With the terminology introduced in the previous section, the proof of Section 2 uses the decay of $I_{n}\lesssim r^{-n}$ and an inspection process on whether $X_{v}\notin A$ to construct a sequence of passage cut-sets. Specifically at each vertex $v\in\mathbb{T}$, starting from the root $\theta$, we inspect each offspring $u\in\{v*1,\,v*2\}$ of $v$. If $X_{u}\notin A$ then $u$ passes the inspection. Otherwise, if $X_{u}\in A$, $u$ does not pass the inspection and the inspection process along the path containing $u$ stops at $u$. For each offspring of $v$ that passes the inspection, we proceed to inspect its own offspring. For example, if $X_{v*1}\notin A$ and $X_{v*2}\notin A$ then we inspect the vertices $v*11$, $v*12$, $v*21$, $v*22$. If $X_{v*1}\notin A$ and $X_{v*2}\in A$, we only inspect $v*11$ and $v*12$. In this manner, the inspection process either continues indefinitely or stops after finitely many steps. Note that we do not inspect the root: $\theta$ is considered passing the inspection whether or not $X_{\theta}\notin A$.

For a vertex $v\in\mathbb{T}$ that passes the inspection, we denote by $\mathscr{O}_{v}$ the number of offspring of $v$ that passes the inspection. The distribution of $\mathscr{O}_{v=\theta}$ is given by

	$\displaystyle\mathbb{P}_{a}\left(\mathscr{O}_{\theta}=2\right)$	$\displaystyle=$	$\displaystyle\mathbb{P}_{a}\left(X_{1}\notin A,\,X_{2}\notin A\right),$
	$\displaystyle\mathbb{P}_{a}(\mathscr{O}_{\theta}=1)$	$\displaystyle=$	$\displaystyle\mathbb{P}_{a}(X_{1}\notin A,\,X_{2}\in A)+\mathbb{P}_{a}(X_{1}\in A,\,X_{2}\notin A),$
	$\displaystyle\mathbb{P}_{a}(\mathscr{O}_{\theta}=0)$	$\displaystyle=$	$\displaystyle\mathbb{P}_{a}(X_{1}\in A,\,X_{2}\in A).$

For $v\in\mathbb{T}\setminus\{\theta\}$, let $B_{v}=[X_{v|1}\notin A,X_{v|2}\notin A,\dots,X_{v}\notin A]$ so that $I_{n}(a,A)=\mathbb{P}_{a}(B_{v})$. Note that this probability only depends on $n=|v|$ due to properties (A) and (B). Let

$$N(a)=\inf\{n\geq 1:~{}I_{n}(a,A)=0\}$$

(4.1)

which could be infinity. For each vertex $v$ with $1\leq|v|=n<N(a)$, the number of offspring passing the inspection has a distribution

	$\displaystyle\mathbb{P}_{a}(\mathscr{O}_{v}=2)$	$\displaystyle=$	$\displaystyle\mathbb{P}_{a}(X_{v*1}\notin A,\,\,X_{v*2}\notin A\ |\ B_{v}),$		(4.2)
	$\displaystyle\mathbb{P}_{a}({{\mathscr{O}}_{v}}=1)$	$\displaystyle=$	$\displaystyle\mathbb{P}_{a}(X_{v*1}\notin A,X_{v*2}\in A\ |\ B_{v})+\mathbb{P}_{a}(X_{v*1}\in A,X_{v*2}\notin A\ |\ B_{v}),$		(4.3)
	$\displaystyle\mathbb{P}_{a}(\mathscr{O}_{v}=0)$	$\displaystyle=$	$\displaystyle\mathbb{P}_{a}(X_{v*1}\in A,\,\,X_{v*2}\in A\ |\ B_{v}).$		(4.4)

Invoking Properties (A) and (B) once again, one can write $\mathscr{O}_{v}\sim\mathscr{O}_{n}$ for $n=|v|.$ Denote $p_{n,i}(a)=\mathbb{P}_{a}(\mathscr{O}_{n}=i)$ for $i=0,1,2$. Let $Z_{n}$ be the total number of individuals at generation $n$ that pass the inspection. Then $Z_{0}=1$ and

$${{Z}_{n+1}}=\sum\limits_{j=1}^{{{Z}_{n}}}{{\mathscr{O}_{n,j}}}\ \ \ \forall\,0\leq n<N(a)$$

where $\mathscr{O}_{n,1}$, $\mathscr{O}_{n,2}$, $\mathscr{O}_{n,3}$,…are i.i.d copies of $\mathscr{O}_{n}$. The inspection process can be viewed as a nonhomogeneous Galton-Watson process with the offspring distribution given above. We will show this process stops eventually.

To complete the proof of Part (i) of Section 2, we use Section 4 to show that the set $A$ is cut-set recurrent with respect to the branching Markov chain $\{X_{v}\}_{v\in\mathbb{T}}$. By definition, the first passage set $\Pi_{A}^{(1)}$ consists, for each $s\in\partial\mathbb{T}$, of the first vertex $v\neq\theta$ such that $X_{v}\in A$ (see Figure 1). In other words, it consists of the vertices at which the inspection process stops. By Section 4, $\Pi_{A}^{(1)}$ is, a.s, a finite set and, thus, must be a cut-set.

Suppose that for some $k\geq 1$, the passage set $\Pi_{A}^{(k)}$ is a cut-set. For each $u\in\mathbb{T}$, denote by $\mathbb{T}^{(u)}$ the subtree of $\mathbb{T}$ rooted at $u$ and write $X^{(u)}_{v}=X_{u*v}$ for $v\in\mathbb{T}$. By conditions (A) and (B), we have that condition (E) is also satisfied by the branching Markov chain $X^{(u)}_{v}$. Indeed, the analogue to (2.1) for this branching Markov process is

	$\displaystyle I_{n}(a^{\prime},A;X^{(u)})$	$\displaystyle\equiv$	$\displaystyle\mathbb{P}(X_{u*s|(|u|+1)}\notin A,X_{u*s|(|u|+2)}\notin A,\dots,X_{u*s|(|u|+n)}\notin A|X_{u}=a^{\prime})$
		$\displaystyle=$	$\displaystyle\mathbb{P}_{a^{\prime}}(X_{s|1}\notin A,X_{s|2}\notin A,\dots,X_{s|n}\notin A)$
		$\displaystyle=$	$\displaystyle I_{n}(a^{\prime},A)$

Given $X_{u}$, by Section 4 the inspection process on $\mathbb{T}^{(u)}$ must terminate a.s. Thus, for each path $s\in\partial\mathbb{T}$ passing through $u$, there exists a random integer $j$, $|u|<j<\infty$ such that $X_{s|j}\in A$. Let $N_{u,s}$ be the smallest value of such $j$’s. Let

$${\Pi^{(1)}_{A}(\mathbb{T}^{(u)})}=\left\{s|_{{N}_{u,s}}:\ s\in{\partial{\mathbb{T}}}~{}~{}\text{passing through }u\right\}$$

(4.6)

corresponding to the first passage set of the tree rooted at $u$ through $A$. Thus, by Section 4, ${\Pi^{(1)}_{A}(\mathbb{T}^{(u)})}$ is a.s. finite and nonempty and separates boundary paths in $\partial\mathbb{T}$ that pass through $u$ from $u$. Note that, by definition, the $k+1$’st passage set is given by

$$\Pi_{A}^{(k+1)}=\bigcup_{u\in\Pi_{A}^{(k)}}{\Pi^{(1)}_{A}(\mathbb{T}^{(u)})}$$

(4.7)

which is nonempty, finite almost surely. Since $\Pi_{A}^{(k)}$ is a cut-set, $\Pi_{A}^{(k+1)}$ is also a cut-set. We conclude that $A$ is cut-set recurrent thus completing the proof Part (i) of Section 2.

We begin the proof of Part (ii) of Section 2 by showing the estimate for the mean cardinality of $\Pi_{A}^{(1)}$, namely

$$\mathbb{E}_{a}[{\rm card}\,\Pi_{A}^{(1)}]\leq 2+\psi(a)\frac{2r}{r-2}$$

(4.8)

and use induction for the other estimate. With $N(a)$ defined in (4.1), if $N(a)=1$ then $\Pi_{A}^{(1)}=\{1,2\}$ a.s. and $\eqref{eq:510214}$ holds. Consider the case $N(a)\geq 2$. The mean total number of individuals passing the inspection process is

$$\mathbb{E}_{a}\left[\sum\limits_{n=0}^{N(a)}{{{Z}_{n}}}\right]=\sum\limits_{n=0}^{N(a)}{\mathbb{E}{{Z}_{n}}}\leq 1+\psi(a)\sum\limits_{n=1}^{N(a)}\left(\frac{2}{r}\right)^{n}\leq 1+\psi(a)\frac{r}{r-2}\,,$$

according to (4.5). Because each vertex in $\Pi_{A}^{(1)}$ is an offspring of a vertex that passes the inspection process, card $\Pi_{A}^{(1)}$ is at most twice the total number of vertices passing the inspection process. Therefore,

$$\mathbb{E}_{a}[{\rm card}\,{\Pi_{A}^{(1)}}]\leq 2\mathbb{E}\left[\sum\limits_{n=0}^{\infty}{{{Z}_{n}}}\right]\leq 2+\psi(a)\frac{2r}{r-2}.$$

Consider now $u\in\Pi_{A}^{(k)}$ for some $k\geq 1$, so that $X_{u}=a^{\prime}\in A.$ Then, by properties (A) and (B) of the branching Markov chain we have

$${{\mathbb{E}}_{a}}[\operatorname{card}({\Pi^{(1)}_{A}(\mathbb{T}^{(u)})})|X_{u}=a^{\prime}]={\mathbb{E}}_{X_{u}=a^{\prime}}[\operatorname{card}({\Pi^{(1)}_{A}(\mathbb{T}^{(u)})})]={{\mathbb{E}}_{{{X}_{\theta}}=a^{\prime}}}[\operatorname{card}(\Pi^{(1)}_{A})]\leq 2+\psi(a^{\prime})\frac{2r}{r-2}~{}~{}~{}$$

Recall that we now assume that $\psi$ is bounded on $A$ and that $M(a)=\sup\left\{\psi(x):\,x\in A\cup\{a\}\,\right\}.$ Then

$${{\mathbb{E}}_{a}}[\operatorname{card}({{\Pi^{(1)}_{A}(\mathbb{T}^{(u)})}})|X_{u}=a^{\prime}]\leq 2+M(a)\frac{2r}{r-2}\equiv\mu(a)$$

(4.9)

The proof of Section 2 is completed by noting that

$$\mathbb{E}_{a}[{\rm card}\,{\Pi_{A}^{(k)}}]=\mathbb{E}_{a}[\mathbb{E}_{a}[{\rm card}\,{\Pi_{A}^{(k)}}|\Pi_{A}^{(k-1)}]]\leq\mu(a)\mathbb{E}_{a}[{\rm card}\,{\Pi_{A}^{(k-1)}}]\leq(\mu(a))^{k}.$$

Thanks to Section 2, we can define a random sequence of nonnegative integers $\{H_{k}\}$ where $H_{0}=1$ and

$$H_{k}=\max\{|v|:~{}v\in\Pi_{A}^{(k)}\}\ \ \ \forall\,k\in\mathbb{N}.$$

(5.1)

One can observe that for any $k\in\mathbb{N}$ and $s\in\partial\mathbb{T}$, there exists $l_{k}=l_{k}(s)\in[k,\,H_{k})$ satisfying $s|l_{k}\in\Pi_{A}^{(k)}$. In order to establish the non-explosive character of the DSY cascade, note that for $k\in\mathbb{N}$ and with $n\geq H_{k}$, we have:

$${{\zeta}_{n}}=\underset{|v|=n}{\mathop{\min}}\,\sum\limits_{j=1}^{n}\frac{T_{v|j}}{\lambda(X_{v|j})}\geq\underset{|v|=n}{\mathop{\min}}\,\sum\limits_{i=1}^{k}{\frac{{{{{T}}}_{v|{{l}_{i}}}}}{\lambda(X_{v|{{l}_{i}}})}}\geq\frac{1}{c}\underset{|v|=n}{\mathop{\min}}\,\sum\limits_{i=1}^{k}{{{{{T}}}_{v|{{l}_{i}}}}}$$

where $l_{i}=l_{i}(s_{v})$ with $s_{v}\in\partial\mathbb{T}$ such that $s_{v}|_{|v|}=v$. In the last inequality we have used that $A=\lambda^{-1}(0,c)$ so that since $X_{v|l_{i}}\in A$ one has $\lambda(X_{v|l_{i}})<c.$ Since $\{T_{v}\}_{v\in\mathbb{T}}$ is independent of $\{X_{v}\}_{v\in\mathbb{T}}$, in order to establish that the DSY cascade is not explosive it suffices to show that almost surely,

$$\underset{k\to\infty}{\mathop{\lim}}\,\underset{|v|=H_{k}}{\mathop{\min}}\,\sum\limits_{i=1}^{k}{{{{{T}}}_{v|{{l}_{i}}}}}=\infty$$

(5.2)

For this, we define a random tree, referred as the “reduced tree”, consisting of the root $\theta$ and vertices in the passage cut-sets $\Pi_{A}^{(1)}$, $\Pi_{A}^{(2)}$, $\Pi_{A}^{(3)}$, …Throughout this section we will use

$$K_{u}=\textrm{card}\,{\Pi^{(1)}_{A}(\mathbb{T}^{(u)})}$$

(5.3)

denote the random number of vertices in the first cut-set of the tree rooted at $u.$ For the reduced tree, the root $\theta$ has $K_{\theta}=\textrm{card}\,{\Pi^{(1)}_{A}}$ offspring, each of which is a vertex in the cut-set ${\Pi^{(1)}_{A}}$. Each vertex $u$ of the first generation of the reduced tree has $K_{u}$ offspring, each of which is a vertex in the cut-set ${\Pi^{(1)}_{A}(\mathbb{T}^{(u)})}$, and so on (Figure 2). Thus, the first generation of the reduced tree consists of vertices in the first cut-set $\Pi_{A}^{(1)}$. The second generation of this tree consists of vertices in the second cut-set $\Pi_{A}^{(2)}$. In general, the $k$’th generation of the reduced tree consists of vertices in the $k$’th cut-set $\Pi_{A}^{(k)}$.

The set of all vertices of the random reduced tree is denoted by $\mathscr{T}\subset\mathbb{V}=\{\theta\}\cup(\cup_{n=1}^{\infty}\mathbb{N}^{n})$. Recall that the configuration of $\mathscr{T}$ is independent of the clocks $\{T_{v}\}_{v\in\mathbb{V}}$ after a natural relabeling of the vertices (Figure 2). Now (5.2) becomes a non-explosion problem of a DSY cascade $\{T_{v}\}_{v\in\mathbb{V}}$ with all intensities equal to 1 on a random tree structure $\mathscr{T}$. The uniform bound obtained in Part (ii) of Section 2 on the expected number of offspring of each vertex turns out to be crucial for our approach. For further criteria for non-explosion of a DSY cascade on a random tree structure, see [part1_2021, Sec. 4].

To show (5.2), we will use a cut-set argument similar to the one used in the proof of Section 2. Let $\varepsilon>0$ be a number to be chosen. Given the random tree $\mathscr{T}$, we start an inspection process of whether $T_{v}\leq\varepsilon$ as follows. At each vertex $v\in\mathscr{T}$, starting from the root $\theta$, we inspect its offspring. If an offspring $u\in\{v*1,\,v*2,\ldots,\,v*K_{v}\}$ satisfies $T_{u}\leq\varepsilon$ then it passes the inspection. Otherwise, it does not pass the inspection and the inspection process along the path containing $u$ stops at $u$. For any vertex that has passed the inspection, we continue to inspect its own offspring. Note that we do not inspect the root: $\theta$ is considered passing the inspection whether or not $T_{\theta}\leq\varepsilon$. In this manner, the inspection process might keep going indefinitely or stop after finitely many steps. To show that the process stops almost surely after finite number of inspections we establish the following more general result.