Searcharxiv⌕ Search

arXiv subjects

Zhenyao Sun

Publications and source records attributed to Zhenyao Sun.

12 recordsLinked to original sources
arXiv2025

On the subcritical self-catalytic branching Brownian motions

The self-catalytic branching Brownian motions (SBBM) are extensions of the classical one-dimensional branching Brownian motions by incorporating pairwise branchings catalyzed by the intersection local times of the particle pairs. These processes naturally arise as the moment duals of certain reaction-diffusion equations perturbed by multiplicative space-time white noise. For the subcritical case of the catalytic branching mechanism, we construct the SBBM allowing an infinite number of initial particles. Additionally, we establish the coming down from infinity (CDI) property for these systems and characterize their CDI rates.

math.PR↗
arXiv2024

Wright-Fisher stochastic heat equations with irregular drifts

Consider the $[0,1]$-valued continuous random field solution $(u_t(x))_{t\geq 0, x\in \mathbb R}$ to the one-dimensional stochastic heat equation \[ \partial_t u_t = \frac{1}{2}\Delta u_t + b(u_t) + \sqrt{u_t(1-u_t)} \dot W, \] where $b(1)\leq 0\leq b(0)$ and $\dot W$ is space-time white noise. In this paper, we establish the weak existence and uniqueness of the above equation for a class of drifts $b(u)$ that may be irregular at the points where the noise coefficient is non-Lipschitz and degenerate, specifically at $u=0$ or $u=1$. This class of drifts includes non-Lipschitz drifts like $b(u) = u^q(1-u)$ for every $q\in (0,1)$, and some discontinuous drifts like $b(u) = \mathbf 1_{(0,1]}(u)-u$. This demonstrates a regularization effect of the multiplicative space-time white noise without the standard assumption that the noise coefficient is Lipschitz and non-degenerate. The method we apply is a further development of a moment duality technique that uses branching-coalescing Brownian motions as the dual particle system. To handle an irregular drift in the above equation, particles in the dual system are allowed to have a number of offspring with infinite expectation, and even an infinite number of offspring with positive probability. We show that, even though the branching mechanism with an infinite number of offspring causes explosions in finite time, immediately after each explosion, the total population comes down from infinity due to the coalescing mechanism. Our results on this dual particle system are of independent interest.

math.PR↗
arXiv2023

Effect of small noise on the speed of reaction-diffusion equations with non-Lipschitz drift

We consider the $[0,1]$-valued solution $(u_{t,x}:t\geq 0, x\in \mathbb R)$ to the one dimensional stochastic reaction diffusion equation with Wright-Fisher noise \[\partial_t u= \partial_x^2 u + f(u) + ε\sqrt{u(1-u)} \dot W.\] Here, $W$ is a space-time white noise, $ε> 0$ is the noise strength, and $f$ is a continuous function on $[0,1]$ satisfying $\sup_{z\in [0,1]}|f(z)|/ \sqrt{z(1-z)} < \infty.$ We assume the initial data satisfies $1 - u_{0,-x} = u_{0,x} = 0$ for $x$ large enough. Recently, it was proved in (Comm. Math. Phys. \textbf{384} (2021), no. 2) that the front of $u_t$ propagates with a finite deterministic speed $V_{f,ε}$, and under slightly stronger conditions on $f$, the asymptotic behavior of $V_{f,ε}$ was derived as the noise strength $ε$ approaches $\infty$. In this paper we complement the above result by obtaining the asymptotic behavior of $V_{f,ε}$ as the noise strength $ε$ approaches $0$: for a given $p\in [1/2,1)$, if $f(z)$ is non-negative and is comparable to $z^p$ for sufficiently small $z$, then $V_{f,ε}$ is comparable to $ε^{-2\frac{1-p}{1+p}}$ for sufficiently small $ε$.

math.PR↗
arXiv2023

On the spectral radius of the $(L,κ)$-lazy Markov chain

We consider an $(L,κ)$-lazy operation on an irreducible Markov transition probability $P$ with state space $S$ where $L \subset S$ and $κ\in[0,1)$. For each $x \in L$ and $y\in S$, this $(L,κ)$-operation replaces $P(x,y)$, the transition probability from $x$ to $y$, by $κ1_{\{x=y\}} + (1-κ)P(x,y)$. We are interested in how $L$ and $κ$ influence the spectral radius $ρ^L_κ$ of this new transition probability. We first show that $ρ^L_κ$ is non-decreasing and continuous in $κ$. We then show that: (1) If $L$ is nonempty and finite, then $P$ being rho-transient is equivalent to that the growth of $\displaystyle (ρ^L_κ)_{κ\in[0,1)}$ exhibits a phase transition: There exists a critical value $κ_c(L) \in (0,1)$ such that $κ\mapsto ρ^L_κ$ is a constant on $[0,κ_c(L)]$ and increases strictly on $[κ_c(L),1)$; (2) For every $κ\in(0,1)$, if $S\setminus L$ is nonempty and finite, then $ρ^L_κ=ρ^S_κ$ if and only if $P$ is not strictly rho-recurrent.

math.PR↗
arXiv2022

On the coming down from infinity of coalescing Brownian motions

Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a necessary and sufficient condition for the number of particles to come down from infinity. We also identify the rate of this coming down from infinity for different initial configurations.

math.PR↗
arXiv2022

Subcritical superprocesses conditioned on non-extinction

We consider a class of subcritical superprocesses $(X_t)_{t\geq 0}$ with general spatial motions and general branching mechanisms. We study the asymptotic behaviors of $\mathbf Q_{t,r}$, the distribution of $X_t$ conditioned on $X_{t+r}$ not being a null measure. We first give the existence of $\lim_{t\to \infty}\mathbf Q_{t,r}$ and $\lim_{r\to \infty}\mathbf Q_{t,r}$, and then show that an $L\log L$-type condition is equivalent to the existence of the double limits: $\lim_{r\to \infty} \lim_{t\to\infty}\mathbf Q_{t, r}$ and $\lim_{t\to \infty} \lim_{r\to\infty}\mathbf Q_{t, r}$. Finally, when the $L\log L$-type condition holds, we show that those double limits, and $\lim_{r,t\to \infty}\mathbf Q_{t,r}$, are the same.

math.PR↗
arXiv2020

Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes, II

This paper is a continuation of our recent paper (Elect. J. Probab. 24 (2019), no. 141) and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes $(X_t)_{t\geq 0}$ with branching mechanisms of infinite second moment. In the aforementioned paper, we proved stable central limit theorems for $X_t(f) $ for some functions $f$ of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for $X_t(f) $ for all functions $f$ of polynomial growth. In this note, we show that the limit stable random variables in the three different regimes are independent, and as a consequence, we get stable central limit theorems for $X_t(f) $ for all functions $f$ of polynomial growth.

math.PR↗
arXiv2019

Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

In this paper, we study the asymptotic behavior of a supercritical $(ξ,ψ)$-superprocess $(X_t)_{t\geq 0}$ whose underlying spatial motion $ξ$ is an Ornstein-Uhlenbeck process on $\mathbb R^d$ with generator $L = \frac{1}{2}σ^2Δ- b x \cdot \nabla$ where $σ, b >0$; and whose branching mechanism $ψ$ satisfies Grey's condition and some perturbation condition which guarantees that, when $z\to 0$, $ψ(z)=-αz + ηz^{1+β} (1+o(1))$ with $α> 0$, $η>0$ and $β\in (0, 1)$. Some law of large numbers and $(1+β)$-stable central limit theorems are established for $(X_t(f) )_{t\geq 0}$, where the function $f$ is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding the branching rate being relatively small, large or critical at a balanced value.

math.PR↗
arXiv2019

Limit theorems for a class of critical superprocesses with stable branching

We consider a critical superprocess $\{X;\mathbf P_μ\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index $γ_0 > 1$. We first show that, under some conditions, $\mathbf P_μ(\|X_t\|\neq 0)$ converges to $0$ as $t\to \infty$ and is regularly varying with index $(γ_0-1)^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\{X_t(f);\mathbf P_μ(\cdot|\|X_t\|\neq 0)\}$, after appropriate rescaling, converges weakly to a positive random variable $\mathbf z^{(γ_0-1)}$ with Laplace transform $E[e^{-u\mathbf z^{(γ_0-1)}}]=1-(1+u^{-(γ_0-1)})^{-1/(γ_0-1)}.$

math.PR↗
arXiv2019

Spine decompositions and limit theorems for a class of critical superprocesses

In this paper, we first establish a decomposition theorem for size-biased Poisson random measures. As consequences of this decomposition theorem, we get a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yaglom's exponential limit law for critical superprocesses.

math.PR↗