Zhenyao Sun

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 \[\partial_t u= \partial_x^2 u + f(u) + \epsilon \sqrt{u(1-u)} \dot W.\] Here, $W$ is a space-time white noise, $\epsilon > 0$ 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 $V_{f,\epsilon}$, and under slightly stronger conditions on $f$, the asymptotic behavior of $V_{f,\epsilon}$ was derived as the noise strength $\epsilon$ approaches $\infty$. In this paper we complement the above result by obtaining the asymptotic behavior of $V_{f,\epsilon}$ as the noise strength $\epsilon$ 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 $V_{f,\epsilon}$ is comparable to $\epsilon^{-2\frac{1-p}{1+p}}$ for sufficiently small $\epsilon$.

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.

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.

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.