We consider the $[0,1]$-valued solution $(u_{t,x}:tgeq 0, xin mathbb R)$ 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 $sup_{zin [0,1]}|f(z)|/ sqrt{z(1-z)} < infty.$ 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. 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 $pin [1/2,1)$, if $f(z)$ is non-negative and is comparable to $z^p$ for sufficiently small $z$, then $V_{f,epsilon}$ is comparable to $epsilon^{-2frac{1-p}{1+p}}$ for sufficiently small $epsilon$.
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 $(X_t)_{tgeq 0}$ with branching mechanisms of infinite se
cond 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.
In this paper, we study the asymptotic behavior of a supercritical $(xi,psi)$-superprocess $(X_t)_{tgeq 0}$ whose underlying spatial motion $xi$ is an Ornstein-Uhlenbeck process on $mathbb R^d$ with generator $L = frac{1}{2}sigma^2Delta - b x cdot a
bla$ where $sigma, b >0$; and whose branching mechanism $psi$ satisfies Greys condition and some perturbation condition which guarantees that, when $zto 0$, $psi(z)=-alpha z + eta z^{1+beta} (1+o(1))$ with $alpha > 0$, $eta>0$ and $betain (0, 1)$. Some law of large numbers and $(1+beta)$-stable central limit theorems are established for $(X_t(f) )_{tgeq 0}$, 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.
We consider a critical superprocess ${X;mathbf P_mu}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index $gamma_0 > 1$. We first show that, under some conditions, $mathbf P_{mu}(|X_t| eq 0)$ conver
ges to $0$ as $tto infty$ and is regularly varying with index $(gamma_0-1)^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of ${X_t(f);mathbf P_mu(cdot||X_t| eq 0)}$, after appropriate rescaling, converges weakly to a positive random variable $mathbf z^{(gamma_0-1)}$ with Laplace transform $E[e^{-umathbf z^{(gamma_0-1)}}]=1-(1+u^{-(gamma_0-1)})^{-1/(gamma_0-1)}.$
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 superprocesse
s. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yagloms exponential limit law for critical superprocesses.
In this note we propose a two-spine decomposition of the critical Galton-Watson tree and use this decomposition to give a probabilistic proof of Yagloms theorem.
