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

Abstract

We consider the $[0,1]$-valued solution $(u_{t,x}:t\ge 0,x\in \mathbb{R})$ 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, $\epsilon >0$ 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 $\epsilon$.

References in corpus (1)

• Shift in the velocity of a front due to a cut-off