The Speed of a Random Front for Stochastic Reaction–Diffusion Equations with Strong Noise

Abstract

We study the asymptotic speed of a random front for solutions $u_t(x)$ to stochastic reaction–diffusion equations of the form

$$\begin{aligned} \partial _tu=\frac{1}{2}\partial _x^2u+f(u)+\sigma \sqrt{u(1-u)}{\dot{W}}(t,x),~t\ge 0,~x\in {\mathbb {R}}, \end{aligned}$$

arising in population genetics. Here, f is a continuous function with $f(0)=f(1)=0$, and such that $|f(u)|\le K|u(1-u)|^\gamma$ with $\gamma \ge 1/2$, and ${\dot{W}}(t,x)$ is a space-time Gaussian white noise. We assume that the initial condition $u_0(x)$ satisfies $0\le u_0(x)\le 1$ for all $x\in {\mathbb {R}}$, $u_0(x)=1$ for $x<L_0$ and $u_0(x)=0$ for $x>R_0$. We show that when $\sigma >0$, for each $t>0$ there exist $R(u_t)<+\infty$ and $L(u_t)<-\infty$ such that $u_t(x)=0$ for $x>R(u_t)$ and $u_t(x)=1$ for $x<L(u_t)$ even if f is not Lipschitz. We also show that for all $\sigma >0$ there exists a finite deterministic speed $V(\sigma )\in {\mathbb {R}}$ so that $R(u_t)/t\rightarrow V(\sigma )$ as $t\rightarrow +\infty$, almost surely. This is in dramatic contrast with the deterministic case $\sigma =0$ for nonlinearities of the type $f(u)=u^m(1-u)$ with $0<m<1$ when solutions converge to 1 uniformly on ${\mathbb {R}}$ as $t\rightarrow +\infty$. Finally, we prove that when $\gamma >1/2$ there exists $c_f\in {\mathbb {R}}$, so that $\sigma ^2V(\sigma )\rightarrow c_f$ as $\sigma \rightarrow +\infty$ and give a characterization of $c_f$. The last result complements a lower bound obtained by Conlon and Doering (J Stat Phys 120(3–4):421–477, 2005) for the special case of $f(u)=u(1-u)$ where a duality argument is available.