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 \[ \partial_tu=\farc{1}{2}\partial_x^2u+f(u)+σ\sqrt{u(1-u)}\dot{W}(t,x),~t\ge 0,~x\in\Rm, \] arising in population genetics. Here, $f$ is a continuous function with $f(0)=f(1)=0$, and such that~$|f(u)|\le K|u(1-u)|^γ$ with~$γ\ge 1/2$, 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 $x\in\Rm$, $u_0(x)=1$ for~$x<L_0$ and $ u_0(x)=0$ for~$x>R_0$. We show that when $σ>0$, 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 $σ>0$ there exists a finite deterministic speed~$V(σ)\in\Rm$ so that~$R(u_t)/t\to V(σ)$ as $t\to+\infty$, almost surely. This is in dramatic contrast with the deterministic case $σ=0$ for nonlinearities of the type $f(u)=u^m(1-u)$ with $0<m<1$ when solutions converge to $1$ uniformly on $\Rm$ as $t\to+\infty$. Finally, we prove that when $γ>1/2$ there exists $c_f\in\Rm$, so that~$σ^2V(σ)\to c_f$ as~$σ\to+\infty$ and give a characterization of $c_f$. The last result complements a lower bound obtained by Conlon and Doering \cite{cd05} for the special case of $f(u)=u(1-u)$ where a duality argument is available.