Small Noise Asymptotics of Traveling Waves

#### Carl Mueller, L. Mytnik, J. Quastel

2008, v.14, №3, 333-342

ABSTRACT

We consider the randomly perturbed Fisher - KPP equation \begin{equation*} \partial_t u = \partial_x^2 u + u(1-u) + \epsilon \sqrt{u(1-u)}\dot W. \end{equation*} where $\dot W = \dot W(t,x)$ is a space-time white noise. We discuss our proof of the Brunet - Derrida conjecture that the speed of traveling fronts for small $\epsilon$ is \begin{equation*} 2-\pi^2 |\log \epsilon^2|^{-2} \end{equation*} with an error of order $(\log|\log\epsilon|)|\log\epsilon|^{-3}$.

Keywords: heat equation,white noise,stochastic partial differentialequations