Effect of Noise on Front Propagation in Reaction-Diffusion equations of KPP type

We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equations $\partial_t u = \partial_x^2 u + u(1-u) + \epsilon \sqrt{u(1-u)}\dot W,$ and $\partial_t u = \partial_x^2 u + u(1-u) + \epsilon \sqrt{u}\dot W,$ where $\dot W= \dot W(t,x)$ is a space-time white noise. We prove the Brunet-Derrida conjecture that the speed of traveling fronts is asymptotically $2-\pi^2 |\log \epsilon^2|^{-2}$ up to a factor of order $(\log|\log\epsilon|)|\log\epsilon|^{-3}$.