Effect of stochastic perturbations for front propagation in Kolmogorov Petrovskii Piscunov equations

Abstract

This article considers equations of Kolmogorov Petrovskii Piscunov type in one space dimension, with stochastic perturbation: ∂tu=κ2uxx+u(1−u)dt+ϵu∂tζu0(x)=1(−∞,−1Nlog2)(x)+12e−Nx1[−1Nlog2,+∞)(x)where the stochastic differential is taken in the sense of Itô and ζ is a Gaussian random field satisfying Eζ=0 and Eζ(s,x)ζ(t,y)=(s∧t)Γ(x−y). Two situations are considered: firstly, ζ is simply a standard Wiener process (i.e. Γ≡1): secondly, Γ∈C∞(R) with lim|z|→+∞|Γ(z)|=0.The results are as follows: in the first situation (standard Wiener process: i.e. Γ(x)≡1), there is a non-degenerate travelling wave front if and only if ϵ22<1, with asymptotic wave speed max2κ(1−ϵ22),1N(1−ϵ22)+κN21{N<2κ(1−ϵ22)}; the noise slows the wave speed. If the stochastic integral is taken instead in the sense of Stratonovich, then the asymptotic wave speed is the classical McKean wave speed and does not depend on ϵ.In the second situation (noise with spatial covariance which decays to 0 at ±∞, stochastic integral taken in the sense of Itô), a travelling front can be defined for all ϵ>0. Its average asymptotic speed does not depend on ϵ and is the classical wave speed of the unperturbed KPP equation.