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: \partial_t u = \left (\frac{\kappa}{2} u_{xx} + u(1-u) \right) dt + \epsilon u \partial_t \zeta where the stochastic differential is taken in the sense of It\^o and $\zeta$ is a Gaussian random field satisfying E [ \zeta ] = 0 and E [ \zeta(s,x)\zeta(t,y) ] = (s \wedge t) \Gamma (x-y). Two situations are considered: firstly, \zeta is simply a standard Wiener process (i.e. $\Gamma \equiv 1$): secondly, \Gamma \in C^\infty (\mathbb{R}) with \int_{-\infty}^\infty |\Gamma(z)| dz < +\infty. The results are as follows: in the first situation (standard Wiener process: i.e. \Gamma(x) \equiv 1), there is a non-degenerate travelling wave front if and only if \frac{\epsilon^2}{2} < 1, with asymptotic wave speed \max\left(\sqrt{2\kappa (1 - \frac{\epsilon^2}{2})}, \frac{1}{N}(1 - \frac{\epsilon^2}{2}) + \frac{\kappa N}{2}\right), the noise slows the wave speed. If the stochastic integral is taken instead in the sense of Stratonovich, then the asymptotic wave speed does not depend on \epsilon. In the second situation, a travelling front can be defined for all \epsilon > 0 and its asymptotic speed does not depend on \epsilon.