Effect of Stochastic Perturbations for Front Propagation in Kolmogorov Petrovskii Piscunov Equations

TL;DR

This paper investigates how stochastic perturbations affect wave front propagation in Kolmogorov-Petrovskii-Piskunov equations, revealing conditions under which noise influences wave speed and front existence.

Contribution

It provides a detailed analysis of stochastic effects on wave fronts in KPP equations, distinguishing between different types of noise and their impact on wave speed and existence.

Findings

Wave speed slows down under Wiener noise when rac{ ext{noise intensity}^2}{2} < 1

Traveling fronts exist for all noise intensities in the smooth kernel case

Wave speed is unaffected by noise in the Stratonovich interpretation

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.