Random Travelling Waves for the KPP Equation with Noise
Journal of Functional Analysis, 128 (2):
439 - 498 (1995)DOI: 10.1006/jfan.1995.1038
Abstract
Consider the stochastic partial differential equation formula where Ẇ =Ẇ(t, x) is two-parameter while noise. Assume that u0 is a continuous function taking values in 0, 1 such that for some constant a $>$ 0, we have (C1) u0(x) = 1 for x $<$ −a.(C2) u0(x) = 0 for x $>$ a. Let the wavefront b(t) = supx ∈ R: u(t, x) $>$ 0. We show that for ϵ small enough and with probability 1, • limt→∞b(t)/t exists and lies in (0, ∞). This limit depends only on ϵ. •The law of v(t, x) ≡ u(t, b(t) + x) tends toward a stationary limit as t → ∞. We also analyze the length of the region a(t), b(t), which is the smallest closed interval containing the points x at which 0 $<$ u(t, x) $<$ 1. We show that the length of this region tends toward a stationary distribution. Thus, the wavefront does not degenerate.