Random Travelling Waves for the KPP Equation with Noise

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) = sup{x ∈ 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.