The Speed of a Random Front for Stochastic Reaction–Diffusion Equations with Strong Noise

We study the asymptotic speed of a random front for solutions u_t(x) to stochastic reaction–diffusion equations of the form ∂tu=12∂x2u+f(u)+σu(1-u)W˙(t,x),t≥0,x∈R,arising in population genetics. Here, f is a continuous function with f(0) = f(1) = 0 , and such that | f(u) | ≤ K| u(1 - u) | ^γ with γ≥ 1 / 2 , and W˙ (t, x) is a space-time Gaussian white noise. We assume that the initial condition u(x) satisfies 0 ≤ u(x) ≤ 1 for all x∈ R, u(x) = 1 for x< L and u(x) = 0 for x> R. We show that when σ> 0 , for each t> 0 there exist R(u_t) < + ∞ and L(u_t) < - ∞ such that u_t(x) = 0 for x> R(u_t) and u_t(x) = 1 for x< L(u_t) even if f is not Lipschitz. We also show that for all σ> 0 there exists a finite deterministic speed V(σ) ∈ R so that R(u_t) / t→ V(σ) as t→ + ∞, almost surely. This is in dramatic contrast with the deterministic case σ= 0 for nonlinearities of the type f(u) = u^m(1 - u) with 0 < m< 1 when solutions converge to 1 uniformly on R as t→ + ∞. Finally, we prove that when γ> 1 / 2 there exists c_f∈ R, so that σ²V(σ) → c_f as σ→ + ∞ and give a characterization of c_f. The last result complements a lower bound obtained by Conlon and Doering (J Stat Phys 120(3–4):421–477, 2005) for the special case of f(u) = u(1 - u) where a duality argument is available.