Effect of small noise on the speed of reaction-diffusion equations with non-Lipschitz drift

We consider the [0,1]-valued solution (u_t,x : t ≥ 0,x ∈ ℝ) to the one dimensional stochastic reaction diffusion equation with Wright-Fisher noise [Formula Presented]. Here, W is a space-time white noise, ϵ > 0 is the noise strength, and f is a continuous function on [0, 1] satisfying [Formula Presented]. We assume the initial data satisfies 1 − u_0,−x = u_0,x = 0 for x large enough. Recently, it was proved in (Comm. Math. Phys. 384 (2021) 699–732) that the front of u_t propagates with a finite deterministic speed V_f,ϵ, and under slightly stronger conditions on f , the asymptotic behavior of V_f,ϵ was derived as the noise strength ϵ approaches ∞. In this paper we complement the above result by obtaining the asymptotic behavior of V_f,ϵ as the noise strength ϵ approaches 0: for a given p ∈ [1/2, 1), if f (z) is non-negative and is comparable to z^p for sufficiently small z, then V_f,ϵ is comparable to [Formula Presented] for sufficiently small ϵ.