Abstract
We consider the [0,1]-valued solution (ut,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 − u0,−x = u0,x = 0 for x large enough. Recently, it was proved in (Comm. Math. Phys. 384 (2021) 699–732) that the front of ut propagates with a finite deterministic speed Vf,ϵ, and under slightly stronger conditions on f , the asymptotic behavior of Vf,ϵ was derived as the noise strength ϵ approaches ∞. In this paper we complement the above result by obtaining the asymptotic behavior of Vf,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 zp for sufficiently small z, then Vf,ϵ is comparable to [Formula Presented] for sufficiently small ϵ.
Original language | English |
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Pages (from-to) | 2382-2414 |
Number of pages | 33 |
Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |
Volume | 60 |
Issue number | 4 |
DOIs | |
State | Published - Nov 2024 |
Keywords
- Reaction-diffusion equations
- Stochastic partial differential equations
- Traveling waves
- White noise
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty