TY - JOUR

T1 - The Speed of a Random Front for Stochastic Reaction–Diffusion Equations with Strong Noise

AU - Mueller, Carl

AU - Mytnik, Leonid

AU - Ryzhik, Lenya

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/6

Y1 - 2021/6

N2 - We study the asymptotic speed of a random front for solutions ut(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(ut) < + ∞ and L(ut) < - ∞ such that ut(x) = 0 for x> R(ut) and ut(x) = 1 for x< L(ut) 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(ut) / 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) = um(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 cf∈ R, so that σ2V(σ) → cf as σ→ + ∞ and give a characterization of cf. 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.

AB - We study the asymptotic speed of a random front for solutions ut(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(ut) < + ∞ and L(ut) < - ∞ such that ut(x) = 0 for x> R(ut) and ut(x) = 1 for x< L(ut) 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(ut) / 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) = um(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 cf∈ R, so that σ2V(σ) → cf as σ→ + ∞ and give a characterization of cf. 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.

UR - http://www.scopus.com/inward/record.url?scp=85105364174&partnerID=8YFLogxK

U2 - 10.1007/s00220-021-04084-0

DO - 10.1007/s00220-021-04084-0

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AN - SCOPUS:85105364174

SN - 0010-3616

VL - 384

SP - 699

EP - 732

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -