The Speed of a Random Front for Stochastic Reaction–Diffusion Equations with Strong Noise
The Speed of a Random Front for Stochastic Reaction–Diffusion Equations with Strong Noise
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DOI：
10.1007/s00220-021-04084-0
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发表时间：
2019-03
影响因子：
2.4
通讯作者：
C. Mueller;L. Mytnik;L. Ryzhik
中科院分区：
文献类型：
--
作者：
C. Mueller;L. Mytnik;L. Ryzhik
我们研究随机前沿的渐近速度，用于解决形式为 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} 的随机反应扩散方程 \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{文档}$$\begin{对齐} \partial _tu=\frac{1}{2}\partial _x^2u+f(u)+\sigma \sqrt{u(1-u)}{\dot{W}}(t,x),~t\ge 0,~x\in {\mathbb {R}}, \end{aligned}$$\end{document}出现在群体遗传学中。这里，f 是一个连续函数，且 是一个时空高斯白噪声。我们假设初始条件满足 for all、for 和 for。我们证明，当，对于每一个存在时，并且对于和对于即使不是利普希茨。我们还表明，对于所有人来说，都存在有限的确定性速度，因此几乎可以肯定。这与当解一致收敛到 1 时类型非线性的确定性情况形成鲜明对比。最后，我们证明当存在时，so that as 并给出一个表征。最后的结果补充了 Conlon 和 Doering (J Stat Phys 120(3–4):421–477, 2005) 对于对偶论证可用的特殊情况获得的下界。
We study the asymptotic speed of a random front for solutionsto stochastic reaction–diffusion equations of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _tu=\frac{1}{2}\partial _x^2u+f(u)+\sigma \sqrt{u(1-u)}{\dot{W}}(t,x),~t\ge 0,~x\in {\mathbb {R}}, \end{aligned}$$\end{document}arising in population genetics. Here,fis a continuous function with, and such thatwith, andis a space-time Gaussian white noise. We assume that the initial conditionsatisfiesfor all,forandfor. We show that when, for eachthere existandsuch thatforandforeven iffis not Lipschitz. We also show that for allthere exists a finite deterministic speedso thatas, almost surely. This is in dramatic contrast with the deterministic casefor nonlinearities of the typewithwhen solutions converge to 1 uniformly onas. Finally, we prove that whenthere exists, so thatasand give a characterization of. 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 ofwhere a duality argument is available.