ON THE VALLEYS OF THE STOCHASTIC HEAT EQUATION

• 成果类型: Article
• 署名作者: Khoshnevisan, Davar; Kim, Kunwoo; Mueller, Carl
• 署名单位: Utah System of Higher Education; University of Utah; Pohang University of Science & Technology (POSTECH); University of Rochester
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1988
• 发表日期: 2024
• 页码: 1177-1198
• 关键词: multiplicative noise; DISSIPATION

摘要

We consider a generalization of the parabolic Anderson model driven by space-time white noise, also called the stochastic heat equation, on the real line: partial derivative(t) u(t, x) = 1/2 partial derivative(2)(x)u(t, x) + sigma(u(t, x)xi(t, x) for t > 0 and x is an element of R. High peaks of solutions have been extensively studied under the name of intermittency, but less is known about spatial regions between peaks, which we may loosely refer to as valleys. We present two results about the valleys of the solution. Our first theorem provides information about the size of valleys and the supremum of the solution u(t, x) over a valley. More precisely, when the initial function u(0)(x) = 1 for all x is an element of R, we show that the supremum of the solution over a valley vanishes as t -> infinity, and we establish an upper bound of exp{-const center dot t(1/3)} for u(t, x) when x lies in a valley. We demonstrate also that the length of a valley grows at least as exp{+const center dot t(1/3)} as t -> infinity. Our second theorem asserts that the length of the valleys are eventually infinite when the initial function u(0, x) has subgaussian tails.