COMPARISON PRINCIPLE FOR STOCHASTIC HEAT EQUATION ON Rd

• 成果类型: Article
• 署名作者: Chen, Le; Huang, Jingyu
• 署名单位: Nevada System of Higher Education (NSHE); University of Nevada Las Vegas; Utah System of Higher Education; University of Utah
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1277
• 发表日期: 2019
• 页码: 989-1035
• 关键词: parabolic anderson model; strict positivity; continuity

摘要

We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on R-d (partial derivative/partial derivative t - 1/2 Delta)u(t, x) = rho(u(t , x)) (M) over dot (t , x), for measure-valued initial data, where (M) over dot is a spatially homogeneous Gaussian noise that is white in time and rho is Lipschitz continuous. These results are obtained under the condition that integral(Rd) (1 + vertical bar xi vertical bar(2))(alpha-1)(f) over cap (d xi) < infinity for some alpha is an element of (0, 1], where <(f)over cap> is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang's condition, that is, alpha = 0. As some intermediate results, we obtain handy upper bounds for L-p (Omega)-moments of u(t, x) for all p >= 2, and also prove that u is a.s. Holder continuous with order alpha - epsilon in space and alpha/2 - epsilon in time for any small epsilon > 0.