REGULARIZATION BY NOISE AND FLOWS OF SOLUTIONS FOR A STOCHASTIC HEAT EQUATION

• 成果类型: Article
• 署名作者: Butkovsky, Oleg; Mytnik, Leonid
• 署名单位: Technion Israel Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1259
• 发表日期: 2019
• 页码: 165-212
• 关键词: white-noise; sdes; uniqueness

摘要

Motivated by the regularization by noise phenomenon for SDEs, we prove existence and uniqueness of the flow of solutions for the non-Lipschitz stochastic heat equation partial derivative u/partial derivative t = 1/2 partial derivative(2)u/partial derivative z(2) + b(u(t,z)) + <(W)single over dot>(t,z), where <(W)single over dot> is a space-time white noise on R+ x R and b is a bounded measurable function on R. As a byproduct of our proof, we also establish the so-called path-by-path uniqueness for any initial condition in a certain class on the same set of probability one. To obtain these results, we develop a new approach that extends Davie's method (2007) to the context of stochastic partial differential equations.