AN OPTIMAL REGULARITY RESULT FOR KOLMOGOROV EQUATIONS AND WEAK UNIQUENESS FOR SOME CRITICAL SPDES

• 成果类型: Article
• 署名作者: Priola, Enrico
• 署名单位: University of Pavia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1482
• 发表日期: 2021
• 页码: 1310-1346
• 关键词: hilbert-spaces; EXISTENCE; Operators; sdes

摘要

We show uniqueness in law for the critical SPDE dX(t) = AX(t) dt + (-A)(1/2) F(X(t)) dt + dW(t), X-0 = x is an element of H, where A : dom(A) subset of H -> H is a negative definite self-adjoint operator on a separable Hilbert space H having A(-1) of trace class and W is a cylindrical Wiener process on H. Here, F : H -> H can be continuous with, at most, linear growth (some functions F which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for three-dimensional stochastic Cahn-Hilliard-type equations which have interesting applications. To get weak uniqueness, we use an infinite dimensional localization principle and also establish a new optimal regularity result for the Kolmogorov equation lambda u - Lu = f associated to the SPDE when F = 0 (lambda > 0, f : H -> R Borel and bounded). In particular, we prove that the first derivative Du(x) belongs to dom((-A)(1/2)), for any x is an element of H, and sup(x is an element of H) vertical bar(-A)(1/2) Du(x)vertical bar H = parallel to(-A)(1/2) Du parallel to(0) <= C parallel to f parallel to(0).