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May 2021 An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs
Enrico Priola
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Ann. Probab. 49(3): 1310-1346 (May 2021). DOI: 10.1214/20-AOP1482

Abstract

We show uniqueness in law for the critical SPDE

dXt=AXtdt+(A)1/2F(X(t))dt+dWt,X0=xH,

where A:dom(A)HH is a negative definite self-adjoint operator on a separable Hilbert space H having A1 of trace class and W is a cylindrical Wiener process on H. Here, F:HH 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 λuLu=f associated to the SPDE when F=0 (λ>0, f:HR Borel and bounded). In particular, we prove that the first derivative Du(x) belongs to dom((A)1/2), for any xH, and supxH|(A)1/2Du(x)|H=(A)1/2Du0Cf0.

Citation

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Enrico Priola. "An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs." Ann. Probab. 49 (3) 1310 - 1346, May 2021. https://doi.org/10.1214/20-AOP1482

Information

Received: 1 December 2019; Revised: 1 September 2020; Published: May 2021
First available in Project Euclid: 7 April 2021

Digital Object Identifier: 10.1214/20-AOP1482

Subjects:
Primary: 35R60, 60H15
Secondary: 35R15

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 3 • May 2021
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