An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs

Abstract

We show uniqueness in law for the critical SPDE $$d{X}_t=A{X}_{t}dt+(-A{)}^{1/2}F(X(t))dt+d{W}_t,X_0=x\in H,$$ where $A$ $:dom(A)\subset H\to H$ 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 $F:H\to H$ 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 also establish a new optimal regularity result for the Kolmogorov equation $\lambda u-Lu=f$ on $H$, where $\lambda >0$, $f:H\to \mathbb{R}$ is Borel and bounded and $L$ is the Ornstein-Uhlenbeck operator related to the SPDE when $F=0$. In particular we show that the first derivative $Du:H\to H$ verifies $Du(x)\in \text{dom}((-A{)}^{1/2})$, for any $x\in H,$ and moreover $$\underset{x\in H}{\sup }|(-A{)}^{1/2}Du(x){|}_H=\Vert (-A{)}^{1/2}Du{\Vert }_0\le C\Vert f{\Vert }_0.$$

arXiv commentto appear in AOP

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• Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift
• Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations
• Strong uniqueness for SDEs in Hilbert spaces with nonregular drift
• On a Class of Martingale Problems on Banach Spaces
• Schauder theorems for a class of (pseudo-)differential operators on finite and infinite dimensional state spaces

Cited by in corpus (1)

• Exponential ergodicity of stochastic heat equations with Hölder coefficients