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

DOI10.1214/23-AOP1650arXiv2210.07096MaRDI QIDQ6413842

E. Priola

Publication date: 13 October 2022

Abstract: We show uniqueness in law for the critical SPDE �egin{eqnarray} label{qq1} dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,;; X_0 =x in H, end{eqnarray} where $A$ $:extdom(A)subsetHoH$ 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 $F:HoH$ can be locally H"older 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. We do not know if uniqueness holds under the sole assumption of continuity of $F$ plus growth condition as stated in [Priola, Ann. of Prob. 49 (2021)]. To get weak uniqueness we use an infinite dimensional localization principle and an optimal regularity result for the Kolmogorov equation $lambdau-Lu=f$ associated to the SPDE when $F=zinH$ is constant and $lambda>0$. This optimal result is similar to a theorem of [Da Prato, J. Evol. Eq. 3 (2003)].

Mathematics Subject Classification ID

Smoothness and regularity of solutions to PDEs (35B65) Stochastic partial differential equations (aspects of stochastic analysis) (60H15) PDEs with randomness, stochastic partial differential equations (35R60)

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