Strong uniqueness for SPDEs and regularity of Kolmogorov equations

Pathwise uniqueness plays a crucial role in the investigation of the existence of strong solutions to Stochastic Differential Equations (SDEs) since the seminal result by Yamada and Watanabe in 1971, where they proved that weak existence and pathwise uniqueness imply strong existence. A few years later, Zvonkin introduced the so-called Zvonkin transformation, which allows to remove a drift term by means of a suitable change of coordinates using the Ito formula, and then applies the result of Yamada and Watanabe to construct strong solutions to a class of SDEs with rough drift coefficient. In recent years, these techniques have been extended to Stochastic Partial Differential Equations (SPDEs) . One of the main tools for proving pathwise uniqueness in infinite dimensions is the so-called Itô-Tanaka trick, which involves replacing the "bad" drift term with the solution to a suitable Kolmogorov equation. In this talk, we will examine these topics for a class of parabolic and hyperbolic SPDEs.