SPD

Abstract

In this talk we discuss our recent progress on maximal regularity of convolutions with respect to Brownian motion. Under certain conditions, we show that stochastic convolutions \[\int_0^t S(t-s) f(s) d W(s)\] satisfy optimal $L^p$-regularity estimates and maximal estimates. Here $S$ is an analytic semigroup on an $L^q$-space. We also provide counterexamples to certain limiting cases and explain the applications to stochastic evolution equations. The results extend and unifies various known maximal $L^p$-regularity results from the literature. In particular, our framework covers and extends the important results of Krylov for the heat semigroup on $\mathbb{R}^d$.