A note on space-time Holder regularity of mild solutions to stochastic Cauchy problems in Lp-spaces

Abstract

This paper revisits the Hölder regularity of mild solutions of parabolic stochastic Cauchy problems in Lebesgue spaces $L^p(\mathcal{O}),$ with $p\ge 2$ and $\mathcal{O}\subset {\mathbb{R}}^d$ a bounded domain. We find conditions on $p,\beta$ and $\gamma$ under which the mild solution has almost surely trajectories in ${\mathcal{C}}^{\beta }([0,T];{\mathcal{C}}^{\gamma }(\bar{\mathcal{O}})).$ These conditions do not depend on the Cameron-Martin Hilbert space associated with the driving cylindrical noise. The main tool of this study is a regularity result for stochastic convolutions in M-type 2 Banach spaces by Z. Brzeźniak (Stoch. Stoch. Rep. Vol. 61, Iss. 3-4, 1997).