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

TL;DR

This paper investigates the space-time Holder regularity of mild solutions to stochastic parabolic Cauchy problems in Lebesgue spaces, establishing conditions for regularity that are independent of the noise's Cameron-Martin space.

Contribution

It provides new conditions on p, β, and γ ensuring regularity of solutions in Lebesgue spaces, extending previous results without dependence on the noise's Hilbert space.

Findings

Conditions on p, β, γ for regularity are established.

Regularity results are independent of the Cameron-Martin space.

Uses a regularity result for stochastic convolutions in M-type 2 Banach spaces.

Abstract

This paper revisits the H\"{o}lder regularity of mild solutions of parabolic stochastic Cauchy problems in Lebesgue spaces $L^p(\mathcal{O}),$ with $p\geq 2$ 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. Brze\'zniak (Stoch. Stoch. Rep. Vol. 61, Iss. 3-4, 1997).