A note on space–time Hölder regularity of mild solutions to stochastic cauchy problems in L^p-spaces

This paper revisits the Hölder regularity of mild solutions of parabolic stochastic Cauchy problems in Lebesgue spaces L^p(O), with p ≥ 2 and O ⊂ ℝ^d a bounded domain. We find conditions on p,β and γ under which the mild solution has almost surely trajectories in C^β([0,T ]; C^γ (Ō). 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 Brzeźniak (Stochastics Stochastics Rep. 61 (1997) 245–295).