Brazilian Journal of Probability and Statistics

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

Rafael Serrano

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This paper revisits the Hölder regularity of mild solutions of parabolic stochastic Cauchy problems in Lebesgue spaces $L^{p}(\mathcal{O})$, with $p\geq2$ 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 Brzeźniak (Stochastics Stochastics Rep. 61 (1997) 245–295).

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Braz. J. Probab. Stat., Volume 29, Number 4 (2015), 767-777.

Received: July 2013
Accepted: April 2014
First available in Project Euclid: 17 September 2015

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Stochastic Cauchy problem additive cylindrical noise Hölder regularity stochastic convolution Lebesgue spaces


Serrano, Rafael. A note on space–time Hölder regularity of mild solutions to stochastic Cauchy problems in $L^{p}$-spaces. Braz. J. Probab. Stat. 29 (2015), no. 4, 767--777. doi:10.1214/14-BJPS245.

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