# $L^p$-valued stochastic convolution integral driven by Volterra noise

## TL;DR

This paper investigates the space-time regularity of solutions to linear SPDEs driven by Volterra noise, demonstrating Hölder continuity in $L^p$ spaces through stochastic convolution analysis.

## Contribution

It introduces a novel approach to establish Hölder regularity of $L^p$-valued solutions using hypercontractivity in Banach spaces for Volterra-driven SPDEs.

## Key findings

- Solutions exhibit space-time Hölder continuity under certain conditions.
- The method applies hypercontractivity to Banach-space valued random variables.
- Regularity results are obtained for specific cases of Volterra noise.

## Abstract

Space-time regularity of linear stochastic partial differential equations is studied. The solution is defined in the mild sense in the state space $L^p$. The corresponding regularity is obtained by showing that the stochastic convolution integrals are H\"{o}lder continuous in a suitable function space. In particular cases, this allows to show space-time H\"{o}lder continuity of the solution. The main tool used is a hypercontractivity result on Banach-space valued random variables in a finite Wiener chaos.