Stochastic integration in quasi-Banach spaces

TL;DR

This paper develops a stochastic integration framework for processes in quasi-Banach spaces using cylindrical Brownian motion, extending known Banach space results, with applications to stochastic heat equations involving Besov regularity.

Contribution

It introduces sufficient conditions for stochastic integrability in quasi-Banach spaces, broadening the scope of stochastic calculus beyond Banach spaces.

Findings

Established stochastic integrability criteria in quasi-Banach spaces.

Applied results to stochastic heat equations with Besov regularity.

Potential for adaptive wavelet methods in SPDEs.

Abstract

In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent $p\in (0,1]$. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.