Stochastic equations with boundary noise

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to semilinear parabolic equations with boundary and interior noise, using an $L^p$ framework and reformulating boundary noise as a perturbation.

Contribution

It introduces a novel approach to handle boundary noise by reformulating it as a perturbation within stochastic evolution equations in extrapolation spaces.

Findings

Proved existence and uniqueness of mild and weak solutions.

Established pathwise regularity results.

Reformulated boundary noise as a perturbation in stochastic evolution equations.

Abstract

We study the wellposedness and pathwise regularity of semilinear non-autonomous parabolic evolution equations with boundary and interior noise in an $L^p$ setting. We obtain existence and uniqueness of mild and weak solutions. The boundary noise term is reformulated as a perturbation of a stochastic evolution equation with values in extrapolation spaces.