Solution theory to Semilinear Hyperbolic Stochastic Partial Differential Equations with polynomially bounded coefficients

TL;DR

This paper establishes existence, uniqueness, and regularity of mild solutions for a class of semilinear hyperbolic stochastic PDEs with polynomially bounded coefficients, under specific spectral and initial data conditions.

Contribution

It provides new conditions for existence and uniqueness of solutions to semilinear hyperbolic SPDEs with polynomially bounded coefficients, including regularity results.

Findings

Existence and uniqueness of mild solutions under spectral measure conditions

Regularity results for solutions in suitable functional classes

Conditions on initial data and stochastic terms for solution well-posedness

Abstract

We study mild solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable hyperbolicity hypotheses on the linear part. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that mild solutions exist and are unique in suitably chosen functional classes. More precisely, function-valued solutions are obtained, as well as a regularity result.