A Unified Approach to Stochastic Evolution Equations Using the Skorokhod Integral

TL;DR

This paper develops a unified framework for solving stochastic evolution equations driven by Gaussian noise, accommodating various noise types and operator orders, by employing the Cameron-Martin Wiener chaos decomposition to handle solutions in weighted spaces.

Contribution

It introduces a novel approach using the Cameron-Martin Wiener chaos decomposition to address stochastic evolution equations with non-square-integrable solutions in weighted spaces.

Findings

The approach effectively handles equations with different noise types.

Solutions are characterized in natural weighted spaces.

The method generalizes previous solutions for stochastic evolution equations.

Abstract

We study stochastic evolution equations driven by Gaussian noise. The key features of the model are that the operators in the deterministic and stochastic parts can have the same order and the noise can be time-only, space-only, or space-time. Even the simplest equations of this kind do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition leads to natural weights and a natural replacement of the square integrability condition.