A Note on variational solutions to SPDE perturbed by Gaussian noise in a general class

TL;DR

This paper investigates the existence and uniqueness of variational solutions to a class of SPDEs driven by Gaussian noise, extending the theory to equations with nonlinear operators and infinite-dimensional Gaussian processes.

Contribution

It establishes conditions for existence and uniqueness of solutions to SPDEs with Gaussian noise and nonlinear operators in a Hilbert space setting.

Findings

Proves existence of variational solutions under monotonicity conditions.

Establishes uniqueness of solutions for the considered class of SPDEs.

Extends variational solution theory to equations driven by infinite-dimensional Gaussian processes.

Abstract

This note deals with existence and uniqueness of (variational) solutions to the following type of stochastic partial differential equations on a Hilbert space H dX(t) = A(t,X(t))dt + B(t,X(t))dW(t) + h(t) dG(t) where A and B are random nonlinear operators satisfying monotonicity conditions and G is an infinite dimensional Gaussian process adapted to the same filtration as the cylindrical Wiener pocess W(t), t >= 0.