Stochastic Integrals and Evolution Equations with Gaussian Random Fields

TL;DR

This paper develops a framework for stochastic integration with Gaussian fields, extending existing methods to random integrands and providing tools for analytical and numerical analysis of related evolution equations.

Contribution

It introduces an extension of the Ito-Skorokhod integral to Gaussian fields using chaos decomposition and Wick products, enhancing the study of stochastic evolution equations.

Findings

Defined the Ito-Skorokhod integral for Gaussian fields.

Provided analytical tools for stochastic differential equations.

Illustrated applications with examples of evolution equations.

Abstract

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of the chaos space of the random field, combined with the Wick product, leads to the \Ito-Skorokhod integral, and provides an efficient tool to study the integral, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields. Some examples of the corresponding stochastic differential equations are also considered.