Stochastic integrals for spde's: a comparison

TL;DR

This paper compares various stochastic integral theories, including Walsh's and Da Prato-Zabczyk's, for analyzing stochastic partial differential equations driven by spatially homogeneous Gaussian noise, highlighting their differences and applications.

Contribution

It provides a comprehensive comparison of stochastic integral frameworks and demonstrates their use in solving stochastic heat and wave equations with Gaussian noise.

Findings

Different theories produce comparable solutions for SPDEs

Theories have distinct mathematical frameworks and applicability

Comparison clarifies the strengths and limitations of each approach

Abstract

We present the Walsh theory of stochastic integrals with respect to martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes and some other approaches to stochastic integration, and we explore the links between these theories. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories.