Stochastic Partial Differential Equations with Unbounded and Degenerate Coefficients

TL;DR

This paper investigates linear second order stochastic partial differential equations with unbounded, degenerate, and non-smooth coefficients, establishing existence, uniqueness, and integrability results, and applies these to nonlinear filtering and degenerate nonlinear SPDEs.

Contribution

It extends DiPerna-Lions theory to SPDEs with unbounded and degenerate coefficients, providing new conditions for solutions and applying them to nonlinear filtering.

Findings

Established existence and uniqueness conditions for degenerate SPDEs.

Proved $L^1$-integrability and maximal principles for solutions.

Applied results to nonlinear filtering problems.

Abstract

In this article, using DiPerna-Lions theory \cite{Di-Li}, we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and uniqueness. Moreover, we also prove the $L^1$-integrability and a general maximal principle for generalized solutions of SPDEs. As applications, we study nonlinear filtering problem and also obtain the existence and uniqueness of generalized solutions for a degenerate nonlinear SPDE.