Carleman Estimate for Stochastic Parabolic Equations and Inverse Stochastic Parabolic Problems

TL;DR

This paper develops a Carleman estimate for stochastic parabolic equations and applies it to solve inverse problems, including determining the process history and identifying sources, with stability and uniqueness results.

Contribution

The paper introduces a novel Carleman estimate for stochastic parabolic equations and uses it to address inverse problems with stability and uniqueness guarantees.

Findings

Established a global Carleman estimate for stochastic parabolic equations.

Derived conditional stability for the process history determination.

Proved uniqueness for the inverse source problem.

Abstract

In this paper, we establish a global Carleman estimate for stochastic parabolic equations. Based on this estimate, we solve two inverse problems for stochastic parabolic equations. One is concerned with a determination problem of the history of a stochastic heat process through the observation at the final time $T$, for which we obtain a conditional stability estimate. The other is an inverse source problem with observation on the lateral boundary. We derive the uniqueness of the source.