Stability and regularization for ill-posed Cauchy problem of a stochastic parabolic differential equation

TL;DR

This paper addresses the ill-posed Cauchy problem for stochastic parabolic equations by establishing stability estimates, analyzing regularization methods, and demonstrating their effectiveness through numerical examples.

Contribution

It introduces a Carleman estimate for stochastic parabolic equations and applies kernel-based Tikhonov regularization with theoretical and numerical validation.

Findings

Established a Carleman estimate for stochastic parabolic equations

Derived stability and convergence rates for Tikhonov regularization

Validated the approach with numerical experiments

Abstract

In this paper, we investigate an ill-posed Cauchy problem involving a stochastic parabolic equation. We first establish a Carleman estimate for this equation. Leveraging this estimate, we derive the conditional stability and convergence rate of the Tikhonov regularization method for the aforementioned ill-posed Cauchy problem. To complement our theoretical analysis, we employ kernel-based learning theory to implement the completed Tikhonov regularization method for several numerical examples.