Carleman estimates for a stochastic degenerate parabolic equation and applications to null controllability and an inverse random source problem

TL;DR

This paper develops new Carleman estimates for stochastic degenerate parabolic equations, enabling null controllability results and stability analysis for inverse problems involving random sources.

Contribution

It introduces two novel Carleman estimates for stochastic degenerate parabolic equations, facilitating controllability and inverse problem solutions.

Findings

Proved null controllability of the forward stochastic degenerate parabolic equation.

Established Lipschitz stability for an inverse problem with a time-dependent random source.

Derived Carleman estimates with singular and regular weight functions.

Abstract

In this paper, we establish two Carleman estimates for a stochastic degenerate parabolic equation. The first one is for the backward stochastic degenerate parabolic equation with singular weight function. Combining this Carleman estimate and an approximate argument, we prove the null controllability of the forward stochastic degenerate parabolic equation with the gradient term. The second one is for the forward stochastic degenerate parabolic equation with regular weighted function, based on which we obtain the Lipschitz stability for an inverse problem of determining a random source depending only on time in the forward stochastic degenerate parabolic equation.