Unique Continuation for Stochastic Heat Equations

TL;DR

This paper proves a unique continuation property for stochastic heat equations, showing solutions are determined by their values on subdomains, and provides quantitative estimates and controllability results for such equations.

Contribution

It establishes a novel unique continuation property for stochastic heat equations and derives related observability and controllability results.

Findings

Unique continuation property proven for stochastic heat equations.

Quantitative estimates provided for convex bounded domains.

Applications include observability and null controllability results.

Abstract

We establish a unique continuation property for stochastic heat equations evolving in a bounded domain $G$. Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of $G$ at any given positive time constant. Further, when $G$ is convex and bounded, we also give a quantitative version of the unique continuation property. As applications, we get an observability estimate for stochastic heat equations, an approximate result and a null controllability result for a backward stochastic heat equation.