Unique continuation for stochastic heat equations

Lü, Qi; Yin, Zhongqi

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 378-398.

We establish a unique continuation property for stochastic heat equations evolving in a domain $G\subset R^n$($n\in N$). 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.

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