Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems

In this paper, we establish a global Carleman estimate for stochastic parabolic equations. Based on this estimate, we study 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

Unique Continuation for Stochastic Heat Equations

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

On terminal value problems for bi-parabolic equations driven by Wiener process and fractional Brownian motions

In this paper, we study two terminal value problems (TVPs) for stochastic bi-parabolic equations perturbed by standard Brownian motion and fractional Brownian motion with Hurst parameter h ∈ ( 1/2 , 1) separately. For each problem, we provide a representation for the mild solution and find the space where the existence of the solution is guaranteed. Additionally, we show clearly that the solution of each problem is not stable, which leads to the ill-posedness of each problem. Finally, we propose two regularization results for both considered problems by using the filter regularization method