Inverse problems for stochastic parabolic equations with additive noise

In this paper, we study two inverse problems for stochastic parabolic equations with additive noise. One is to determinate the history of a stochastic heat process and the random heat source simultaneously by the observation at the final time 𝑇. For this inverse problem, we obtain a conditional stability result. The other one is an inverse source problem to determine two kinds of sources simultaneously by the observation at the final time and on the lateral boundary. The main tool for solving the inverse problems is a new global Carleman estimate for the stochastic parabolic equation.

Rescaling nonlinear noise for 1D stochastic parabolic equations

We show regularization effect of nonlinear gradient noise to the solution of 1D stochastic parabolic equation. We demonstrate convergence to a martingale (independent upon space variable) when we rescale noise at the extremum points of the process.