Boundary stabilization of the stochastic heat equation by proportional feedbacks

In this work we design an explicit random deterministic, finite-dimensional stabilizing boundary feedback to the null solution for the heat equation with noise perturbation. The simple form of the feedback allows us to write the solution of the corresponding closed-loop equation in a mild formulation via a kernel and use some techniques from the existing literature in order to show the stability of it. As far as we know, the present work is the first result on boundary feedback stabilization for stochastic parabolic-type equations, with the stability guaranteed independent of how large the level of the noise is.