Null controllability for one-dimensional stochastic heat equations with mixed Dirichlet-dynamic boundary conditions

In this paper, we study the null controllability of one-dimensional forward and backward linear stochastic heat equations with mixed Dirichlet-dynamic boundary conditions. Our equations incorporate noise not only within the domain but also at the boundary, represented by a two-dimensional Brownian motion. The primary tool will be global Carleman estimates, which yield the appropriate observability inequalities for the related adjoint systems. Hence, by classical duality arguments, we establish the corresponding null controllability results. Specifically, we first establish a Carleman estimate for a general adjoint backward stochastic heat equation using a weighted identity method. This approach combines two weighted identities: one for a stochastic parabolic operator and the other for a stochastic transport operator. Subsequently, we derive a Carleman estimate for a general adjoint forward stochastic heat equation by employing a duality method.