Parabolic stochastic partial differential equations with dynamical boundary conditions

Abstract

We consider systems of nonlinear parabolic stochastic partial differential equations with dynamical boundary conditions. These boundary conditions are qualitatively different from the standard, like Dirichlet, or Neumann, or Robin boundary conditions. Such conditions contain a time derivative and can be used to describe mathematical models with a dynamics on the boundary. In our model the noise is acting in the domain but also on the boundary and is presented as the temporal generalized derivative of an infinite-dimensional Wiener processes. In addition, we have coefficients for the spatial differential operators depending on space and time. We prove existence and uniqueness of mild solutions to these stochastic partial differential equations and study the properties of these solutions. The stochastic equation on the boundary contains a parameter. If this parameter becomes small, then the equation on the boundary has an interpretation in a fast time scale. We prove the relative compactness of the distribution of the solution if the parameter mentioned tends to 0.