Construction and analysis of sticky reflected diffusions

Abstract

We give a Dirichlet form approach for the construction of distorted Brownian motion in a bounded domain $\Omega$ of $\mathbb{R}^d$, $d \geq 1$, with boundary $\Gamma$, where the behavior at the boundary is sticky. The construction covers the case of a static boundary behavior as well as the case of a diffusion on the hypersurface $\Gamma$ (for $d \geq 2)$. More precisely, we consider the state space $\overline{\Omega}=\Omega \stackrel{.}{\cup} \Gamma$, the process is a diffusion process inside $\Omega$, the occupation time of the process on the boundary $\Gamma$ is positive and the process may diffuse on $\Gamma$ as long as it sticks on the boundary. The problem is formulated in an $L^2$-setting and the construction is formulated under weak assumptions on the coefficients and $\Gamma$. In order to analyze the process we assume a $C^2$-boundary and some weak differentiability conditions. In this case, we deduce that the process is also a solution to a given SDE for quasi every starting point in $\overline{\Omega}$ with respect to the underyling Dirichlet form. Under the addtional condition that $\{ \varrho =0 \}$ is of capacity zero, we prove ergodicity of the constructed process and consequently, we verify that the boundary behavior is indeed sticky. Moreover, we show ($\mathcal{L}^p$-)strong Feller properties which allow to characterize the constructed process even for every starting point in $\overline{\Omega} \backslash \{ \varrho=0\}$.