Construction of distorted Brownian motion with permeable sticky behaviour on sets with Lebesgue measure zero

The starting point is a gradient Dirichlet form with respect to $\varrho\lambda^d$ on the space $L^2({\mathbb{R}}^d, \varrho\mu)$. Here $\lambda^d$ is the Lebesgue measure on ${\mathbb R}^d$, $\varrho$ a strictly positive density and $\mu$ puts weight on a set $A\subset {\mathbb R}^d$ with Lebesgue measure zero. We show that the Dirichlet form admits an associated stochastic process $X$. We derive an explicit representation of the corresponding generator if $A$ is a Lipschitz boundary. This representation together with the Fukushima decomposition identifies $X$ as a distorted Brownian motion with drift given by the logarithmic derivative of $\varrho$ in ${\mathbb R}^d \setminus A$. Furthermore, we prove $X$ to be irreducible and recurrent. Finally, via ergodicity we prove positive s\'ejour time of $X$ on $A$. Hence we obtain a stochastic process $X$ with permeable sticky behaviour on $A$.