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 \(L^2({\mathbb{R}}^d, \varrho\mu)\). 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 séjour time of \(X\) on \(A\). Hence we obtain a stochastic process \(X\) with permeable sticky behaviour on \(A\).