| Abstract: |
| The starting point is a gradient Dirichlet form with respect to $\lambda^d_\varrho$ on the space $L^2({\mathbb R}^d, \mu_\varrho)$. 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 show positive s\`{e}jour time of $X$ on $A$. Hence we obtain a stochastic process $X$ with permeable sticky behaviour on $A$. |
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