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Skew Brownian diffusions across Koch interfaces. (English) Zbl 1367.60102

The authors consider a skew Brownian motion defined on a domain in \(\mathbb R^2\) which is bounded by a pre-Koch interface. Various boundary conditions on the domain are discussed and the asymptotic behaviour of the process is studied when the skewness parameter changes or the thickness of the boundary layer converges to zero. The paper uses the language of Dirichlet forms. In the first sections, the notions of a (pre-)Koch curve, positive continuous additive functionals and the corresponding potential theory are discussed. Then, using the toy case of the annulus, the collapse of the thin boundary layer is discussed from a probabilistic point of view. The main part of the paper consists of Sections 5 and 6, where the transmission condition on irregular interfaces is discussed (using Dirichlet forms and suitable Sobolev spaces). In order to describe the convergence behaviour, the Mosco convergence is used. The main result (Theorem 10 in Section 6) finally identifies the limiting process on the collapsed domain \(\overline\Omega\) – it is a elastic/partially reflected Brownian motion, a reflecting Brownian motion, or an absorbing Brownian motion – according to the way one shrinks the layer around \(\Omega\).

MSC:

60J65 Brownian motion
60J60 Diffusion processes
60J55 Local time and additive functionals
35J25 Boundary value problems for second-order elliptic equations
28A80 Fractals
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