Reflecting diffusions on Lipschitz domains with cusps -- analytic construction and Skorohod representation
type: Item
instance of: scholarly article
title: Reflecting diffusions on Lipschitz domains with cusps -- analytic construction and Skorohod representation
zbMATH Open document ID: 0840.60070
DOI: 10.1007/BF01053454
author: Masatoshi Fukushima
author: Matsuyo Tomisaki
published in: Potential Analysis
publication date: 1996-02-01T00:00:00Z
review text: Let \(D \subset R^d\) be a Euclidean Lipschitz domain and let \(\Delta : = \sum_{ 1 \leq i, j \leq n} a_{ij} (x) \cdot {\partial^2 \over \partial x_i \cdot \partial x_j}\) be an elliptic operator in \(D\) whose coefficients are measurable, symmetric such that \[ c^{-1} |\xi |^2 \leq \sum_{1 \leq i,j \leq n} a_{ij} (x) \xi_i \xi_j \leq c |\xi |^2, \quad \text{for some constant } c > 1, \tag{1} \] and let \({\mathcal E}(u,v)\) be the Dirichlet form on \(H(D)\) associated with \(\Delta\), \[ {\mathcal E} (u,v) = \int_D \sum_{1 \leq i,j \leq n} a_{ij} (x) \cdot \partial_i u(x) \partial_j v(x) dx, \quad \forall u, v \in H^1 (D). \tag{2} \] In order to construct the reflecting diffusion process generated by \(\Delta\) in the closure \(\overline D\) of \(D\), the authors investigate two separated interesting problems, the main results of both of them are presented in subsection 2 while details of proofs are treated respectively in subsections 3-4 and in the last subsections 5-6. By introducing the notion of cusps boundary points (Assumptions A1--A2) they are able to modify a Sobolev-inequality due to Moser and obtain from that a Green function \(\{G_\lambda (x,y) : \lambda > 0\}\) on \(\overline D \times \overline D\) (off the diagonal) (Theorem 2.1). The standard program is the construction of the resolvent (and its dual) from the above mentioned Green function; then the Feller transition semigroup on \(C_0 (\overline D)\) and finally the associated Hunt process \(M = \{\Omega, X_t, P_x\) \((x \in \overline D)\}\) are obtained, which is actually a conservatrice diffusion (Theorem 2.2). Note that similar results for the case where \(D\) is bounded are also obtained previously by \textit{R. F. Bass} and \textit{P. Hsu} [Ann. Probab. 19, No. 2, 486-508 (1991; Zbl 0732.60090)]. While the first problem is more or less of analytical nature the second one is typically of probabilistic nature. The reflecting diffusion process \(M\) is now regarded as the solution of a ``semi-mart…
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 819161
zbMATH Keywords: elliptic operator
zbMATH Keywords: Dirichlet form
zbMATH Keywords: reflecting diffusion process
reviewed by: Xuan Loc Nguyen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01053454
OpenAlex ID: W4249270166
description: scientific article; zbMATH DE number 819161