Reflecting diffusions on Lipschitz domains with cusps -- analytic construction and Skorohod representation
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-martingale problem'' and so admits the following Skorokhod representation (Theorem 2.3) for a.s. \(P^x\) \((x \in \overline D)\): \[ X_t^{(i)} - X_0^{(i)} = M_t^{(i)} + \sum^d_{j = 1} \int^t_0 \partial a_{ij} (X_s) ds + \sum^d_{j = 1} \int^t_0 a_{ij} (X_s) n_j (X_s) dL_s,\;t \geq 0,\;1 \leq i \leq n, \tag{3} \] where \(L_t\) denotes the unique AF in the strict sense which is a local time of \(X_t\) associated with the surface measure \(\sigma\) on the boundary \(\partial D\) and where \(M_t^{(i)}\) \((1 \leq i \leq n)\) are continuous AF in the strict sense such that \[ E_x (M_t^{(i)}) = 0, \quad E_x (M_t^{(i)} M_t^{ (j)}) = 2E_x \left[ \int^t_0 a_{ij} (X_s) ds \right], \quad t \geq 0,\;x \in \overline D. \] Note that the case where \(D\) is a two-dimensional cusp was formulated as a ``submartingale problem'' and treated previously by means of conformal maps by \textit{R. D. DeBlassie} and \textit{E. H. Toby} [Probab. Theory Relat. Fields 94, No. 4, 505-524 (1993; Zbl 0791.60070) and Trans. Am. Math. Soc. 339, No. 1, 297-321 (1993; Zbl 0781.60065)].
scientific article; zbMATH DE number 819161
scientific article