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February, 1981 The Degenerate Neumann Problem and Degenerate Diffusions with Venttsel's Boundary Conditions
Kunio Nishioka
Ann. Probab. 9(1): 103-118 (February, 1981). DOI: 10.1214/aop/1176994511

Abstract

A stochastic solution of the Neumann problem is obtained, when the second order elliptic operator $L$ is degenerate at the boundary of the domain. Let $D$ be a domain in $R^n$ with the smooth boundary $\partial D$, and the second order elliptic operator $L$ be defined in $R^n$. We construct a diffusion $X^r(t) = (\mathscr P\underset{\smile}^r, D(\mathscr P\underset{\smile}^r))$ in $\bar{D}$ such that (i) $D(\mathscr P\underset{\smile}^r) \supset D(A^r) = \{f \in C^2(D); \partial f/\partial v = 0$ for $x \in D\}$, (ii) $f \in D(A^r) \Rightarrow \mathscr P\underset{\smile}^r f = Lf$. With that diffusion, the stochastic solution of our Neumann problem is defined, and the existence and the uniqueness conditions of that are obtained. The analytic meaning of our stochastic solution is explained. The diffusions in $\bar{D}$, satisfying the other Venttsel boundary conditions are also constructed, which are useful for the degenerate third boundary value problems.

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Kunio Nishioka. "The Degenerate Neumann Problem and Degenerate Diffusions with Venttsel's Boundary Conditions." Ann. Probab. 9 (1) 103 - 118, February, 1981. https://doi.org/10.1214/aop/1176994511

Information

Published: February, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0453.60060
MathSciNet: MR606800
Digital Object Identifier: 10.1214/aop/1176994511

Subjects:
Primary: 60H99
Secondary: 60J45

Keywords: compatible condition , degenerate Neumann problem , degenerate Venttsel's boundary conditions , Fichera problem , invariant measure , stochastic solution

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 1 • February, 1981
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