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January, 2014 On general boundary conditions for one-dimensional diffusions with symmetry
Masatoshi FUKUSHIMA
J. Math. Soc. Japan 66(1): 289-316 (January, 2014). DOI: 10.2969/jmsj/06610289

Abstract

We give a simple proof of the symmetry of a minimal diffusion $X^0$ on a one-dimensional open interval $I$ with respect to the attached canonical measure $m$ along with the identification of the Dirichlet form of $X^0$ on $L^2(I; m)$ in terms of the triplet $(s,m,k)$ attached to $X^0$. The $L^2$-generators of $X^0$ and its reflecting extension $X^r$ are then readily described. We next use the associated reproducing kernels in connecting the $L^2$-setting to the traditional $C_b$-setting and thereby deduce characterizations of the domains of $C_b$-generators of $X^0$ and $X^r$ by means of boundary conditions. We finally identify the $C_b$-generators for all other possible symmetric diffusion extensions of $X^0$ and construct by that means all diffusion extensions of $X^0$ in [IM2].

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Masatoshi FUKUSHIMA. "On general boundary conditions for one-dimensional diffusions with symmetry." J. Math. Soc. Japan 66 (1) 289 - 316, January, 2014. https://doi.org/10.2969/jmsj/06610289

Information

Published: January, 2014
First available in Project Euclid: 24 January 2014

zbMATH: 1296.60211
MathSciNet: MR3161402
Digital Object Identifier: 10.2969/jmsj/06610289

Subjects:
Primary: 60J60
Secondary: 31C25, 60J50

Rights: Copyright © 2014 Mathematical Society of Japan

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Vol.66 • No. 1 • January, 2014
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