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Transactions of the American Mathematical Society

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Application of the dual-process method to the study of a certain singular diffusion

Author: David Williams
Journal: Trans. Amer. Math. Soc. 237 (1978), 101-110
MSC: Primary 60J35; Secondary 60H10
MathSciNet review: 0464409
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Abstract: This paper should be regarded as a sequel to a paper by Holley, Stroock and the author. Its primary purpose is to provide further illustration of the application of the dual-process method. The main result is that if $ d \geqslant 2$ and $ \varphi $ is the characteristic function of an aperiodic random walk on $ {{\mathbf{Z}}^d}$, then there is precisely one Feller semigroup on the d-dimensional torus with generator extending $ A = \{ 1 - \varphi (\theta )\} \Delta $. A necessary and sufficient condition for the associated Feller process to leave the singular point 0 is determined. This condition provides a criterion for uniqueness in law of a stochastic differential equation which is naturally associated with the process.

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Keywords: Dual process, diffusion, Markov chain, Feller property, Bochner map, stochastic differential equation, Girsanov's example
Article copyright: © Copyright 1978 American Mathematical Society

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