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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Application of the dual-process method to the study of a certain singular diffusion
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by David Williams PDF
Trans. Amer. Math. Soc. 237 (1978), 101-110 Request permission


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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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 237 (1978), 101-110
  • MSC: Primary 60J35; Secondary 60H10
  • DOI:
  • MathSciNet review: 0464409