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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
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by Alexander Grigor’yan PDF
Bull. Amer. Math. Soc. 36 (1999), 135-249 Request permission


We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion. It is shown that both properties have various analytic characterizations, in terms of the heat kernel, Green function, Liouville properties, etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds, etc., which are related to recurrence and non-explosion.
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Additional Information
  • Alexander Grigor’yan
  • MR Author ID: 203816
  • Email:
  • Received by editor(s): October 1, 1997
  • Received by editor(s) in revised form: September 2, 1998
  • Published electronically: February 19, 1999
  • Additional Notes: Research supported by the EPSRC Fellowship B/94/AF/1782 (United Kingdom).
  • © Copyright 1999 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 36 (1999), 135-249
  • MSC (1991): Primary 58G32, 58G11
  • DOI:
  • MathSciNet review: 1659871