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Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds


Author: Alexander Grigor'yan
Journal: Bull. Amer. Math. Soc. 36 (1999), 135-249
MSC (1991): Primary 58G32, 58G11
DOI: https://doi.org/10.1090/S0273-0979-99-00776-4
Published electronically: February 19, 1999
MathSciNet review: 1659871
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Abstract: 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
Email: a.grigoryan@ic.ac.uk

DOI: https://doi.org/10.1090/S0273-0979-99-00776-4
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).
Article copyright: © Copyright 1999 American Mathematical Society

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