Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
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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.References
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Additional Information
- Alexander Grigor’yan
- MR Author ID: 203816
- Email: a.grigoryan@ic.ac.uk
- 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: https://doi.org/10.1090/S0273-0979-99-00776-4
- MathSciNet review: 1659871