Scaling coupling of reflecting Brownian motions and the hot spots problem

Author:
Mihai N. Pascu

Journal:
Trans. Amer. Math. Soc. **354** (2002), 4681-4702

MSC (2000):
Primary 60J65; Secondary 60J45, 35B05, 35K05

DOI:
https://doi.org/10.1090/S0002-9947-02-03020-9

Published electronically:
May 7, 2002

MathSciNet review:
1926894

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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a new type of coupling of reflecting Brownian motions in smooth planar domains, called *scaling coupling*. We apply this to obtain monotonicity properties of antisymmetric second Neumann eigenfunctions of convex planar domains with one line of symmetry. In particular, this gives the proof of the hot spots conjecture for some known types of domains and some new ones.

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Additional Information

**Mihai N. Pascu**

Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Address at time of publication:
Department of Mathematics, Purdue Uniwersity, West Lafayette, Indiana 47907-1395

Email:
pascu@math.purdue.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03020-9

Keywords:
Coupling of diffusions,
reflecting Brownian motion,
hot spots conjecture,
eigenfunctions,
Neumann problem,
Laplacian

Received by editor(s):
September 9, 2001

Received by editor(s) in revised form:
January 12, 2002

Published electronically:
May 7, 2002

Additional Notes:
I would like to thank Richard Bass, Krzysztof Burdzy, Nicolae N. Pascu and Nicolae R. Pascu for several helpful discussions

Article copyright:
© Copyright 2002
American Mathematical Society