Brownian functionals on hypersurfaces

in Euclidean space

Authors:
Kimberly K. J. Kinateder and Patrick McDonald

Journal:
Proc. Amer. Math. Soc. **125** (1997), 1815-1822

MSC (1991):
Primary 60J65, 58G32

DOI:
https://doi.org/10.1090/S0002-9939-97-03741-6

MathSciNet review:
1371132

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Abstract | References | Similar Articles | Additional Information

Abstract: Using the first exit time for Brownian motion from a smoothly bounded domain in Euclidean space, we define two natural functionals on the space of embedded, compact, oriented, unparametrized hypersurfaces in Euclidean space. We develop explicit formulas for the first variation of each of the functionals and characterize the critical points.

**[AS]**M. Aizenman and B. Simon,*Brownian motion and Harnack inequality for Schrodinger operators*, Comm. Pure and Appl. Math.**35**(1982), 209-273. MR**84a:35062****[AL]**F. J. Almgren, Jr. and E. H. Lieb,*Symmetric decreasing rearrangement is sometimes continuous*, J. Amer. Math. Soc.**2**(1989), 683-773. MR**90f:49038****[FM]**S. J. Fromm and P. McDonald,*A symmetry problem from probability*, Proc. Amer. Math. Soc. (to appear).**[GS]**P. R. Garabedian and M. Schiffer,*Convexity of domain functionals*, J. Analyse Math.**2**(1953), 281-368. MR**15:627a****[P]**G. Polya,*Torsional rigidity, principal frequency, electrostatic capacity and symmetrization*, Quart. Appl. Math.**6**(1948), 267-277. MR**10:206b****[S]**J. Serrin,*A symmetry problem in potential theory*, Arch. Rat. Mech. and Anal.**43**(1971), 304-318. MR**48:11545**

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Additional Information

**Kimberly K. J. Kinateder**

Affiliation:
Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Address at time of publication:
Department of Mathematics, Wright State University, Dayton, Ohio 45435

Email:
kjk@euler.math.wright.edu

**Patrick McDonald**

Affiliation:
Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Address at time of publication:
Department of Mathematics, New College of University of South Florida, Sarasota, Florida 34243

Email:
pmacdona@virtu.sar.usf.edu, pmacdona@virtu.sar.usf.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03741-6

Keywords:
Brownian motion,
exit times,
variational calculus

Received by editor(s):
August 9, 1995

Received by editor(s) in revised form:
December 2, 1995

Communicated by:
Richard T. Durrett

Article copyright:
© Copyright 1997
American Mathematical Society