Hypersurfaces in and the variance

of exit times for Brownian motion

Authors:
Kimberly K. J. Kinateder and Patrick McDonald

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2453-2462

MSC (1991):
Primary 60J65, 58G32

DOI:
https://doi.org/10.1090/S0002-9939-97-03925-7

MathSciNet review:
1401746

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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.

**[AL]**F. J. Almgren, Jr. and E. H. Lieb,*Symmetric decreasing rearrangement is sometimes continuous*, J. Am. Math. Soc.**2**(1989), 683-773. MR**90f:49038****[B]**C. Bandle,*Isoperimetric Inequalities and Applications*, Pitman Publishing Inc., Marshfield, Mass., 1980. MR**81e:35095****[GS]**P. R. Garabedian and M. Schiffer,*Convexity of domain functionals*, J. Anal. Math.**2**(1953), 281-368. MR**15:627a****[GNN]**B. Gidas, W.-M. Ni and L. Nirenberg,*Symmetry and related properties via the maximum principle*, Comm. Math. Phys.**68**(1979), 209-243. MR**80h:35043****[FM]**S. J. Fromm and P. McDonald,*A symmetry problem from probability*, Comm. PDE (submitted).**[KM]**K. K. J. Kinateder and P. McDonald,*Brownian functionals on hypersurfaces in Euclidean space*, Proc. Amer. Math. Soc. (to appear).**[S]**J. Serrin,*A symmetry problem in potential theory*, Arch. Rational Mech. Anal.**43**(1971), 304-318. MR**48:11545**

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Additional Information

**Kimberly K. J. Kinateder**

Affiliation:
Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435-0001

Email:
kjk@euler.wright.edu

**Patrick McDonald**

Affiliation:
Department of Mathematics, University of South Florida, Sarasota, Florida

DOI:
https://doi.org/10.1090/S0002-9939-97-03925-7

Keywords:
Brownian motion,
exit times,
variance,
variational calculus,
free boundary problems

Received by editor(s):
December 2, 1995

Received by editor(s) in revised form:
March 5, 1996

Communicated by:
Stanley Sawyer

Article copyright:
© Copyright 1997
American Mathematical Society