|
Hypoelliptic random heat kernels: A case study
Author(s):
Richard
B.
Sowers
Journal:
Proc. Amer. Math. Soc.
129
(2001),
2451-2460.
MSC (1991):
Primary 60H15
Posted:
December 7, 2000
MathSciNet review:
1823931
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We consider the fundamental solution of a simple hypoelliptic stochastic partial differential equation in which the first-order term is modulated by white noise. We derive some short-time asymptotic formulæ. We discover that the form of the dominant short-time asymptotics depends nontrivially upon the interplay between the geometry of the noisy first-order term and the geometry defined by the hypoelliptic operator.
References:
-
- 1.
- G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. École Norm. Sup. (4) 21 (1988), 307-331. MR 89k:60087
- 2.
- I. Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115, Academic Press, 1984. MR 86g:58140
- 3.
- K. D. Elworthy, Stochastic differential equations on manifolds, London Mathematical Society Lecture Note Series, 70, Cambridge University Press, New York, 1982. MR 84d:58080
- 4.
- Y. Kannai, Off diagonal short time asymptotics for solutions of diffusion equations, Communications in P.D.E.'s 2 (1977), 781-830. MR 58:29247
- 5.
- H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, New York, 1990. MR 91m:60107
- 6.
- R. Léandre, Majoration en temps petit de la densité d'une diffusion dégénérée, Probability Theory and Related Fields 74 (1987), 289-294. MR 88c:60144
- 7.
- -, Minoration en temps petit de la densité d'une diffusion dégénérée, Journal of Functional Analysis 74 (1987), 399-414. MR 88k:60147
- 8.
- -, Développement asymptotique de la densité de diffusions dégénérées, Forum Math. 4 (1992), 45-75. MR 93d:60100.
- 9.
- L. Mesnager, Estimation en temps petit de densités conditionelles dan des problemes de filtrage nonlineare, Ph.D. Thesis, Université de Paris-Sud, 1996.
- 10.
- S. A. Molchanov, Diffusion processes, and Riemannian geometry, Uspehi Mat. Nauk 30 (1975), 3-59. MR 54:1404
- 11.
- R. B. Sowers, Recent results on the short-time geometry of random heat kernels, Math. Res. Lett. 1 (1994), 663-675. MR 95m:58133
- 12.
- -, Short-time geometry of random heat kernels, Mem. Amer. Math. Soc. 132 (1998). MR 98i:60060
- 13.
- H. Zhang, Développement en temps petit de la solution de l'équation de Zakai et résolution numérique par maillage adaptatif, Ph.D. Thesis, Université de Provence-Centre Saint-Charles, 1992.
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (1991):
60H15
Retrieve articles in all Journals with
MSC (1991):
60H15
Additional Information:
Richard
B.
Sowers
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 60201
Email:
r-sowers@math.uiuc.edu
DOI:
10.1090/S0002-9939-00-05822-6
PII:
S 0002-9939(00)05822-6
Keywords:
Fundamental solution,
hypoellipticity,
stochastic PDE's
Received by editor(s):
March 7, 1999
Received by editor(s) in revised form:
December 6, 1999
Posted:
December 7, 2000
Additional Notes:
The author would like to thank the anonymous referee for a very careful reading of the manuscript. The author received support from NSF DMS-9726739 and NSF DMS-9615877 during the preparation of this work.
Communicated by:
Claudia Neuhauser
Copyright of article:
Copyright
2000,
American Mathematical Society
|
