Hypoelliptic random heat kernels: A case study

Sowers, Richard

doi:10.1090/S0002-9939-00-05822-6

Hypoelliptic random heat kernels: A case study
HTML articles powered by AMS MathViewer

by Richard B. Sowers PDF

Proc. Amer. Math. Soc. 129 (2001), 2451-2460 Request permission

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

G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 3, 307–331 (French). MR 974408, DOI 10.24033/asens.1560
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
K. D. Elworthy, Stochastic differential equations on manifolds, London Mathematical Society Lecture Note Series, vol. 70, Cambridge University Press, Cambridge-New York, 1982. MR 675100, DOI 10.1017/CBO9781107325609
Y. Kannai, Off diagonal short time asymptotics for fundamental solutions of diffusion equations, Comm. Partial Differential Equations 2 (1977), no. 8, 781–830. MR 603299, DOI 10.1080/03605307708820048
Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1990. MR 1070361
Rémi Léandre, Majoration en temps petit de la densité d’une diffusion dégénérée, Probab. Theory Related Fields 74 (1987), no. 2, 289–294 (French, with English summary). MR 871256, DOI 10.1007/BF00569994
Rémi Léandre, Minoration en temps petit de la densité d’une diffusion dégénérée, J. Funct. Anal. 74 (1987), no. 2, 399–414 (French). MR 904825, DOI 10.1016/0022-1236(87)90031-0
Rémi Léandre, Développement asymptotique de la densité d’une diffusion dégénérée, Forum Math. 4 (1992), no. 1, 45–75 (French). MR 1142473, DOI 10.1515/form.1992.4.45
L. Mesnager, Estimation en temps petit de densités conditionelles dan des problemes de filtrage nonlineare, Ph.D. Thesis, Université de Paris-Sud, 1996.
S. A. Molčanov, Diffusion processes, and Riemannian geometry, Uspehi Mat. Nauk 30 (1975), no. 1(181), 3–59 (Russian). MR 0413289
Richard B. Sowers, Recent results on the short-time geometry of random heat kernels, Math. Res. Lett. 1 (1994), no. 6, 663–675. MR 1306012, DOI 10.4310/MRL.1994.v1.n6.a4
Richard B. Sowers, Short-time geometry of random heat kernels, Mem. Amer. Math. Soc. 132 (1998), no. 629, viii+130. MR 1401494, DOI 10.1090/memo/0629
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.