Electronic Communications in Probability

Small time asymptotics of Ornstein-Uhlenbeck densities in Hilbert spaces

Terence Jegaraj

Abstract

We show that Varadhan's small time asymptotics for densities of the solution of a stochastic differential equation in $\mathbb{R}^n$ carries over to a Hilbert space-valued Ornstein-Uhlenbeck process whose transition semigroup is strongly Feller and symmetric. In the Hilbert space setting, densities are with respect to a Gaussian invariant measure.

Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 53, 552-559.

Dates
Accepted: 9 December 2009
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465234762

Digital Object Identifier
doi:10.1214/ECP.v14-1510

Mathematical Reviews number (MathSciNet)
MR2570678

Zentralblatt MATH identifier
1196.60055

Rights

Citation

Jegaraj, Terence. Small time asymptotics of Ornstein-Uhlenbeck densities in Hilbert spaces. Electron. Commun. Probab. 14 (2009), paper no. 53, 552--559. doi:10.1214/ECP.v14-1510. https://projecteuclid.org/euclid.ecp/1465234762

References

• A. Chojnowska-Michalik and B. Goldys. On regularity properties of nonsymmetric Ornstein-Uhlenbeck semigroup in L^p spaces. Stoch. Stoch. Rep. 59, No. 3-4 (1996), 183-209.
• A. Chojnowska-Michalik and B. Goldys. Symmetric Ornstein-Uhlenbeck semigroups and their generators. Probab. Theory Relat. Fields 124 (2002), 459-486.
• G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge University Press (1992).
• G. Da Prato and J. Zabczyk. Second Order Partial Differential Equations in Hilbert Spaces. Cambridge University Press (2002).
• P. Lax. Functional Analysis. John Wiley and Sons (2002).
• J.Norris. Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. Acta Math. 179 (1997), 79-103.
• A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag (1983).
• S.R.S. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20 (1967), 431-455.
• J. Zabczyk. Topics in Stochastic Systems. Asymptotics and Regularity. Flight Systems Research Laboratory, University of California, Los Angeles. Technical Report No. 1-Z-4015-88 (1988).