The Annals of Probability

On the small time asymptotics of diffusion processes on Hilbert spaces

T. S. Zhang

Full-text: Open access

Abstract

In this paper,we establish a small time large deviation principle and obtain the following small time asymptotics:

\lim_{t \to 0}2t \log P(X_0 \in B, X_t \in C) = -d^2 (B, C),

for diffusion processes on Hilbert spaces, where $d(B,C)$ is the intrinsic metric between two subsets $B$ and $C$ associated with the diffusions. The case of perturbed Ornstein–Uhlenbeck processes is treated separately at the end of the paper.

Article information

Source
Ann. Probab., Volume 28, Number 2 (2000), 537-557.

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1019160252

Digital Object Identifier
doi:10.1214/aop/1019160252

Mathematical Reviews number (MathSciNet)
MR1782266

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60F10: Large deviations
Secondary: 31C25: Dirichlet spaces

Keywords
Dirichlet form intrinsic metric large deviation stochastic evolution equation Girsanov transform

Citation

Zhang, T. S. On the small time asymptotics of diffusion processes on Hilbert spaces. Ann. Probab. 28 (2000), no. 2, 537--557. doi:10.1214/aop/1019160252. https://projecteuclid.org/euclid.aop/1019160252


Export citation

References

  • [1] Aida, S. and Kawabi, X. (1999). Short time asymptotics of certain infinite dimensional diffusion processes. Preprint.
  • [2] Aida, S. and Shigekawa, I. (1994). Logarithmic Sobolev inequalities and spectral gaps: perturbation theory. J.Funct.Anal.448-475.
  • [3] Albeverio, S. and R ¨ockner, M. (1991). Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab.Theory Related Fields 89 347-386.
  • [4] Albeverio, S., R ¨ockner, M. and Zhang, T. S. (1993). Girsanov transform for symmetric diffusions with infinite dimensional state space. Ann.Probab.21 961-978.
  • [5] Bogachev, V. I. and R ¨ockner, M. (1999). Mehler formula and capacities for infinite dimensional Ornstein-Uhlenbeck processes with general linear drift. Osaka Math.J.To appear.
  • [6] Chow, P. L. and Menaldi, J. L. (1990). Exponential estimates in exit probabilityfor some diffusion processes in Hilbert spaces, Stochastics Stochastics Rep. 29 377-393.
  • [7] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press.
  • [8] Davies, E. B. (1989). Heat Kernels and Spectral Theory. Cambridge Univ. Press.
  • [9] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques. Jones and Bartlett, Boston.
  • [10] Fang, S. (1994). On the Ornstein-Uhlenbeck process. Stochastics Stochastics Rep. 46 141-159.
  • [11] Fang, S. and Zhang, T. S. (1997). On the small time behavior of Ornstein-Uhlenbeck processes with unbounded linear drifts. Preprint. Probab.Theory Related Fields. To appear.
  • [12] Fukushima, M. Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin.
  • [13] Iscoe, I. and McDonald D. (1989). Large deviations for l2-valued Ornstein-Uhlenbeck processes. Ann.Probab.17 58-73.
  • [14] Kuo, H. H. (1977). Gaussian Measures in Banach Spaces. Lecture Notes in Math. Springer, Berlin.
  • [15] Lyons, T. J. and Zhang, T. S. (1994). Decomposition of Dirichlet processes and its applications. Ann.Probab.22 494-524.
  • [16] Lyons, T. J. and Zheng, W. A. (1988). A crossing estimate for the canonical process on a Dirichlet space and tightness result. Colloque Paul L´evysur les processes stochastique. Asterisque 157-158 248-272.
  • [17] Ma,M. and R ¨ockner, M. (1992). An Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin.
  • [18] Pardoux, E. (1979). Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 127-167.
  • [19] R ¨ockner, M. and Zhang, T. S. (1992). Uniqueness of generalized Schr¨odinger operators and applications. J.Funct.Anal.105 187-231.
  • [20] R ¨ockner, M. and Zhang, T. S. (1994). Uniqueness of generalized Schr¨odinger operators II. J.Funct.Anal.119 455-467.
  • [21] Stroock, D. (1984). An Introduction to the Theory of Large Deviations. Springer, Berlin.
  • [22] Takeda, M. (1989). On a martingale method for symmetric diffusion processes and its applications. Osaka J.Math.26 605-625.
  • [23] Tessitore, G. and Zabczyk, J. (1998). Strict positivityfor stochastic heat equations. Stochastic Process.Appl.77 83-98.
  • [24] Varadhan, S. R. S. (1967). Diffusion processes in small time intervals. Comm.Pure Appl. Math. 20 659-685.