Potential Analysis

, Volume 11, Issue 3, pp 235–247

Exponential Decay of Lifetimes and a Theorem of Kac on Total Occupation Times

  • Masayoshi Takeda
Article

Abstract

Let \((\tfrac{1}{2}D,H^1 (R^d ))\) be the Dirichlet integral and \((B_t ,P_z^W )\) the Brownian motion on R. Let μ be a finite positive measure in the Kato class and \(A_t^\mu \) the additive functional associated with μ. We prove that for a regular domain D of Rd
$$\begin{gathered} \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\log P_z^W (A_{\tau _D }^\mu > \beta )\;\; = \;\; - \inf \left\{ {\tfrac{1}{2}D(u,u):u \in C_0^\infty (D)\int_D {u^2 {\text{d}}} \mu = 1} \right\} \hfill \\ {\text{ for any }}x \in D, \hfill \\ \end{gathered} $$
where τD is the exit time from D. As an application, we consider the integrability of Wiener functional exp (\(A_{\tau _D }^\mu \)).
Exponenial Decay of Lifetime Large Deviation Additive Functional Time Change Dirichlet Form. 

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References

  1. 1.
    Albeverio, S., Blanchard, P. and Ma, Z. M.: 'Feynmann-Kac semigroups in terms of signed smooth measures', Random Partial Differential Equations,(eds.) U. Hornung et al., Birkhäuser, 1989.Google Scholar
  2. 2.
    Albeverio, S. and Ma, Z. M.: 'Perturbation of Dirichlet forms-Lower boundedness, closability, and form cores', J.Funct.Anal. 99, 332-356 (1991).Google Scholar
  3. 3.
    Brasche, J., Exner, P., Kuperin, Yu. and Seba, P.: 'Schrödinger operators with singular interactions', J.Math.Anal.Appl. 184, 112-139 (1994).Google Scholar
  4. 4.
    Chung, K. L. and Zhao, Z.: From Brownian Motion to Schr ödinger's Equation, Springer, Berlin, Heidelberg, New York, 1995.Google Scholar
  5. 5.
    Davies, E. B.: 'L 1 properties of second order elliptic operators', Bull.London Math.Soc. 17, 417-436 (1985).Google Scholar
  6. 6.
    Davies, E. B.: Heat Kernels and Spectral Theory, Cambridge, Cambridge, 1989.Google Scholar
  7. 7.
    Donsker, M. D. and Varadhan, S. R. S.: 'Asymptotic evaluation of certain Wiener integrals for large time', Proceedings of International Conference on Function Space, (ed.) A. M. Arthure, Oxford, 1974.Google Scholar
  8. 8.
    Fitzsimmons, P. J. and Getoor, R. K.: 'On the potential theory of symmetric Markov processes', Math.Ann. 281, 495-512 (1988).Google Scholar
  9. 9.
    Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, Berlin, 1994.Google Scholar
  10. 10.
    Kac, M.: 'On some connections between probability theory and differential equations', Proc.2nd Berk.Symp.Math.Statist. Probability, 189-215 (1950).Google Scholar
  11. 11.
    Kawabata, T. and Takeda,M.: 'On uniqueness problem for local Dirichlet forms', Osaka J.Math. 33, 881-893 (1996).Google Scholar
  12. 12.
    Maz'ja, V. G.: Sobolev Spaces, Springer, Berlin, Heidelberg, New York, 1985.Google Scholar
  13. 13.
    McKean, H. P.: Stochastic Integrals, Academic Press, New York, 1969.Google Scholar
  14. 14.
    Rogers, L. C. G. and Williams, D.: Diffusions, Markov Processes, and Martingale, Vol I, Wiley, New York, 1994.Google Scholar
  15. 15.
    Sharpe, M.: General Theory of Markov Processes, Academic Press, New York, 1988.Google Scholar
  16. 16.
    Stollmann, P. and Voigt, V.: 'Perturbation of Dirichlet forms by measures', preprint, Potential Analysis 5, 109-138 (1996).Google Scholar
  17. 17.
    Sturm, K. T.: 'Analysis on local Dirichlet spacesI, Recurrence, conservativeness and L pLiouville properties', J.Reine Angew.Math. 456, 173-196 (1994).Google Scholar
  18. 18.
    Takeda, M.: 'A large deviation for symmetric Markov processes with finite lifetime', Stochastics and Stochastic Reports 59, 143-167 (1996).Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Masayoshi Takeda
    • 1
  1. 1.Department of MathematicsTohoku UniversitySendaiJapan. e-mail

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