Skip to main content
SpringerLink
Log in

Asymptotic Properties of Generalized Feynman–Kac Functionals

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

Let (\({\mathcal{E}},{\mathcal{F}}\)) be a regular Dirichlet form on L2(X;m) and {Px}x ∈ X the Hunt process generated by (\({\mathcal{E}},{\mathcal{F}}\)). Let μ be a signed 'smooth measure' associated with (\({\mathcal{E}},{\mathcal{F}}\)) and Aμt the continuous additive functional corresponding to the measure μ. Under some conditions on (\({\mathcal{E}},{\mathcal{F}}\)) and μ, we shall prove that

$$\mathop {\lim }\limits_{t \to \infty } \frac{1}{t}\log E_x \left( {\exp \left( { - A_t^\mu } \right)} \right)$$
$$ = - \mathop {\mathop {\inf }\limits_{u \in \mathcal{F}^\mu } }\limits_{\left\| u \right\|_2 = 1} \left( {\mathcal{E}\left( {u,u} \right) + \int_X {\tilde u\tilde u} d\mu } \right)forallx \in X,$$

where \(\mathcal{F}^\mu = \left\{ {u \in \mathcal{F}:\tilde u \in L^2 \left( {X;\left| \mu \right|} \right)} \right\}\)

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albeverio, S., Blanchard, P. and Ma, Z. M.: 'Feynman-Kac semigroups in terms of signed smooth measures', in U. Hornung et al. (eds.), Random Partial Differential Equations, Birkhöuser, 1991.

  2. Albeverio, S., Gesztesy, F., Hoegh-Krohn, R. and Holden, H.: Solvable Models in Quantum Mechanics, Springer, 1988.

  3. Albeverio, S. and Ma, Z. M.: 'Perturbation of Dirichlet forms — Lower boundedness, closability, and form cores', J. Funct. Ananl. 99 (1991), 332–356.

    Google Scholar 

  4. Antonie, J. P., Gesztesy, F. and Shabani, J.: 'Exactly solvable models of sphere interactions in quantum mechanics', J. Phys. A20 (1987), 3687–3712.

    Google Scholar 

  5. Biroli, M. and Mosco, U.: 'Formes de Dirichlet et estimations structurelles dans les millieux discontinuous', C. R. Acad. Sci. Paris 313 (1991), 593–598.

    Google Scholar 

  6. Blanchard, P. and Ma, Z. M.: 'Semigroup of Schrödinger operators with potentials given by Radon measures', in S. Albeverio et al. (eds.), Stochastic Processes — Physics and Geometry, World Scient., Singapore, 1990.

    Google Scholar 

  7. Brasche, J., Exner, P., Kuperin, Yu. and Seba, P.: 'Schrödinger operators with singular interactions', J. Math. Anal. Appl. 184 (1994), 112–139.

    Google Scholar 

  8. Carmona, R., Masters, W. C. and Simon, B.: 'Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions', J. Funct. Anal. 91 (1990), 117–142.

    Google Scholar 

  9. Chung, K. L.: 'Doubly-Feller process with multiplicative functional', Seminar on Stochastic Processes, Birkhäuser (1986), 63–78.

  10. Chung, K. L. and Getoor, R. K.: 'The condenser problem', Ann. Prob. 5 (1986), 82–86.

    Google Scholar 

  11. Davies, E. B.: Heat Kernels and Spectral Theory, Cambridge, 1989.

  12. Donsker, M. D. and Varadhan, S. R. S.: 'Asymptotic evaluation of certain Wiener integrals for large time', Proceedings of International Conference on Function Space, A. M. Arthure (ed.), Oxford, 1974.

  13. Donsker, M. D. and Varadhan, S. R. S.: 'Asymptotic evaluation of certain Markov process expectations for large time I', Comm. Pure Appl. Math. 28 (1975), 1–47.

    Google Scholar 

  14. Donsker, M. D. and Varadhan, S. R. S.: 'Asymptotic evaluation of certain Markov process expectations for large time III', Comm. Pure Appl. Math. 29 (1976), 389–461.

    Google Scholar 

  15. Fukushima, M.: 'A note on irreducibility and ergodicity of symmetric Markov processes', Springer Lecture Notes in Physics 173 (1982).

  16. Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, 1994.

  17. Kac, M.: 'On some connections between probability theory and differential equations', Proc. 2nd Berk. Symp. Math. Statist. Probability (1950), 189–215.

  18. Ma, Z. M. and Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer, 1992.

  19. Sharpe, M.: General Theory of Markov Processes, Academic Press, 1988.

  20. Simon, B.: 'A canonical decomposition for quadratic forms with applications to monotone convergence theorems', J. Funct. Anal. 28 (1978), 377–385.

    Google Scholar 

  21. Simon, B.: 'Schrödinger semigroups', Bull. Am. Math. Soc. 7 (1982), 447–536.

    Google Scholar 

  22. Stollmann, P. and Voigt, V.: 'Perturbation of Dirichlet forms be measures', Potential Analysis 5 (1996), 109–138.

    Google Scholar 

  23. Stroock, D.: Probability Theory, An Analytic View, Cambridge, 1993.

  24. Sturm, K. T.: 'Schrödinger operators and Feynman-Kac semigroups with arbitrary nonnegative potentials', Expo. Math. 12 (1994), 385–411.

    Google Scholar 

  25. Sturm, K. T.: 'Schrödinger semigroups on manifolds', J. Funct. Anal. 118 (1993), 309–350.

    Google Scholar 

  26. Sturm, K. T.: 'On the Lp-spectrum of uniformly elliptic operators on Riemannian manifolds', J. Funct. Anal. 118 (1993), 442–453.

    Google Scholar 

  27. Sturm, K. T.: 'Analysis on local Dirichlet spaces — I. Recurrence, conservativeness and Lp-Liouville properties', J. reine angew. Math. 456 (1994), 173–196.

    Google Scholar 

  28. Sturm, K. T.: 'Analysis on local Dirichlet space — II. Gaussian upper bounds for fundamental solutions of parabolic equations', Osaka J. Math. 32 (1995), 275–312.

    Google Scholar 

  29. Sturm, K. T.: 'On the geometry defined by Dirichlet forms, Seminer on Stochastic Analysis, Randam Fields and Applications', Bolthausen, (ed.), Progress in Probability 36, Birkhäuser (1995), 231–242.

  30. Takeda, M.: 'A large deviation for symmetric Markov processes with finite life time', Stochastics and Stochastic Reports 59 (1996), 143–167.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Science, Faculty of Engineering Science, Osaka University, Japan

    Masayoshi Takeda

Authors
  1. Masayoshi Takeda
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Takeda, M. Asymptotic Properties of Generalized Feynman–Kac Functionals. Potential Analysis 9, 261–291 (1998). https://doi.org/10.1023/A:1008656907265

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008656907265

Search

Navigation