Summary
In the work of Donsker and Varadhan, Fukushima and Takeda and that of Deuschel and Stroock it has been shown, that the lower bound for the large deviations of the empirical distribution of an ergodic symmetric Markov process is given in terms of its Dirichlet form. We give a short proof generalizing this principle to general state spaces that include, in particular, infinite dimensional and non0metrizable examples. Our result holds w.r.t. quasi-every starting point of the Markov process. Moreover we show the corresponding weak upper bound w.r.t. quasi-every starting point.
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[A-HK] Albeverio, S., Høegh-Krohn, R.: Dirichlet forms and diffusion processes on rigged Hilbert spaces. Z. Wahrscheinlichkeitstheor. Verw. Geb.40, 1–57 (1977)
[A-R1] Albeverio, S., Röckner, M.: Classical Dirichlet forms on topological vector spaces-the construction of the associated Diffusion process. Probab. Theory Relat. Fields83, 405–434 (1989)
[A-R2] Albeverio, S., Röckner, M.: Classical Dirichlet forms on topological vector spaces-closability and a Cameron-Martin formula. Probab. Theory Relat. Fields88, 395–436 (1990)
[A-R3] Albeverio, S., Röckner, M.: New developments in the theory and application of Dirichlet forms. In: Albeverio, S. et al.: Stochastic processes, physics and geometry, Ascona/Locarno, Switzerland, 4–9 July 1988, pp. 27–76, Singapore: World Scientific 1990
[A-R4] Albeverio, S., Röckner, M.: Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Relat. Fields89, 347–386 (1991)
[A-R-Z] Albeverio, S., Röckner, M., Zhang, T.S.: Girsanov transform for symmetric diffusions with infinite dimensional state space. Preprint no. 165, Bonn: SFB 256 (1991). (to appear in Ann. Probab.)
[B-R] Bogachev, V.I., Röckner, M.: Mehler formula and capacities for infinite dimensional Ornstein-Uhlenbeck processes with general linear drift. (Preprint 1993)
[D-St] Deuschel, J.D., Stroock, D.W.: Large deviations. (Pure Appl. Math., vol. 137). Boston New York London Tokyo: Academic Press 1989
[Do-V1] Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Wiener integrals for large time. In: Arthur, A.M. (ed.) Functional integration and its applications. Proc. of the Int. Conf. held at Cumberland Lodge, Windsor Great Park, London 1974, p. 15–32 Oxford: Clarendon Press 1975
[Do-V2] Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, I. Comun. Pure Appl. Math.28, 1–47 (1975)
[Do-V3] Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, IV. Commun. Pure Appl. Math.36, 183–212 (1983)
[F1] Fukushima, M.: Dirichlet forms and Markov processes. (Mathematical Library, vol. 23) Amsterdam Oxford New York: North-Holland 1980
[F2] Fukushima, M.: A note on irreducibility and ergodicity of symmetric Markov processes. In: Albeverio, S. et al.: Stochastic processes in quantum theory and statistical physics, Marseille 1981 (Lect. Notes Phys., vol. 173, pp. 200–207) Berlin Heidelberg New York: Springer 1982
[F-T] Fukushima, M., Takeda, M.: A transformation of a symmetric Markov process and the Donsker-Varadhan theory. Osaka J. Math.21, 311–326 (1984)
[M-O-R] Ma, Z.M., Overbeck, L., Röckner, M.: Markov processes associated with Semi-Dirichletforms. (Preprint 1993)
[M-R] Ma, Z.M., Röckner, M.: An introduction to the theory of (Non-Symmetric) Dirichlet forms (Universitext). Berlin Heidelberg New York: Springer 1992
[Mü] Mück, S.: Dissertation, University of Bonn. (In preparation)
[P] Pazy, A.: Semigroups of linear operators and applications to partial differential equations (Appl. Math. Scie., vol. 44) Berlin Heidelberg New York: Springer 1983
[Re-S1] Reed, M., Simon, B.: Methods of modern mathematical physics II. Fourier analysis, self-adjointness. New York San Francisco London: Academic Press 1975
[Re-S2] Reed, M., Simon, B.: Methods of moderm mathematical physics IV. Analysis of operators. New York San Francisco London: Academic Press 1978
[R1] Röckner, M.: Trances of harmonic functions and a new path space for the free quantum field. J. Funct. Anal.79, 211–249 (1988)
[R2] Röckner, M.: On the transition function of the infinite dimensional Ornstein-Uhlenbeck process given by the free quantum field. In: Kral, J. et al.: Potential theory, Prague 1987, pp. 277–294. New York: Plenum Press 1988
[R-Z] Röckner, M., Zhang, T.S.: Uniqueness of generalized Schrödinger operators and applications. J. Funct. Anal.105, 187–231 (1992)
[Sh] Sharpe, M.: General theory of Markov processes (Pure Appl. Math., vol. 133). Boston New York London Tokyo: Academic Press 1988
[S] Simon, B.: TheP(φ)2 (quantum) field theory. New York: Princeton University Press 1974
[W1] Wu, Liming: Some general methods of large deviations and applications. (Preprint 1991)
[W2] Wu, Liming: Large deviations for essentially irreducible Markov processes, II. continuous time case. (Preprint 1992)
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This research was supported by the Graduiertenkolleg “Algebraische, analytische und geometrische Methoden und ihre Wechselwirkung in der modernen Mathematik”, Bonn
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Mück, S. Large deviations w.r.t. quasi-every starting point for symmetric right processes on general state spaces. Probab. Th. Rel. Fields 99, 527–548 (1994). https://doi.org/10.1007/BF01206231
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DOI: https://doi.org/10.1007/BF01206231