×

Laplace approximation for stochastic line integrals. (English) Zbl 1181.60038

Given a diffusion process \((z_t)_{t\in {\mathbb R}_+}\) on a Riemannian manifold \(M\), the author considers the current process \((Y_t)_{t\in {\mathbb R}_+}\) mapping a \(1\)-form \(\alpha\) to the martingale part \(Y_t ( \alpha )\) of the stochastic line integral \(\int_{z[0,t]} \alpha\), and the empirical measure \(l_t = \int_0^t \delta_{z_s} ds\). The main result of the paper is a precise Laplace asymptotics for the pair \(t^{-1}(Y_t , l_t )\), which can essentially be described as \[ \lim_{t\to \infty} e^{-tK_F} E_x \left[ \exp \left( t F \left( {1 \over t} Y_t , {1 \over t} l_t \right) \right) \right] = \sum_{(\omega , \mu ) } D_{F,\omega} h_\omega ( x) \int_M {1 \over h_\omega} d\mu, \] where \(F\) is a sufficiently regular functional, the sum is taken among the equilibrium states \((\omega , \mu )\) of the system, \(h_\omega \in C_+ (M)\), \(D_{F,\omega}>0\) is a constant, and \[ K_F = \lim_{t\to \infty} {1 \over t} E_x \left[ \exp \left( t F \left( {1 \over t} Y_t , {1 \over t} l_t \right) \right) \right] , \] under a non-degeneracy condition on the Hessian of \(F-I\). The Laplace approximation for stochastic line integrals of periodic diffusions follows as a particular case, and an application to the convergence of path measures is given.

MSC:

60F10 Large deviations
58J65 Diffusion processes and stochastic analysis on manifolds
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60B10 Convergence of probability measures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Avellaneda, M.: Large deviation estimates and the homological behavior of Brownian motion on manifolds. Ph.D. thesis. University of Minnesota, USA (1985)
[2] Bolthausen, E.; Deuschel, J. D.; Schmock, U., Convergence of path measures arising from a mean field or polaron type interaction, Probab. Theory Relat. Fields, 95, 3, 283-310 (1993) · Zbl 0791.60093
[3] Bolthausen, E., Deuschel, J.D., Tamura, Y.: Laplace approximations for large deviations of nonreversible Markov processes on compact state spaces part II: the degenerate Hessian case (preprint) · Zbl 0838.60023
[4] Bolthausen, E.; Deuschel, J. D.; Tamura, Y., Laplace approximations for large deviations of nonreversible Markov processes. The nondegenerate case, Ann. Probab., 23, 1, 236-267 (1995) · Zbl 0838.60023
[5] Deuschel, J. D.; Stroock, D. W., Large deviations (1989), Boston: Academic Press, Boston · Zbl 0705.60029
[6] Doukhan, P.: Mixing. Properties and examples. Lecture Notes in Statistics, vol. 85. Springer, New York (1994) · Zbl 0801.60027
[7] Ellis, R. S.; Rosen, J. S., Asymptotic analysis of Gaussian integrals. I. Isolated minimal points, Trans. Am. Math. Soc., 273, 2, 447-481 (1982) · Zbl 0521.28009
[8] Gaffney, M. P., Asymptotic distributions associated with the Laplacian for forms, Comm. Pure. Appl. Math., 11, 535-545 (1958) · Zbl 0102.09604
[9] Hörmander, L., Linear partial differential operators (1963), Heidelberg: Springer, Heidelberg · Zbl 0108.09301
[10] Ikeda, N., Limit theorems for a class of random currents, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), 181-193 (1987), Boston: Academic Press, Boston · Zbl 0638.60041
[11] Ikeda, N.; Manabe, S., Integral of differential forms along the path of diffusion processes, Publ. RIMS, Kyoto Univ., 15, 827-852 (1979) · Zbl 0462.60056
[12] Ikeda, N.; Ochi, Y., Central limit theorems and random currents, Lect. Notes Contr. Inform. Sci., 78, 195-205 (1986) · Zbl 0589.60066
[13] Kato, T., Perturbation theory for linear operators (1980), Heidelberg: Springer, Heidelberg · Zbl 0435.47001
[14] Kusuoka, S.; Tamura, Y., Symmetric Markov processes with mean field potentials, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34, 2, 371-389 (1987) · Zbl 0638.60106
[15] Kusuoka, S.; Tamura, Y., Precise estimate for large deviation of Donsker-Varadhan type, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 38, 3, 533-565 (1991) · Zbl 0752.60026
[16] Kuwada, K., On large deviations for random currents induced from stochastic line integrals, Forum Math., 18, 639-676 (2006) · Zbl 1112.60005
[17] Kuwada, K., Sample path large deviations for a class of random currents, Stochastic Process. Appl., 108, 203-228 (2003) · Zbl 1075.60509
[18] Manabe, S., Large deviation for a class of current-valued processes, Osaka J. Math., 29, 89-102 (1992) · Zbl 0758.60024
[19] Maz’ya, V.G., Shaposhnikova, T.O.: Theory of multipliers in spaces of differentiable functions. Monographs and Studies in Mathematics, vol. 23. Pitman (Advanced Publishing Program), Boston, MA (1985) · Zbl 0645.46031
[20] Moser, J., A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa, 20, 3, 265-315 (1966) · Zbl 0144.18202
[21] Ochi, Y., Limit theorems for a class of diffusion processes, Stoch., 15, 251-269 (1985) · Zbl 0583.60073
[22] Pinsky, R.G.: Positive harmonic functions and diffusion, Cambridge Studies in Advanced Mathematics, vol. 45. Cambridge University Press, Cambridge (1995) · Zbl 0858.31001
[23] Yoshida, N., Sobolev spaces on a Riemannian manifold and their equivalence, J. Math. Kyoto univ., 32, 3, 621-654 (1992) · Zbl 0771.58005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.