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Limit theorems for a class of diffusion processes. (English) Zbl 0583.60073
Author’s abstract: We consider a diffusion process \(\{\) x(t)\(\}\) on a compact Riemannian manifold with generator \(\Delta /2+b\). A current- valued continuous stochastic process \(\{X_ t\}\) corresponds to \(\{\) x(t)\(\}\) by considering the stochastic line integral \(X_ t(\alpha)\) along \(\{\) x(t)\(\}\) for every smooth 1-form \(\alpha\). Furthermore \(\{X_ t\}\) is decomposed into the martingale part and the bounded variation part as a current-valued continuous process. We show the central limit theorems for \(\{X_ t\}\) and the martingale part of \(\{X_ t\}\). Occupation time laws for recurrent diffusions and homogenization problems of periodic diffusions are closely related to these theorems.
Reviewer: S.Eloshvili

60J60 Diffusion processes
58J65 Diffusion processes and stochastic analysis on manifolds
Full Text: DOI
[1] Billingsley P., Convergence of Probability Measures (1968) · Zbl 0172.21201
[2] Bensoussan A., Asymptotic Analysis for Periodic Structure (1978)
[3] de Rham G., Variétes Différentiates (1960)
[4] Gel’fand I. M., Generalized Functions 4 (1964)
[5] Gikhman I. I., The Theory of Stochastic Processes (1974) · Zbl 0288.60032 · doi:10.1007/978-3-642-61943-4
[6] Ikeda N., Publ. Res. Inst. Math. Sci. Kyoto University 15 pp 827– (1979) · Zbl 0462.60056 · doi:10.2977/prims/1195187879
[7] Ikeda N., Stochastic Differential Equations and Diffusion Processes (1981) · Zbl 0495.60005
[8] Itô, K. ”Foundations of stochastic differential equations in infinite dimensional spaces”. CBMS-NSF. Regional Conference Series in Applied Mathematics · Zbl 0547.60064
[9] Itô K., Lecture Notes in Mathematics 1021 (1983)
[10] Kobayashi S., Foundations of Differential Geometry 1 (1963) · Zbl 0119.37502
[11] Manabe S., Osaka J. Math 19 pp 429– (1982)
[12] Mitoma I., Ann. Probability 11 pp 989– (1983) · Zbl 0527.60004 · doi:10.1214/aop/1176993447
[13] Nakao S., Stochastic calculus for continuous additive functionals of zero energy, to appear · Zbl 0604.60068
[14] Totoki H., Mem. Fac. Sci. Kyushu Univ 15 pp 178– (1961)
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