Summary
Let Φ be a weighted Schwartz's space of rapidly decreasing functions, Φ′ the dual space and ℒ(t) a perturbed diffusion operator with polynomial coefficients from Φ into itself. It is proven that ℒ(t) generates the Kolmogorov evolution operator from Φ into itself via stochastic method. As applications, we construct a unique solution of a Langevin's equation on Φ′:
whereW(t) is a Φ′ Brownian motion and ℒ*(t) is the adjoint of ℒ(t) and show a central limit theorem for interacting multiplicative diffusions.
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Billingsley, P.: Convergence of probability measures. New York London Sydney Toronto: Wiley 1968
Dawson, D.A.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys.31, 29–85 (1983)
Gelfand, I.M., Vilenkin, N.Ya.: Generalized functions 4. New York London: Academic Press 1964
Gihman, I.I., Skorohod, A.V.: Stochastic differential equations. Berlin Heidelberg New York: Springer 1972
Graham, H., Schenzle, A.: Carleman imbedding of multiplicative stochastic processes. Phys. Rev. A25, 1731–1754 (1982)
Hitsuda, M., Mitoma, I.: Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions. J. Multivariate Anal.19, 311–328 (1986)
Holley, R.A., Stroock, D.W.: Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions. Publ. RIMS. Kyoto Univ.14, 741–788 (1978)
Holváth, J.: Topological vector spaces and distributions 1. Reading, Mass.: Addison-Wesley 1966
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Tokyo: North-Holland-Kodansha 1981
Itô, K.: Distribution-valued processes arising from independent Brownian motions. Math. Z.182, 17–33 (1982)
Itô, K.: Infinite dimensional Ornstein-Uhlenbeck processes. Kinokuniya, Tokyo: Taniguchi Symp. SA, Katata, 197–224 (1984)
Itô, K., Nawata, M.: Regularization of linear random functionals. (Lect. Notes Math., vol. 1021), pp. 257–267. Berlin Heidelberg New York: Springer 1983
Kato, T.: Perturbation theory for linear operators. Berlin Heidelberg New York: Springer 1976
Komatsu, H.: Semi-groups of operators in locally convex spaces. J. Math. Soc. Japan16, 232–262 (1964)
Kunita, H.: Stochastic integrals based on martingales taking values in Hilbert space. Nagoya Math. J.38, 41–52 (1970)
Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. (Lect. Notes Math., vol. 1097). Berlin Heidelberg New York Tokyo: Springer 1984
Kusuoka, S., Tamura, Y.: Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo Sect. 1 A Math.31, 223–245 (1984)
McKean, H.P.: Propagation of chaos for a class of non linear parabolic equations. Lect. Ser. Differ. Eq. Catholic Univ.,7, 41–57 (1967)
Mitoma, I.: On the norm continuity ofL′-valued Gaussian processes. Nagoya Math. J.82, 209–220 (1981)
Mitoma, I.: On the sample continuity ofL′-processes. J. Math. Soc. Japan35, 629–636 (1983)
Mitoma, I.: Tightness of probabilities onC([0,1]:L′ andD([0,1]:L′. Ann. Probab.11, 989–999 (1983)
Mitoma, I.: An ∞-dimensional inhomogeneous Langevin's equation. J. Funct. Anal.61, 342–359 (1985)
Shiga, T., Tanaka, H.: Central limit theorem for a system of Markovian particles with mean field interactions. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 439–459 (1985)
Sznitman, A.S.: A fluctuation result for non linear diffusions. (preprint)
Tanaka, H., Hitsuda, M.: Central limit theorem for a simple diffusion model of interacting particles. Hiroshima Math. J.11, 415–423 (1981)
Tanaka, H.: Limit theorems for certain diffusion processes with interactions. Kinokuniya, Tokyo: Taniguchi Symp. SA, Katata, 469–488 (1984)
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Mitoma, I. Generalized ornstein-uhlenbeck process having a characteristic operator with polynomial coefficients. Probab. Th. Rel. Fields 76, 533–555 (1987). https://doi.org/10.1007/BF00960073
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DOI: https://doi.org/10.1007/BF00960073