A class of processes on the path space over a compact Riemannian manifold with unbounded diffusion

Löbus, Jörg-Uwe

doi:10.1090/S0002-9947-04-03439-7

A class of processes on the path space over a compact Riemannian manifold with unbounded diffusion
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by Jörg-Uwe Löbus

Trans. Amer. Math. Soc. 356 (2004), 3751-3767

DOI: https://doi.org/10.1090/S0002-9947-04-03439-7

Published electronically: January 13, 2004

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Abstract:

A class of diffusion processes on the path space over a compact Riemannian manifold is constructed. The diffusion of such a process is governed by an unbounded operator. A representation of the associated generator is derived and the existence of a certain local second moment is shown.

References

Shigeki Aida, Logarithmic derivatives of heat kernels and logarithmic Sobolev inequalities with unbounded diffusion coefficients on loop spaces, J. Funct. Anal. 174 (2000), no. 2, 430–477. MR 1768982, DOI 10.1006/jfan.2000.3592
S. Albeverio and M. Röckner, Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Related Fields 89 (1991), no. 3, 347–386. MR 1113223, DOI 10.1007/BF01198791
Nicolas Bouleau and Francis Hirsch, Dirichlet forms and analysis on Wiener space, De Gruyter Studies in Mathematics, vol. 14, Walter de Gruyter & Co., Berlin, 1991. MR 1133391, DOI 10.1515/9783110858389
Bruce K. Driver, A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold, J. Funct. Anal. 110 (1992), no. 2, 272–376. MR 1194990, DOI 10.1016/0022-1236(92)90035-H
Bruce K. Driver and Michael Röckner, Construction of diffusions on path and loop spaces of compact Riemannian manifolds, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 5, 603–608 (English, with English and French summaries). MR 1181300
Andreas Eberle, Diffusions on path and loop spaces: existence, finite-dimensional approximation and Hölder continuity, Probab. Theory Related Fields 109 (1997), no. 1, 77–99. MR 1469921, DOI 10.1007/s004400050126
J. Eells and K. D. Elworthy, Stochastic dynamical systems, Control theory and topics in functional analysis (Internat. Sem., Internat. Centre Theoret. Phys., Trieste, 1974) Internat. Atomic Energy Agency, Vienna, 1976, pp. 179–185. MR 0516923
Masatoshi Fukushima, Y\B{o}ichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354, DOI 10.1515/9783110889741
Elton P. Hsu, Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold, J. Funct. Anal. 134 (1995), no. 2, 417–450. MR 1363807, DOI 10.1006/jfan.1995.1152
Shigeo Kusuoka, Dirichlet forms and diffusion processes on Banach spaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 1, 79–95. MR 657873
Jörg-Uwe Löbus, A flat copy of the Ornstein-Uhlenbeck operator on the path space over a Riemannian manifold, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), no. 3, 351–394. MR 1930958, DOI 10.1142/S0219025702000900
Paul Malliavin, Formules de la moyenne, calcul de perturbations et théorèmes d’annulation pour les formes harmoniques, J. Functional Analysis 17 (1974), 274–291 (French). MR 0385932, DOI 10.1016/0022-1236(74)90041-x
Zhi Ming Ma and Michael Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375, DOI 10.1007/978-3-642-77739-4
Nolio Okada, On the Banach-Saks property, Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), no. 7, 246–248. MR 774563
Michael Röckner and Byron Schmuland, Tightness of general $C_{1,p}$ capacities on Banach space, J. Funct. Anal. 108 (1992), no. 1, 1–12. MR 1174156, DOI 10.1016/0022-1236(92)90144-8
M. Röckner and T. S. Zhang, Finite-dimensional approximation of diffusion processes of infinite-dimensional spaces, Stochastics Stochastics Rep. 57 (1996), no. 1-2, 37–55. MR 1407946, DOI 10.1080/17442509608834050