Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes

Inahama, Yuzuru

doi:10.1090/S0002-9947-2015-06290-4

Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes
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Abstract:

Since T. Lyons invented rough path theory, one of its most successful applications is a new proof of Freidlin-Wentzell’s large deviation principle for diffusion processes. In this paper we extend this method to the case of pinned diffusion processes under a mild ellipticity assumption. Besides rough path theory, our main tool is quasi-sure analysis, which is a kind of potential theory in Malliavin calculus.

References

Shigeki Aida, Vanishing of one-dimensional $L^2$-cohomologies of loop groups, J. Funct. Anal. 261 (2011), no. 8, 2164–2213. MR 2824575, DOI 10.1016/j.jfa.2011.06.003
I. Bailleul, Large deviation principle for bridges of degenerate diffusion processes. Preprint 2013. Arxiv Math: 1303.2854.
P. Baldi and M. Sanz, Une remarque sur la théorie des grandes déviations, Séminaire de Probabilités, XXV, Lecture Notes in Math., vol. 1485, Springer, Berlin, 1991, pp. 345–348 (French). MR 1187792, DOI 10.1007/BFb0100868
Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, 2nd ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998. MR 1619036, DOI 10.1007/978-1-4612-5320-4
S. Dereich, Rough paths analysis of general Banach space-valued Wiener processes, J. Funct. Anal. 258 (2010), no. 9, 2910–2936. MR 2595729, DOI 10.1016/j.jfa.2010.01.018
Avner Friedman, Stochastic differential equations and applications. Vol. 2, Probability and Mathematical Statistics, Vol. 28, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0494491
Peter Friz and Harald Oberhauser, A generalized Fernique theorem and applications, Proc. Amer. Math. Soc. 138 (2010), no. 10, 3679–3688. MR 2661566, DOI 10.1090/S0002-9939-2010-10528-2
Peter Friz and Nicolas Victoir, Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 703–724 (English, with English and French summaries). MR 2144230, DOI 10.1016/j.anihpb.2004.05.003
P. Friz and N. Victoir, Large deviation principle for enhanced Gaussian processes. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), no. 6, 775–785.
Peter K. Friz and Nicolas B. Victoir, Multidimensional stochastic processes as rough paths, Cambridge Studies in Advanced Mathematics, vol. 120, Cambridge University Press, Cambridge, 2010. Theory and applications. MR 2604669, DOI 10.1017/CBO9780511845079
Fuqing Gao and Jiagang Ren, Large deviations for stochastic flows and their applications, Sci. China Ser. A 44 (2001), no. 8, 1016–1033. MR 1857556, DOI 10.1007/BF02878977
Y. Higuchi, Master Theses (in Japanese), Graduate School of Engineering Sciences, Osaka University, 2006.
Pei Hsu, Brownian bridges on Riemannian manifolds, Probab. Theory Related Fields 84 (1990), no. 1, 103–118. MR 1027823, DOI 10.1007/BF01288561
Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
Yuzuru Inahama, Quasi-sure existence of Brownian rough paths and a construction of Brownian pants, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 4, 513–528. MR 2282717, DOI 10.1142/S0219025706002512
Yuzuru Inahama and Hiroshi Kawabi, Large deviations for heat kernel measures on loop spaces via rough paths, J. London Math. Soc. (2) 73 (2006), no. 3, 797–816. MR 2241981, DOI 10.1112/S0024610706022654
Shigeo Kusuoka and Daniel W. Stroock, Precise asymptotics of certain Wiener functionals, J. Funct. Anal. 99 (1991), no. 1, 1–74. MR 1120913, DOI 10.1016/0022-1236(91)90051-6
M. Ledoux, Z. Qian, and T. Zhang, Large deviations and support theorem for diffusion processes via rough paths, Stochastic Process. Appl. 102 (2002), no. 2, 265–283. MR 1935127, DOI 10.1016/S0304-4149(02)00176-X
Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310. MR 1654527, DOI 10.4171/RMI/240
Terry Lyons and Zhongmin Qian, System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002. Oxford Science Publications. MR 2036784, DOI 10.1093/acprof:oso/9780198506485.001.0001
Paul Malliavin, Stochastic analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 313, Springer-Verlag, Berlin, 1997. MR 1450093, DOI 10.1007/978-3-642-15074-6
Paul Malliavin and David Nualart, Quasi-sure analysis of stochastic flows and Banach space valued smooth functionals on the Wiener space, J. Funct. Anal. 112 (1993), no. 2, 287–317. MR 1213140, DOI 10.1006/jfan.1993.1034
Annie Millet and Marta Sanz-Solé, Large deviations for rough paths of the fractional Brownian motion, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 2, 245–271 (English, with English and French summaries). MR 2199801, DOI 10.1016/j.anihpb.2005.04.003
Jia Gang Ren, Analyse quasi-sûre des équations différentielles stochastiques, Bull. Sci. Math. 114 (1990), no. 2, 187–213 (French, with English summary). MR 1056161
Ichiro Shigekawa, Stochastic analysis, Translations of Mathematical Monographs, vol. 224, American Mathematical Society, Providence, RI, 2004. Translated from the 1998 Japanese original by the author; Iwanami Series in Modern Mathematics. MR 2060917, DOI 10.1090/mmono/224
Hiroshi Sugita, Positive generalized Wiener functions and potential theory over abstract Wiener spaces, Osaka J. Math. 25 (1988), no. 3, 665–696. MR 969026
Hiroshi Sugita, Hu-Meyer’s multiple Stratonovich integral and essential continuity of multiple Wiener integral, Bull. Sci. Math. 113 (1989), no. 4, 463–474. MR 1029620
S. Takanobu and S. Watanabe, Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations, Asymptotic problems in probability theory: Wiener functionals and asymptotics (Sanda/Kyoto, 1990) Pitman Res. Notes Math. Ser., vol. 284, Longman Sci. Tech., Harlow, 1993, pp. 194–241. MR 1354169
Shinzo Watanabe, Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab. 15 (1987), no. 1, 1–39. MR 877589
Shinzo Watanabe, Itô calculus and Malliavin calculus, Stochastic analysis and applications, Abel Symp., vol. 2, Springer, Berlin, 2007, pp. 623–639. MR 2397809, DOI 10.1007/978-3-540-70847-6_{2}9