Doob decomposition, Dirichlet processes, and entropies on Wiener space. (English) Zbl 1497.60065
Chen, Zhen-Qing (ed.) et al., Dirichlet forms and related topics, in honor of Masatoshi Fukushima’s beiju, IWDFRT 2022, Osaka, Japan, August 22–26,2022. Singapore: Springer. Springer Proc. Math. Stat. 394, 119-141 (2022).
Summary: As an extension of the Doob-Meyer decomposition of a semimartingale and the Fukushima representation of a Dirichlet process, we introduce a general Doob decomposition in continuous time, where a square-integrable process is represented as the sum of a martingale and a process with “vanishing local risk”. For a probability measure \(Q\) on Wiener space, we discuss how entropy conditions on \(Q\) formulated with respect to Wiener measure \(P\) are connected with the Doob decomposition of the coordinate process \(W\) under \(Q\). The situation is well understood if the relative entropy \(H(Q|P)\) is finite; in this case the decomposition is classical and yields an immediate proof of Talagrand’s transport inequality on Wiener space. To go beyond this restriction, we consider the specific relative entropy \(h(Q|P)\) on Wiener space that was introduced by N. Gantert in [Einige große Abweichungen der Brownschen Bewegung. Bonn: Univ. Bonn, Math.-Naturwiss. Fak. (1991; Zbl 0746.60026)]. We discuss its interplay with the Doob decomposition of \(W\) under \(Q\) and a corresponding version of Talagrand’s inequality, with special emphasis on the case where \(W\) is a Dirichlet process under \(Q\).
For the entire collection see [Zbl 1493.11005].
For the entire collection see [Zbl 1493.11005].
MSC:
| 60G44 | Martingales with continuous parameter |
| 60G48 | Generalizations of martingales |
| 60J65 | Brownian motion |
| 60F10 | Large deviations |
| 46G12 | Measures and integration on abstract linear spaces |
Keywords:
optimal transport; coupling on Wiener space; relative entropy; Talagrand’s inequality; Dirichlet processes; Doob decomposition; Fukushima decompositionCitations:
Zbl 0746.60026References:
| [1] | Acciaio, B.; Backhoff Veraguas, J.; Zalashko, A., Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization, Stoch. Process. Appl., 9, 3, 203-228 (2019) · Zbl 1435.90012 |
| [2] | Bertoin, J., Les Processus de Dirichlet en tant qu’Espaces de Banach, Stochastics, 18, 155-188 (1986) · Zbl 0602.60069 · doi:10.1080/17442508608833406 |
| [3] | F. Coquet, A. Jakubowski, J. Mémin, L. Slominski, Natural decomposition of processes and weak dirichlet processes, in In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX. Lecture Notes in Mathematics, vol. 1874 (Springer, Berlin, 2006), pp. 81-116 · Zbl 1139.60019 |
| [4] | C. Dellacherie, P.-A. Meyer, Probabilités et Potentiel, Ch. V-VIII (Hermann, Paris, 1980) · Zbl 0464.60001 |
| [5] | Feyel, D.; Üstünel, AS, Monge-Kantorovich measure transportation and Monge-Ampere equation on Wiener Space, Probab. Theor. Relat. Fields, 128, 3, 347-385 (2004) · Zbl 1055.60052 · doi:10.1007/s00440-003-0307-x |
| [6] | Föllmer, H., The exit measure of a supermartingale, Z. Wahrscheinlichkeitstheorie Verw. Geb., 21, 154-166 (1972) · Zbl 0231.60033 · doi:10.1007/BF00532472 |
| [7] | H. Föllmer, Dirichlet processes, in Stochastic Integrals, ed. by D. Williams. Lecture Notes in Mathematics, vol. 851 (Springer, 1981), pp. 476-478 · Zbl 0462.60046 |
| [8] | H. Föllmer, Random fields and diffusion processes, in Ecole d’ été de Probabilités de Saint-Flour XV-XVII, 1985-87. Lecture Notes in Mathematics, vol. 1362 (Springer, Berlin, 1988), pp. 101-203 · Zbl 0661.60063 |
| [9] | H. Föllmer, Optimal coupling on Wiener space and an extension of Talagrand’s transport inequality, in Stochastic Analysis, Stochastic Control, and Stochastic Optimization—A Commemorative Volume to Honor Professor Mark H. Davis’s Contributions, ed. by G. Yin, T. Zariphopoulou (Springer, 2022) |
| [10] | M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland Mathematical Library, vol. 23 (North-Holland Publishing Co., Amsterdam, 1980) · Zbl 0422.31007 |
| [11] | N. Gantert, Einige grosse Abweichungen der Brownschen Bewegung. Ph.D. thesis. Universität Bonn, 1991 · Zbl 0746.60026 |
| [12] | Gantert, N., Self-similarity of Brownian motion and a large deviation principle for random fields on a binary tree, Probab. Theory Relat. Fields, 98, 7-20 (1994) · Zbl 0794.60014 · doi:10.1007/BF01311346 |
| [13] | Gozzi, F.; Russo, F., Weak Dirichlet processes with a stochastic control perspective, Stoch. Process. Appl., 116, 11, 1563-1583 (2006) · Zbl 1113.60036 · doi:10.1016/j.spa.2006.04.009 |
| [14] | Graversen, SE; Rao, M., Quadratic variation and energy, Nagoya Math. J., 100, 163-180 (1985) · Zbl 0586.60070 · doi:10.1017/S0027763000000283 |
| [15] | Lassalle, R., Causal transference plans and their Monge-Kantorovich problems, Stoch. Anal. Appl., 36, 1, 1-33 (2018) · Zbl 1407.91037 · doi:10.1080/07362994.2017.1356732 |
| [16] | Lehec, J., Representation formula for the entropy and functional inequalities, Ann. Inst. Henri Poincaré Probab. Stat., 49, 3, 885-899 (2013) · Zbl 1279.39011 · doi:10.1214/11-AIHP464 |
| [17] | Talagrand, M., Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6, 3, 587-600 (1996) · Zbl 0859.46030 · doi:10.1007/BF02249265 |
| [18] | C. Villani, Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338 (Springer, 2009) · Zbl 1156.53003 |
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.
