[Calcul de variations stochastiques et l'inéqualité de Harnack sur l'espace de chemins riemanniens]
On décrit l'espace tangent à l'espace de chemins riemanniens comme un espace de processus tangents localisé sur des fueuilles browniennes ; le fibré de repères adaptés sur l'espace de chemins riemanniens et son équation de structure sont donnés. Le calcul de variations stochastiques permet de dériver l'inégalité de Harnack–Bismut pour le semigroupe de Norris.
We describe the tangent space of Riemannian path space as a space of tangent processes localized on Brownian sheets; the bundle of adapted frames above a Riemannian path space and its structural equation are given. The stochastic calculus of variations allows us to derive Harnack–Bismut inequality for the Norris semigroup.
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@article{CRMATH_2002__335_10_817_0,
author = {Ana-Bela Cruzeiro and Paul Malliavin},
title = {Stochastic calculus of variations and {Harnack} inequality on {Riemannian} path spaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {817--820},
publisher = {Elsevier},
volume = {335},
number = {10},
year = {2002},
doi = {10.1016/S1631-073X(02)02561-X},
language = {en},
}
TY - JOUR AU - Ana-Bela Cruzeiro AU - Paul Malliavin TI - Stochastic calculus of variations and Harnack inequality on Riemannian path spaces JO - Comptes Rendus. Mathématique PY - 2002 SP - 817 EP - 820 VL - 335 IS - 10 PB - Elsevier DO - 10.1016/S1631-073X(02)02561-X LA - en ID - CRMATH_2002__335_10_817_0 ER -
Ana-Bela Cruzeiro; Paul Malliavin. Stochastic calculus of variations and Harnack inequality on Riemannian path spaces. Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 817-820. doi : 10.1016/S1631-073X(02)02561-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02561-X/
[1] A Weitzenböck formula for the damped O–U operator in adapted differential geometry, C. R. Acad. Sci. Paris, Série I, Volume 332 (2001), pp. 447-452
[2] A probabilistic Weitzenböck formula on Riemannian path space, J. Analyse Math, Volume 80 (2000), pp. 87-100
[3] Renormalized differential geometry on path space: structural equation, curvature, J. Funct. Anal, Volume 139 (1996), pp. 119-181
[4] Frame bundle of Riemannian path space and Ricci tensor in adapted differential geometry, J. Funct. Anal, Volume 177 (2000), pp. 219-253
[5] A.B. Cruzeiro, X. Zhang, Finite dimensional approximation of Riemannian path space geometry, J. Funct. Anal., to appear
[6] Measures and Differential Equations in Infinite-Dimensional Space, Math. Appl, Kluwer Academic, 1991
[7] Construction of diffusions on path space and on loop space of compact Riemannian manifold, C. R. Acad. Sci. Paris, Série I, Volume 320 (1995), pp. 1249-1254
[8] Heat equation derivative formulas for vector bundles, J. Funct. Anal, Volume 183 (2001), pp. 42-108
[9] Markovian connection, curvatures and Weitzenböck formula on Riemannian path space, J. Funct. Anal, Volume 181 (2001), pp. 476-507
[10] Stochastic analysis on the path space of a Riemannian manifold, J. Funct. Anal, Volume 118 (1993), pp. 249-274
[11] Le processus de Ornstein–Uhlenbeck sur l'espace des chemins et le probleme des martingales, J. Funct. Anal, Volume 144 (1997), pp. 20-45
[12] Stochastic Analysis, Grundlehren Math, Springer, 1997
[13] Twisted sheets, J. Funct. Anal, Volume 132 (1995), pp. 273-334
[14] Stochastic Analysis on Riemannian Path Space, American Mathematical Society, 2000
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