Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups

Baudoin, Fabrice; Gordina, Maria; Melcher, Tai

doi:10.1090/S0002-9947-2012-05778-3

Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups
HTML articles powered by AMS MathViewer

by Fabrice Baudoin, Maria Gordina and Tai Melcher PDF

Trans. Amer. Math. Soc. 365 (2013), 4313-4350 Request permission

Abstract:

We study heat kernel measures on sub-Riemannian infinite- dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $L^p$-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvature-dimension estimate which holds on approximating finite-dimensional projection groups. Such estimates were first introduced by Baudoin and Garofalo.

References

Shigeki Aida and Hiroshi Kawabi, Short time asymptotics of a certain infinite dimensional diffusion process, Stochastic analysis and related topics, VII (Kusadasi, 1998) Progr. Probab., vol. 48, Birkhäuser Boston, Boston, MA, 2001, pp. 77–124. MR 1915450
Dominique Bakry, Fabrice Baudoin, Michel Bonnefont, and Bin Qian, Subelliptic Li-Yau estimates on three dimensional model spaces, Potential theory and stochastics in Albac, Theta Ser. Adv. Math., vol. 11, Theta, Bucharest, 2009, pp. 1–10. MR 2681833
D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206 (French). MR 889476, DOI 10.1007/BFb0075847
Dominique Bakry and Michel Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality, Rev. Mat. Iberoam. 22 (2006), no. 2, 683–702. MR 2294794, DOI 10.4171/RMI/470
Dominique Bakry and Zhongmin M. Qian, Harnack inequalities on a manifold with positive or negative Ricci curvature, Rev. Mat. Iberoamericana 15 (1999), no. 1, 143–179. MR 1681640, DOI 10.4171/RMI/253
Fabrice Baudoin and Michel Bonnefont, Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality, J. Funct. Anal. 262 (2012), no. 6, 2646–2676. MR 2885961, DOI 10.1016/j.jfa.2011.12.020
Fabrice Baudoin, Michel Bonnefont and Nicola Garofalo, A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality, arXiv:1007.1600, 2010.
Fabrice Baudoin and Nicola Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, arXiv:1101.3590v1, 2011.
Fabrice Baudoin and Josef Teichmann, Hypoellipticity in infinite dimensions and an application in interest rate theory, Ann. Appl. Probab. 15 (2005), no. 3, 1765–1777. MR 2152244, DOI 10.1214/105051605000000214
Vladimir I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998. MR 1642391, DOI 10.1090/surv/062
Matthew Cecil, The Taylor map on complex path groups, J. Funct. Anal. 254 (2008), no. 2, 318–367. MR 2376574, DOI 10.1016/j.jfa.2007.09.018
Bruce K. Driver, Towards calculus and geometry on path spaces, Stochastic analysis (Ithaca, NY, 1993) Proc. Sympos. Pure Math., vol. 57, Amer. Math. Soc., Providence, RI, 1995, pp. 405–422. MR 1335485, DOI 10.1090/pspum/057/1335485
Bruce K. Driver, Integration by parts and quasi-invariance for heat kernel measures on loop groups, J. Funct. Anal. 149 (1997), no. 2, 470–547. MR 1472366, DOI 10.1006/jfan.1997.3103
Bruce K. Driver, Heat kernels measures and infinite dimensional analysis, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 101–141. MR 2039953, DOI 10.1090/conm/338/06072
Bruce K. Driver and Maria Gordina, Heat kernel analysis on infinite-dimensional Heisenberg groups, J. Funct. Anal. 255 (2008), no. 9, 2395–2461. MR 2473262, DOI 10.1016/j.jfa.2008.06.021
Bruce K. Driver and Maria Gordina, Integrated Harnack inequalities on Lie groups, J. Differential Geom. 83 (2009), no. 3, 501–550. MR 2581356
Bruce K. Driver and Maria Gordina, Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups, Probab. Theory Related Fields 147 (2010), no. 3-4, 481–528. MR 2639713, DOI 10.1007/s00440-009-0213-y
Maria Gordina, Heat kernel analysis and Cameron-Martin subgroup for infinite dimensional groups, J. Funct. Anal. 171 (2000), no. 1, 192–232. MR 1742865, DOI 10.1006/jfan.1999.3505
Maria Gordina and Tai Melcher, A subelliptic Taylor isomorphism on infinite–dimensional Heisenberg groups, to appear in Probab. Theory Related Fields, (2011).
Mikhael Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79–323. MR 1421823
Leonard Gross, Potential theory on Hilbert space, J. Functional Analysis 1 (1967), 123–181. MR 0227747, DOI 10.1016/0022-1236(67)90030-4
Martin Hairer and Jonathan C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math. (2) 164 (2006), no. 3, 993–1032. MR 2259251, DOI 10.4007/annals.2006.164.993
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726, DOI 10.1017/CBO9780511526169
Hui Hsiung Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR 0461643
Michel Ledoux, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 2, 305–366 (English, with English and French summaries). Probability theory. MR 1813804
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
Paul Malliavin, Hypoellipticity in infinite dimensions, Diffusion processes and related problems in analysis, Vol. I (Evanston, IL, 1989) Progr. Probab., vol. 22, Birkhäuser Boston, Boston, MA, 1990, pp. 17–31. MR 1110154
Jonathan C. Mattingly and Étienne Pardoux, Malliavin calculus for the stochastic 2D Navier-Stokes equation, Comm. Pure Appl. Math. 59 (2006), no. 12, 1742–1790. MR 2257860, DOI 10.1002/cpa.20136
Tai Melcher, Heat kernel analysis on semi-infinite Lie groups, J. Funct. Anal. 257 (2009), no. 11, 3552–3592. MR 2572261, DOI 10.1016/j.jfa.2009.08.003
Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
Hiroshi Sugita, Properties of holomorphic Wiener functions—skeleton, contraction, and local Taylor expansion, Probab. Theory Related Fields 100 (1994), no. 1, 117–130. MR 1292193, DOI 10.1007/BF01204956
Hiroshi Sugita, Regular version of holomorphic Wiener function, J. Math. Kyoto Univ. 34 (1994), no. 4, 849–857. MR 1311623, DOI 10.1215/kjm/1250518889
Feng-Yu Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997), no. 3, 417–424. MR 1481127, DOI 10.1007/s004400050137
Feng-Yu Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007), no. 4, 1333–1350. MR 2330974, DOI 10.1214/009117906000001204
Feng-Yu Wang, Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl. (9) 94 (2010), no. 3, 304–321 (English, with English and French summaries). MR 2679029, DOI 10.1016/j.matpur.2010.03.001