Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Hypoelliptic heat kernels on infinite-dimensional Heisenberg groups


Authors: Bruce K. Driver, Nathaniel Eldredge and Tai Melcher
Journal: Trans. Amer. Math. Soc. 368 (2016), 989-1022
MSC (2010): Primary 58J35; Secondary 58J65, 60B15
DOI: https://doi.org/10.1090/tran/6461
Published electronically: June 11, 2015
MathSciNet review: 3430356
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the law of a hypoelliptic Brownian motion on an infinite-dimensional Heisenberg group based on an abstract Wiener space. We show that the endpoint distribution, which can be seen as a heat kernel measure, is absolutely continuous with respect to a certain product of Gaussian and Lebesgue measures, that the heat kernel is quasi-invariant under translation by the Cameron-Martin subgroup, and that the Radon-Nikodym derivative is Malliavin smooth.


References [Enhancements On Off] (What's this?)

  • [1] Andrei Agrachev, Ugo Boscain, Jean-Paul Gauthier, and Francesco Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, J. Funct. Anal. 256 (2009), no. 8, 2621-2655. MR 2502528 (2010c:58042), https://doi.org/10.1016/j.jfa.2009.01.006
  • [2] Martín Argerami, Approximating a Hilbert-Schmidt operator, Mathematics Stack Exchange, URL:http://math.stackexchange.com/q/373117 (version: 2013-04-26).
  • [3] Fabrice Baudoin, Michel Bonnefont, and Nicola Garofalo, A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality, Math. Ann. 358 (2014), no. 3-4, 833-860. MR 3175142, https://doi.org/10.1007/s00208-013-0961-y
  • [4] Fabrice Baudoin and Nicola Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, ArXiv e-prints (2011).
  • [5] Fabrice Baudoin, Maria Gordina, and Tai Melcher, Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups, Trans. Amer. Math. Soc. 365 (2013), no. 8, 4313-4350. MR 3055697, https://doi.org/10.1090/S0002-9947-2012-05778-3
  • [6] 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 (2006g:60080), https://doi.org/10.1214/105051605000000214
  • [7] Vladimir I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998. MR 1642391 (2000a:60004)
  • [8] Chin-Huei Chang, Der-Chen Chang, Peter Greiner, and Hsuan-Pei Lee, The positivity of the heat kernel on Heisenberg group, Anal. Appl. (Singap.) 11 (2013), no. 5, 1350019, 15. MR 3104104, https://doi.org/10.1142/S021953051350019X
  • [9] Daniel Dobbs and Tai Melcher, Smoothness of heat kernel measures on infinite-dimensional Heisenberg-like groups, J. Funct. Anal. 264 (2013), no. 9, 2206-2223. MR 3029152, https://doi.org/10.1016/j.jfa.2013.02.013
  • [10] 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 (99a:60054a), https://doi.org/10.1006/jfan.1997.3103
  • [11] 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 (2005e:58058), https://doi.org/10.1090/conm/338/06072
  • [12] 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 (2010f:60010), https://doi.org/10.1016/j.jfa.2008.06.021
  • [13] Bruce K. Driver and Maria Gordina, Integrated Harnack inequalities on Lie groups, J. Differential Geom. 83 (2009), no. 3, 501-550. MR 2581356 (2011d:58086)
  • [14] 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 (2011k:58052), https://doi.org/10.1007/s00440-009-0213-y
  • [15] Nathaniel Eldredge, Precise estimates for the subelliptic heat kernel on $ H$-type groups, J. Math. Pures Appl. (9) 92 (2009), no. 1, 52-85 (English, with English and French summaries). MR 2541147 (2010h:35399), https://doi.org/10.1016/j.matpur.2009.04.011
  • [16] Xavier Fernique, Intégrabilité des vecteurs gaussiens, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1698-A1699 (French). MR 0266263 (42 #1170)
  • [17] Bernard Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), no. 1-2, 95-153. MR 0461589 (57 #1574)
  • [18] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. MR 0222474 (36 #5526)
  • [19] Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726 (99f:60082)
  • [20] James Kuelbs and Wenbo Li, A functional LIL for stochastic integrals and the Lévy area process, J. Theoret. Probab. 18 (2005), no. 2, 261-290. MR 2137043 (2006c:60040), https://doi.org/10.1007/s10959-003-2604-9
  • [21] Hui Hsiung Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR 0461643 (57 #1628)
  • [22] Shigeo Kusuoka and Daniel W. Stroock, Precise asymptotics of certain Wiener functionals, J. Funct. Anal. 99 (1991), no. 1, 1-74. MR 1120913 (93a:60085), https://doi.org/10.1016/0022-1236(91)90051-6
  • [23] Michel Ledoux, A note on large deviations for Wiener chaos, Séminaire de Probabilités, XXIV, 1988/89, Lecture Notes in Math., vol. 1426, Springer, Berlin, 1990, pp. 1-14. MR 1071528 (91i:60017), https://doi.org/10.1007/BFb0083753
  • [24] Paul Lévy, Le mouvement brownien plan, Amer. J. Math. 62 (1940), 487-550 (French). MR 0002734 (2,107g)
  • [25] Paul Lévy, Wiener's random function, and other Laplacian random functions, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, pp. 171-187. MR 0044774 (13,476b)
  • [26] Wenbo V. Li, Small ball probabilities for Gaussian Markov processes under the $ L_p$-norm, Stochastic Process. Appl. 92 (2001), no. 1, 87-102. MR 1815180 (2001m:60090), https://doi.org/10.1016/S0304-4149(00)00072-7
  • [27] George Lowther, Does infinite-dimensional Brownian motion live in hyperplanes?, MathOverflow, URL:http://mathoverflow.net/q/102963 (version: 2012-07-24).
  • [28] Françoise Lust-Piquard, A simple-minded computation of heat kernels on Heisenberg groups, Colloq. Math. 97 (2003), no. 2, 233-249. MR 2031850 (2005g:22006), https://doi.org/10.4064/cm97-2-9
  • [29] 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 (93b:60132)
  • [30] Roger Mansuy and Marc Yor, Aspects of Brownian motion, Universitext, Springer-Verlag, Berlin, 2008. MR 2454984 (2010a:60278)
  • [31] 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 (2007j:60082), https://doi.org/10.1002/cpa.20136
  • [32] Michel Métivier, Semimartingales, de Gruyter Studies in Mathematics, vol. 2, Walter de Gruyter & Co., Berlin-New York, 1982. A course on stochastic processes. MR 688144 (84i:60002)
  • [33] Jennifer Randall, The heat kernel for generalized Heisenberg groups, J. Geom. Anal. 6 (1996), no. 2, 287-316. MR 1469125 (99b:22018), https://doi.org/10.1007/BF02921603
  • [34] N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884 (95f:43008)
  • [35] Marc Yor, Les filtrations de certaines martingales du mouvement brownien dans $ {\bf R}^{n}$, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), Lecture Notes in Math., vol. 721, Springer, Berlin, 1979, pp. 427-440 (French). MR 544812 (81b:60045)
  • [36] Marc Yor, Remarques sur une formule de Paul Lévy, Seminar on Probability, XIV (Paris, 1978/1979) (French), Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 343-346. MR 580140 (82c:60144)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 58J35, 58J65, 60B15

Retrieve articles in all journals with MSC (2010): 58J35, 58J65, 60B15


Additional Information

Bruce K. Driver
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
Email: bdriver@math.ucsd.edu

Nathaniel Eldredge
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853 – and – School of Mathematical Sciences, University of Northern Colorado, Greeley, Colorado 80639
Email: neldredge@unco.edu

Tai Melcher
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: melcher@virginia.edu

DOI: https://doi.org/10.1090/tran/6461
Keywords: Heisenberg group, hypoelliptic, heat kernel, smooth measures
Received by editor(s): December 12, 2013
Published electronically: June 11, 2015
Additional Notes: The first author’s research was supported in part by NSF Grant DMS-1106270.
The second author’s research was supported in part by NSF Grant DMS-0739164.
The third author’s research was supported in part by NSF Grants DMS-0907293 and DMS-1255574.
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society