## Coupling in the Heisenberg group and its applications to gradient estimates.(English)Zbl 1441.60013

A coupling of two probability measures $$\mu_1$$ and $$\mu_2$$, defined on respective measure spaces $$(\Omega_1, \mathcal{A}_1)$$ and $$(\Omega_2, \mathcal{A}_2)$$, is a measure $$\mu$$ on the product space $$(\Omega_1\times \Omega_2, \mathcal{A}_1 \times\mathcal{A}_2)$$ with marginals $$\mu_1$$ and $$\mu_2$$. In this paper, coupling of two Markov processes having the same generator, but starting from different points joining together (coupling) at some random time is considered, and how these can be used to obtain total variation bounds and prove gradient estimates for harmonic functions on $$\mathbf{H}^3$$. The paper is organized as follows. Section 2 gives basics on sub-Riemannian manifolds and the Heisenberg group $$\mathbf{H}^3$$ including Brownian motion on $$\mathbf{H}^3$$. In Section 3, the non-Markovian coupling of Brownian motions in $$\mathbf{H}^3$$ is constructed and describes its properties. In Section 4, the authors prove the gradient estimates for harmonic functions for the hypoelliptic Laplacian which is the generator of Brownian motion in the Heisenberg group.

### MSC:

 60D05 Geometric probability and stochastic geometry 60H30 Applications of stochastic analysis (to PDEs, etc.)
Full Text:

### References:

 [1] Bakry, D., Baudoin, F., Bonnefont, M. and Chafaï, D. (2008). On gradient bounds for the heat kernel on the Heisenberg group. J. Funct. Anal.255 1905–1938. · Zbl 1156.58009 [2] Banerjee, S. and Kendall, W. S. (2016). Coupling the Kolmogorov diffusion: Maximality and efficiency considerations. Adv. in Appl. Probab.48 15–35. [3] Banerjee, S. and Kendall, W. S. (2017). Rigidity for Markovian maximal couplings of elliptic diffusions. Probab. Theory Related Fields168 55–112. · Zbl 1374.60148 [4] Baudoin, F., Bonnefont, M. and Garofalo, N. (2014). A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality. Math. Ann.358 833–860. · Zbl 1287.53025 [5] Baudoin, F. and Garofalo, N. (2017). Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. J. Eur. Math. Soc. (JEMS) 19 151–219. · Zbl 1359.53018 [6] Ben Arous, G., Cranston, M. and Kendall, W. S. (1995). Coupling constructions for hypoelliptic diffusions: Two examples. In Stochastic Analysis. Proc. Sympos. Pure Math.57 193–212. Amer. Math. Soc., Providence, RI. · Zbl 0823.60068 [7] Bonfiglioli, A., Lanconelli, E. and Uguzzoni, F. (2007). Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer, Berlin. · Zbl 1128.43001 [8] Calin, O., Chang, D.-C. and Greiner, P. (2007). Geometric Analysis on the Heisenberg Group and Its Generalizations. AMS/IP Studies in Advanced Mathematics40. Amer. Math. Soc., Providence, RI; International Press, Somerville, MA. · Zbl 1132.53001 [9] Chang, S.-C., Tie, J. and Wu, C.-T. (2010). Subgradient estimate and Liouville-type theorem for the CR heat equation on Heisenberg groups. Asian J. Math.14 41–72. · Zbl 1214.32011 [10] Cheng, S. Y. and Yau, S. T. (1975). Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math.28 333–354. · Zbl 0312.53031 [11] Cranston, M. (1991). Gradient estimates on manifolds using coupling. J. Funct. Anal.99 110–124. · Zbl 0770.58038 [12] Cranston, M. (1992). A probabilistic approach to gradient estimates. Canad. Math. Bull.35 46–55. · Zbl 0788.60089 [13] Driver, B. K. and Melcher, T. (2005). Hypoelliptic heat kernel inequalities on the Heisenberg group. J. Funct. Anal.221 340–365. · Zbl 1071.22005 [14] Goldstein, S. (1978/79). Maximal coupling. Z. Wahrsch. Verw. Gebiete46 193–204. · Zbl 0398.60097 [15] Gordina, M. and Laetsch, T. (2016). Sub-Laplacians on sub-Riemannian manifolds. Potential Anal.44 811–837. · Zbl 1338.35447 [16] Griffeath, D. (1974/75). A maximal coupling for Markov chains. Z. Wahrsch. Verw. Gebiete31 95–106. · Zbl 0301.60043 [17] Hajłasz, P. and Koskela, P. (2007). Sobolev Met Poincaré. Mem. Amer. Math. Soc.145. [18] Hörmander, L. (1967). Hypoelliptic second order differential equations. Acta Math.119 147–171. [19] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics113. Springer, New York. · Zbl 0734.60060 [20] Karhunen, K. (1947). Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys.1947 79. [21] Kendall, W. S. (1986). Nonnegative Ricci curvature and the Brownian coupling property. Stochastics19 111–129. · Zbl 0584.58045 [22] Kendall, W. S. (1989). Coupled Brownian motions and partial domain monotonicity for the Neumann heat kernel. J. Funct. Anal.86 226–236. · Zbl 0684.60060 [23] Kendall, W. S. (2007). Coupling all the Lévy stochastic areas of multidimensional Brownian motion. Ann. Probab.35 935–953. · Zbl 1133.60033 [24] Kendall, W. S. (2010). Coupling time distribution asymptotics for some couplings of the Lévy stochastic area. In Probability and Mathematical Genetics. London Mathematical Society Lecture Note Series378 446–463. Cambridge Univ. Press, Cambridge. · Zbl 1213.60131 [25] Kuwada, K. (2007). On uniqueness of maximal coupling for diffusion processes with a reflection. J. Theoret. Probab.20 935–957. · Zbl 1137.60043 [26] Kuwada, K. (2010). Duality on gradient estimates and Wasserstein controls. J. Funct. Anal.258 3758–3774. · Zbl 1194.53032 [27] Kuwada, K. and Sturm, K.-T. (2007). A counterexample for the optimality of Kendall–Cranston coupling. Electron. Commun. Probab.12 66–72 (electronic). · Zbl 1134.60011 [28] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. · Zbl 1160.60001 [29] Li, H.-Q. (2006). Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg. J. Funct. Anal.236 369–394. · Zbl 1106.22009 [30] Loève, M. (1948). Sur l’équivalence asymptotique des lois. C. R. Math. Acad. Sci. Paris227 1335–1337. · Zbl 0034.22501 [31] Montgomery, R. (2002). A Tour of Subriemannian Geometries, Their Geodesics and Applications. Mathematical Surveys and Monographs91. Amer. Math. Soc., Providence, RI. · Zbl 1044.53022 [32] Neuenschwander, D. (1996). Probabilities on the Heisenberg Group: Limit theorems and Brownian motion. Lecture Notes in Math.1630. Springer, Berlin. · Zbl 0870.60007 [33] Pitman, J. W. (1976). On coupling of Markov chains. Z. Wahrsch. Verw. Gebiete35 315–322. · Zbl 0356.60003 [34] Sverchkov, M. Y. and Smirnov, S. N. (1990). Maximal coupling for processes in $$D[0,∞ ]$$. Dokl. Akad. Nauk SSSR311 1059–1061. · Zbl 0755.60026 [35] Wang, L. (2008). Karhunen–Loeve Expansions and Their Applications. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)—London School of Economics and Political Science (United Kingdom). [36] Yau, S. T. (1975). Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math.28 201–228. · Zbl 0291.31002 [37] Yau, S. T. and Schoen, R. (1994). Lectures on Differential Geometry. International Press, Boston, MA. · Zbl 0830.53001 [38] Yor, M. · Zbl 0751.60077
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.