# zbMATH — the first resource for mathematics

Radial processes for sub-Riemannian Brownian motions and applications. (English) Zbl 1451.53046
The authors study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô’s formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. They deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and $$H$$-type groups, one can push the analysis further and, taking advantage of the recently proved sub-Laplacian comparison theorems, can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, they also prove Cheng’s type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.
##### MSC:
 53C17 Sub-Riemannian geometry 35H20 Subelliptic equations 58J65 Diffusion processes and stochastic analysis on manifolds
Full Text:
##### References:
 [1] A. Agrachev, Any sub-Riemannian metric has points of smoothness, Dokl. Akad. Nauk 424 (2009), no. 3, 295-298. · Zbl 1253.53029 [2] A. Agrachev, D. Barilari, and U. Boscain, A comprehensive introduction to sub-Riemannian geometry, Cambridge Studies in Advanced Mathematics, vol. 181, Cambridge University Press, Cambridge, 2020, From the Hamiltonian viewpoint, With an appendix by Igor Zelenko. · Zbl 07073879 [3] A. Agrachev and P. W. Y. Lee, Bishop and Laplacian comparison theorems on three-dimensional contact sub-Riemannian manifolds with symmetry, J. Geom. Anal. 25 (2015), no. 1, 512-535. · Zbl 1341.53058 [4] E. Barletta and S. Dragomir, Jacobi fields of the Tanaka-Webster connection on Sasakian manifolds, Kodai Math. J. 29 (2006), no. 3, 406-454. · Zbl 1133.53037 [5] F. Baudoin, Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations, Geometry, analysis and dynamics on sub-Riemannian manifolds. Vol. 1, EMS Ser. Lect. Math., Eur. Math. Soc., Zürich, 2016, pp. 259-321. · Zbl 1379.53035 [6] F. Baudoin, Stochastic analysis on sub-Riemannian manifolds with transverse symmetries, Ann. Probab. 45 (2017), no. 1, 56-81. · Zbl 1386.58019 [7] F. Baudoin, Q. Feng, and M. Gordina, Integration by parts and quasi-invariance for the horizontal Wiener measure on foliated compact manifolds, J. Funct. Anal. 277 (2019), no. 5, 1362-1422. · Zbl 1427.53028 [8] F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 1, 151-219. · Zbl 1359.53018 [9] F. Baudoin, E. Grong, K. Kuwada, and A. Thalmaier, Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations, Calc. Var. Partial Differential Equations 58 (2019), no. 4, Paper No. 130, 38. · Zbl 1419.53034 [10] F. Baudoin, E. Grong, G. Molino, and L. Rizzi, Comparison theorems on H-type sub-Riemannian manifolds, arXiv:1909.03532 (2019). [11] I. Chavel, Riemannian geometry—a modern introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. · Zbl 0810.53001 [12] J. Cheeger and S. T. Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981), no. 4, 465-480. · Zbl 0481.35003 [13] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, extended ed., De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011. · Zbl 1227.31001 [14] M. Gordina and T. Laetsch, A convergence to Brownian motion on sub-Riemannian manifolds, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6263-6278. · Zbl 1372.60119 [15] R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. · Zbl 0414.53043 [16] E. Grong and A. Thalmaier, Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part I, Math. Z. 282 (2016), no. 1-2, 99-130. · Zbl 1361.53028 [17] E. Grong and A. Thalmaier, Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part II, Math. Z. 282 (2016), no. 1-2, 131-164. · Zbl 1361.53029 [18] E. Grong and A. Thalmaier, Stochastic completeness and gradient representations for sub-Riemannian manifolds, Potential Anal. 51 (2019), no. 2, 219-254. · Zbl 07086015 [19] E. P. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002. · Zbl 0994.58019 [20] K. Ichihara, Comparison theorems for Brownian motions on Riemannian manifolds and their applications, J. Multivariate Anal. 24 (1988), no. 2, 177-188. · Zbl 0635.60091 [21] N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka Math. J. 14 (1977), no. 3, 619-633. · Zbl 0376.60065 [22] W. S. Kendall, The radial part of Brownian motion on a manifold: a semimartingale property, Ann. Probab. 15 (1987), no. 4, 1491-1500. · Zbl 0647.60086 [23] P. W. Y. Lee, Bishop and Laplacian comparison theorems on three-dimensional contact sub-Riemannian manifolds with symmetry, J. Geom. Anal. 25 (2015), no. 1, 512-535. · Zbl 1341.53058 [24] L. Rifford and E. Trélat, Morse-Sard type results in sub-Riemannian geometry, Math. Ann. 332 (2005), no. 1, 145-159. · Zbl 1069.53033 [25] K.-T. Sturm, Sharp estimates for capacities and applications to symmetric diffusions, Probab. Theory Related Fields 103 (1995), no. 1, 73-89. · Zbl 0828.60062 [26] A. · Zbl 0931.41017
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.