Shy and fixed-distance couplings of Brownian motions on manifolds

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Pascu, Mihai N.¹  Popescu, Ionel^2,3 

[1] Transilvania Univ Brasov, Fac Math & Comp Sci, Str Iuliu Maniu 50, Brasov 500091, Romania

[2] Georgia Inst Technol, Sch Math, 686 Cherry St, Atlanta, GA 30332 USA

[3] Romanian Acad, Simion Stoilow Inst Math, 21 Calea Grivitei, Bucharest, Romania

关键词: Brownian motion;    Manifolds;    Couplings;   

DOI  :  10.1016/j.spa.2015.09.014

来源: Elsevier

【 摘 要 】

In this paper we introduce three Markovian couplings of Brownian motions on smooth Riemannian manifolds without boundary which sit at the crossroad of two concepts. The first concept is the one of shy coupling put forward in Benjamin et al. (2007) and the second concept is the lower bound on the Ricci curvature and the connection with couplings made in von Renesse and Sturm (2005). The first construction is the shy coupling, the second one is a fixed-distance coupling and the third is a coupling in which the distance between the processes is a deterministic exponential function of time. The result proved here is that an arbitrary Riemannian manifold satisfying some technical conditions supports shy couplings. If in addition, the Ricci curvature is non-negative, there exist fixed-distance couplings. Furthermore, if the Ricci curvature is bounded below by a positive constant, then there exists a coupling of Brownian motions for which the distance between the processes is a decreasing exponential function of time. The constructions use the intrinsic geometry, and relies on an extension of the notion of frames which plays an important role for even dimensional manifolds. In fact, we provide a wider class of couplings in which the distance function is deterministic in Theorem 5 and Corollary 9. As an application of the fixed-distance coupling we derive a maximum principle for the gradient of harmonic functions on manifolds with non-negative Ricci curvature. As far as we are aware of, these constructions are new, though the existence of shy couplings on manifolds is suggested by Kendall in Kendall (2009). (C) 2015 Elsevier B.V. All rights reserved.

【 授权许可】

Free