Intrinsic Coupling on Riemannian Manifolds and Polyhedra

Max-K. von Renesse (Technical University Berlin)


Starting from a central limit theorem for geometric random walks we give an elementary construction of couplings between Brownian motions on Riemannian manifolds. This approach shows that cut locus phenomena are indeed inessential for Kendall's and Cranston's stochastic proof of gradient estimates for harmonic functions on Riemannian manifolds with lower curvature bounds. Moreover, since the method is based on an asymptotic quadruple inequality and a central limit theorem only it may be extended to certain non smooth spaces which we illustrate by the example of Riemannian polyhedra. Here we also recover the classical heat kernel gradient estimate which is well known from the smooth setting.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 411-435

Publication Date: June 8, 2004

DOI: 10.1214/EJP.v9-205


Burago, Dmitri; Burago, Yuri; Ivanov, Sergei. A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. xiv+415 pp. ISBN: 0-8218-2129-6 MR1835418 (2002e:53053)

Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp. MR0233396 (38 #1718)

Blum, Gilles. A note on the central limit theorem for geodesic random walks. Bull. Austral. Math. Soc. 30 (1984), no. 2, 169--173. MR0759783 (86a:60106)

Chen, Jingyi; Hsu, Elton P. Gradient estimates for harmonic functions on manifolds with Lipschitz metrics. Canad. J. Math. 50 (1998), no. 6, 1163--1175. MR1657771 (99m:53065)

Cranston, Michael; Kendall, Wilfrid S.; March, Peter. The radial part of Brownian motion. II. Its life and times on the cut locus. Probab. Theory Related Fields 96 (1993), no. 3, 353--368. MR1231929 (94g:60154)

Cranston, Michael. Gradient estimates on manifolds using coupling. Journal of Functional Analysis 99 (1991), no. 1, 110-124. MR1120916 (93a:58175)

Durrett, Richard. Stochastic calculus. A practical introduction. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. x+341 pp. ISBN: 0-8493-8071-5 MR1398879 (97k:60148)

Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085 (88a:60130)

Jørgensen, Erik. The central limit problem for geodesic random walks. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 1--64. MR0400422 (53 #4256)

Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1 MR0959133 (89k:60044)

Kendall, Wilfrid S. Nonnegative Ricci curvature and the Brownian coupling property. Stochastics 19 (1986), no. 1-2, 111--129. MR0864339 (88e:60092)

Le, Hui Ling; Barden, Dennis. Itô correction terms for the radial parts of semimartingales on manifolds. Probab. Theory Related Fields 101 (1995), no. 1, 133--146. MR1314177 (96a:58211)

Mosco, Umberto. Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123 (1994), no. 2, 368--421. MR1283033 (95d:47088)

Malliavin, Paul; Stroock, Daniel W. Short time behavior of the heat kernel and its logarithmic derivatives. J. Differential Geom. 44 (1996), no. 3, 550--570. MR1431005 (98c:58164)

Petrunin, A. Parallel transportation for Alexandrov space with curvature bounded below. Geom. Funct. Anal. 8 (1998), no. 1, 123--148. MR1601854 (98j:53048)

Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4 MR0532498 (81f:60108)

v. Renesse, Max-K. Comparison properties of diffusion semigroups on spaces with lower curvature bounds. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2001. Bonner Mathematische Schriften [Bonn Mathematical Publications], 355. Universität Bonn, Mathematisches Institut, Bonn, 2003. ii+90 pp. MR2013040 (2004g:58050)

Wang, Feng Yu. Successful couplings of nondegenerate diffusion processes on compact manifolds. (Chinese) Acta Math. Sinica 37 (1994), no. 1, 116--121. MR1272513 (95d:58145)

Yau, Shing Tung. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28 (1975), 201--228. MR0431040 (55 #4042)

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.