 |
|
|
| | | | | |
|
|
|
|
|
Intrinsic Coupling on Riemannian Manifolds and Polyhedra
|
Max-K. von Renesse, Technical University Berlin
|
Abstract
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 lo-
cus phenomena are indeed inessential for Kendall's and Cranston's
stochastic proof of gradient estimates for harmonic functions on Rie-
mannian 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: PDF
Pages: 411-435
Published on: June 8, 2004
|
Bibliography
TITLE HERE
| 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)
|
|
|
|
|
|
|
|
|
| | | | |
Electronic Journal of Probability. ISSN: 1083-6489 |
|
Context