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Gradient estimates on manifolds using coupling. (English) Zbl 0770.58038
Using a probabilistic coupling technique on a complete Riemannian manifold \(M\), an estimate is given for the gradient of solutions to \((1/2\Delta+Z)u=0\). Here \(\Delta\) is the Laplacian and \(Z\) a given vector field. The bound in the gradient estimate depends on the bounds of \(Z\) and on a lower bound on the Ricci curvature of \(M\).

58J05 Elliptic equations on manifolds, general theory
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
60G46 Martingales and classical analysis
Full Text: DOI
[1] Greene, R.E; Wu, H, Function theory on manifolds which possess a pole, (1979), Springer-Verlag New York · Zbl 0414.53043
[2] Kendall, W, Stochastic differential geometry, a coupling property, and harmonic maps, J. London math. soc., 33, 554-566, (1986), (2) · Zbl 0573.58029
[3] Ikeda, N; Watanabe, S, Stochastic differential equations and diffusion processes, (1981), Elsevier-North Holland New York · Zbl 0495.60005
[4] Kendall, W, Nonnegative Ricci curvature and the Brownian coupling property, Stochastics, 19, 111-129, (1986) · Zbl 0584.58045
[5] Kendall, W, Coupled Brownian motions and partial domain monotonicity for the Neumann heat kernel, J. funct. anal., 86, No. 2, 226-236, (1989) · Zbl 0684.60060
[6] Lindvall, T; Rogers, L, Coupling of multi-dimensional diffusions by reflection, Ann. probab., 14, 860-872, (1986) · Zbl 0593.60076
[7] Lyons, T; Sullivan, D, Function theory, random paths, and covering spaces, J. differential geom., 19, 299-323, (1984) · Zbl 0554.58022
[8] Malliavin, P, Asymptotic of the Green’s function of a Riemannian manifold and Itô’s stochastic integrals, (), 381-383 · Zbl 0449.60040
[9] Yau, S.T, Harmonic functions on complete Riemannian manifolds, Comm. pure appl. math., 28, 201-228, (1975) · Zbl 0291.31002
[10] Yau, S.T, On the heat kernel of a complete Riemannian manifold, J. math. pures appl., 57, 191-201, (1978) · Zbl 0405.35025
[11] Cheeger; Ebin, Comparison theorems in Riemannian geometry, (1975), North-Holland New York · Zbl 0309.53035
[12] Hsu, On the distance function of a Riemannian manifold, (1989), preprint
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