The Annals of Probability

Diffusion processes and heat kernels on metric spaces

K. T. Sturm

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We present a general method to construct $m$-symmetric diffusion processes $(X_t, \mathbf{P}_x)$ on any given locally compact metric space $(X, d)$ equipped with a Radon measure $m$. These processes are associated with local regular Dirichlet forms which are obtained as $\Gamma$-limits of approximating nonlocal Dirichlet forms. This general method works without any restrictions on $(X, d, m)$ and yields processes which are well defined for quasi every starting point.

The second main topic of this paper is to formulate and exploit the so-called Measure Contraction Property. This is a condition on the original data $(X, d, m)$ which can be regarded as a generalization of curvature bounds on the metric space $(X, d)$. It is a bound for distortions of the measure $m$ under contractions of the state space $X$ along suitable geodesics or quasi geodesics w.r.t. the metric $d$. In the case of Riemannian manifolds, this condition is always satisfied. Several other examples will be discussed, including uniformly elliptic operators, operators with weights, certain subelliptic operators, manifolds with boundaries or corners and glueing together of manifolds.

The Measure Contraction Property implies upper and lower Gaussian estimates for the heat kernel and a Harnack inequality for the associated harmonic functions. Therefore, the above-mentioned diffusion processes are strong Feller processes and are well defined for every starting point.

Article information

Ann. Probab., Volume 26, Number 1 (1998), 1-55.

First available in Project Euclid: 31 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 31C25: Dirichlet spaces 58G32 60G07: General theory of processes 49Q20: Variational problems in a geometric measure-theoretic setting

Diffusion process Feller process heat kernel Dirichlet form $\Gamma$-convergence variational limit intrinsic metric stochastic differential geometry Riemannian manifold Lipschitz manifold Poincaré inequality Gaussian estimate


Sturm, K. T. Diffusion processes and heat kernels on metric spaces. Ann. Probab. 26 (1998), no. 1, 1--55. doi:10.1214/aop/1022855410.

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  • BALLMANN, W. 1995. Lectures on Spaces of Nonpositive Curvature. DMV Seminar 25. Birkhauser, Boston. ¨
  • BIROLI, M. and MOSCO, U. 1995. A Saint-Vernant principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl. 169 125 181.
  • BOULEAU, N. and HIRSCH, F. 1991. Dirichlet Forms and Analysis on Wiener Space. de Gruyter, Berlin.
  • CHAVEL, I. 1993. Riemannian Geometry: A Modern Introduction. Cambridge Univ. Press.
  • DAL MASO, G. 1993. An Introduction to -Convergence. Birkhauser, Boston. ¨
  • ETHIER, S. N. and KURTZ, T. G. 1986. Markov Processes. Wiley, New York.
  • FUKUSHIMA, M., OSHIMA, Y. and TAKEDA, M. 1994. Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin.
  • GHYS, E. and DE LA HARPE, P. 1990. Sur les Groupes Hyperboliques d'apres Mikhael Gromov. Birkhauser, Boston. ¨Z
  • GROMOV, M. 1981. Structures Metriques pour les Varietes Riemanniennes J. Lafontaine and ´ ´ ´ ´. P. Pansu, eds.. Cedic F. Nathan.
  • JERISON, D. 1986. The Poincare inequality for vector fields satisfying an Hormander's condition. ´ ¨ Duke J. Math. 53 503 523.
  • JOST, J. 1994. Equilibrium maps between metric spaces. Calc. Var. 2 173 204.
  • JOST, J. 1996. Generalized harmonic maps between metric spaces. In Geometric Analysis andthe Calculus of Variations for Stefan Hildebrandt J. Jost, ed.. International Press, Boston. Z.
  • JOST, J. 1997. Generalized Dirichlet forms and harmonic maps. Calc. Var. 5 1 19.
  • KOREVAAR, N. J. and SCHOEN, R. M. 1993. Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1 561 659.()
  • MA,and ROCKNER, M. 1992. Introduction to the Theory of Non-Symmetric Dirichlet Forms. ¨ Universitext. Springer, Berlin. Z.
  • MOSCO, U. 1994. Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123 368 421.
  • MOSER, J. 1964. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 101 134.
  • RINOW, M. 1961. Die innere Geometrie der metrischen Raume. Grundlehren 105. Springer, ¨ Berlin.
  • STURM, K. T. 1994. Analysis on local Dirichlet spaces I. Recurrence, conservativeness and L p-Liouville properties. J. Reine Angew. Math. 456 173 196.
  • STURM, K. T. 1995a. Analysis on local Dirichlet spaces II. Upper Gaussian estimates for the fundamental solution of parabolic equations. Osaka J. Math. 32 275 312.
  • STURM, K. T. 1995b. On the geometry defined by Dirichlet forms. In Seminar on StochasticAnalysis, Random Fields and Applications E. Bolthausen et al., eds. 231 242. Birkhauser, Boston. ¨ Z.
  • STURM, K. T. 1996. Analysis on local Dirichlet spaces III. The parabolic Harnack inequality. J. Math. Pures Appl. 75 273 297.
  • STURM, K. T. 1997. How to construct diffusion processes on metric spaces. Potential Analysis. To appear.