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)

Digital Object Identifier

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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