Diffusion processes and heat kernels on metric spaces

Abstract

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.