Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion. It is shown that both properties have various analytic characterizations, in terms of the heat kernel, Green function, Liouville properties etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds etc. which are related to recurrence and non-explosion.

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