Abstract. We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and nonexplosion. 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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The behavior of the Green function G(x, y, t) of the Cauchy problem for the heat equation on a connected, noncompact, complete Riemannian manifold is investigated. For manifolds with boundary it is assumed that the Green function satisfies a Neumann condition on the boundary.

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