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

Abstract

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.

Stochastic completeness for graphs with curvature dimension conditions

“…So far, the optimal volume growth condition for stochastic completeness was given by Grigor'yan [Gri86]. We refer to [Gri99] for the literature on stochastic completeness of Riemannian manifolds. These results have been generalized to a quite general setting, namely, regular strongly local Dirichlet forms by Sturm [Stu94].…”

Conformal type of ends of revolution in space forms of constant sectional curvature

“…Then, by using Corollary 2.7 of [Gim14], Σ has positive Cheeger constant h(Σ) > 0, in particular the end of revolution E has also positive Cheeger constant h(E) > 0, and therefore E has positive fundamental tone λ * (E) which implies that E is non-parabolic (see [Gri99]) in contradiction with Theorem A.…”

“…When Σ is immersed in R 3 or in S 3 the stochastic completeness of its ends is straight forward according that every parabolic surface is stochastically complete (see [Gri99]for instance). For surfaces Σ in H 3 such that every of its ends with respect to some compact Ω ⊂ Σ is an end of revolution in H 3 , we are going to show that there exist a 1-superharmonic function satisfying the hypothesis of proposition 2.13 (and hence Σ is stochastically complete).…”

“…In particular in [Tro99] sufficient and necessary conditions for the parabolicity of a manifold with a warped cylindrical end were provided, in [Gri99] rotationally symmetric manifolds were analyzed, and in the examples of [HP11,MP10] certain surfaces of revolution in R 3 have been studied from an extrinsic approach. Our first result characterizes the conformal type of complete ends of revolution in R 3 or in S 3 Theorem A.…”

Поведение Ограниченных Решений Квазилинейных Эллиптических Уравнений

“…Вопросы, рассматриваемые в этом сообщении, исследовались рядом авторов [1]- [5]. Договоримся о следующих обозначениях: Br = {x : |x| < r} и Sr = {x : |x| = r}.…”