Curvature, geodesics and the Brownian motion on a Riemannian manifold I—Recurrence properties

Abstract

Let M be an n-dimensional, complete, connected and locally compact Riemannian manifold and g be its metric. Denote by Δ M the Laplacian on M.The Brownian motion on the Riemannian manifold M is defined to be the unique minimal diffusion process (X t , ζ, P x , x e M) associated to the Laplacian Δ M where ζ is the explosion time i.e. if ζ(ω) < +oo, lim.X r ί (<y) t-*ζ = oo. It should be remarked that ζ -+oo a.s. if M is compact. See Mckean [7], page 90. The Brownian motion X on M is said to be recurrent if for every open subset U of M P x {ω\X t {ω) e U for some ί>0} = lonM;otherwise it will be called transient. It has been known that the Brownian motion on a compact Riemannian manifold is recurrent. See for example Mckean [7] page 99. In this paper we shall restrict our consideration to non compact case and clarify the relation between the recurrent and transient properties of the process X and geodesies, curvature of M.In the first section we shall summarize recurrence and transience of the Brownian motion on a rotationally symmetric Riemannian manifold (See Section 1 for the precise definition). Section 2 is devoted to the discussion for the general case. Some examples will be shown in the last section.

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

“…The following tests were proved by many authors in various settings ( [1], [139], [111], [99], [100], [75] …”

Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below

“…Icihara's condition in [9] then also gives parabolicity. To apply Milnor's condition, (see [20]), we calculate the arc length presentation:…”

“…In particular the capacity condition (b) implies that (M, g) is parabolic if it has vanishing capacity-a condition which we will apply in Sect. 9.…”

“…Parabolicity of those surfaces is known already from criteria due to by Milnor and Ichihara (see [20] and [9]). The examples we have in mind are classical but suitably modified to provide well defined (and simple) extrinsic balls (discs) to exemplify our analysis.…”

On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form

“…It is well known that complete surfaces with finite total curvature are parabolic, see [19] and the proof of [24, theorem 12.2]. Taking into account that surfaces with positive fundamental tone are hyperbolic surfaces, see [16], one concludes that surfaces with finite total curvature has zero fundamental tone as well as the surfaces with tamed second fundamental form with quadratic extrinsic area growth.…”