The radial part of Brownian motion II. Its life and times on the cut locus

Abstract

This paper is a sequel to Kendall (1987), which explained how the It6 formula for the radial part of Brownian motion X on a Riemannian manifold can be extended to hold for all time including those times at which X visits the cut locus. This extension consists of the subtraction of a correction term, a continuous predictable non-decreasing process L which changes only when X visits the cut locus. In this paper we derive a representation of L in terms of measures of local time of X on the cut locus. In analytic terms we compute an expression for the singular part of the Laplacian of the Riemannian distance function. The work uses a relationship of the Riemannian distance function to convexity, first described by Wu (1979) and applied to radial parts of F-martingales in Kendall (1993). (1991): 58G32, 60H10, 60J45 Mathematics Subject Classification

Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds

“…We say a SDE is complete at one point if there is a point x 0 in M with ξ(x 0 ) = ∞. From the theorem we have the following corollary, which is known for elliptic diffusions without condition (4).…”

Brownian Motion and the Distance to a Submanifold

“…Since the boundaries of regular domains are included as examples, this could yield applications related to the study of reflected Brownian motion. The main references here are [17], [4] and [6]. The articles [17] and [4] approach geometric local time in the general context of continuous semimartingales from the point of view of Tanaka's formula while [6] approaches the topic for the special case of Brownian motion using Markov process theory.…”

“…Such inequalities were originally studied by Gross in [11] and we refer to the article [7] for the special case of the heat kernel measure. We then prove more general estimates using the Itô-Tanaka formula of Section 2, which is derived from the formula of Barden and Le [17] and which reduces to the formula of Cranston, Kendall and March [6] in the one-point case. The basic method is similar to that of Hu in [14], who studied (uniform) exponential integrability for diffusions in R m for C 2 functions satisfying another Lyapunov-like condition.…”

Hessian formulas and estimates for parabolic Schrödinger operators

“…where L Cut t is a non-negative term, vanishing off the cut locus, ∆r is the Laplacian of the distance function off the cut locus of y 0 and vanishes on the cut locus. See [CKM93], especially for h = 0. This can be obtained also by taking a smooth approximation r ǫ of r and applying to them Itô's formula, and the following distributional inequality [Yau76].…”