On the occupation times of cones by Brownian motion

Summary We study some features concerning the occupation timeA _t of a d-dimensional coneC by Brownian motion. In particular, in the case whereC is convex, we investigate the asymptotic behaviour ofP(A₁u0, when the Brownian motion starts at the vertex ofC. We also give the precise integral test, which decides whether a.s., lim inf _t A _t/(tf(t))=0 or for a decreasing functionf.     

On sojourn of Brownian motion inside moving boundaries

We investigate the large scale structure of certain sojourn sets of one dimensional Brownian motion within two-sided moving boundaries. The macroscopic Hausdorff dimension, upper mass dimension and logarithmic density of these sets, are computed. We also give a uniform macroscopic dimension result for the Brownian level sets.     

On the semimartingale representation of reflecting Brownian motion in a cusp

Summary LetC be the symmetric cusp {(x, y)²:–x yx ,x0} where >1. In this paper we decide whether or not reflecting Brownian motion inC has a semimartingale representation. Here the reflecting Brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary. Our results carry through for a wide class of asymmetric cusps too.     

On the variances of occupation times of conditioned Brownian motion

We extend some bounds on the variance of the lifetime of two--dimensional Brownian motion, conditioned to exit a planar domain at a given point, to certain domains in higher dimensions. We also give a short ``analytic' proof of some existing results.

     

Integration by parts on the law of the reflecting Brownian motion

We prove an integration by parts formula on the law of the reflecting Brownian motion in the positive half line, where B is a standard Brownian motion. In other terms, we consider a perturbation of X of the form X^ε=X+εh with h smooth deterministic function and ε>0 and we differentiate the law of X^ε at ε=0. This infinitesimal perturbation changes drastically the set of zeros of X for any ε>0. As a consequence, the formula we obtain contains an infinite-dimensional generalized functional in the sense of Schwartz, defined in terms of Hida's renormalization of the squared derivative of B and in terms of the local time of X at 0. We also compute the divergence on the Wiener space of a class of vector fields not taking values in the Cameron-Martin space.     

On tail probabilities and first passage times for fractional Brownian motion

In the paper we present a method of simulation of ruin probability over infinite horizon for fractional Brownian motion with parameter of self-similarity H >½. We derive some theoretical results which show how fast the method works. As an application of our method we numerically compute the Pickands constant.     

Pathwise uniqueness for reflecting Brownian motion in Euclidean domains

For a bounded C ^1,α domain in ℝ ^d we show that there exists a strong solution to the multidimensional Skorokhod equation and that weak uniqueness holds for this equation. These results imply that pathwise uniqueness and strong uniqueness hold for the Skorokhod equation. Received: 3 February 1999 / Revised version: 2 September 1999 /?Published online: 11 April 2000     

Domain functionals and exit times for Brownian motion

Two domain functionals describing the averaged expectation of exit times and averaged variance of exit times of Brownian motion from a domain, respectively, are studied. We establish the variational formulas for maximizing the functionals over domains with a volume constraint, and characterize the stationary points and maximizers.

     

Contours of Brownian processes with several-dimensional times

Summary Two generalisations of Brownian motion to several-dimensional time are considered and the topology of their level sets is analysed. It is shown that for these maps non-trivial contours are quite rare — their union has Lebesgue measure zero. The boundedness of all contours is established for the generalisation due to Lévy. For the other, the Brownian sheet, a partial result concerning the behaviour of the zero contour near the boundary is established.Most of the results in this paper were obtained in the course of an S.R.C studentship at the University of Oxford, and appear in the ensuing D. Phil. thesis. I wish to acknowledge the encouragement of my supervisor John Kingman, and the stimulus of correspondence with J.B. Walsh and R. Pyke.     

An occupation time identity for reflecting Brownian motion with drift

Summary This note is about an occupation time identity derived in [14] for reflecting Brownian motion with drift ]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>-\mu<0,$ RBM($-\mu$), for short. The identity says that for RBM($-\mu$) in stationary state ]]>(I^{+}_t, I^{-}_t) \rr (t-G_t,D_t-t),\qquad t\in \mathbb{R},$$ where $G_t$ and $D_t$ denote the starting time and the ending time, respectively, of an excursion from 0 to 0 (straddling $t$) and $I^{+}_t$ and $I^{-}_t$ are the occupation times above and below, respectively, of the observed level at time $t$ during the excursion. Due to stationarity, the common distribution does not depend on $t.$ In fact, it is proved in [9] that the identity is true, under some assumptions, for all recurrent diffusions and stationary processes. In the null recurrent diffusion case the common distribution is not, of course, a probability distribution. The aim of this note is to increase understanding of the identity by studying the RBM($-\mu$) case via Ray--Knight theorems.     

Moments of first hitting times for birth-death processes on trees

An explicit and recursive representation is presented for moments of the first hitting times of birth-death processes on trees. Based on that, the criteria on ergodicity, strong ergodicity, and l-ergodicity of the processes as well as a necessary condition for exponential ergodicity are obtained.     

Hilbert-valued self-intersection local times for planar Brownian motion

Dynkin's construction for self-intersection local time of a planar Wiener process is extended to Hilbert-valued weights. In Dynkin's construction, the weight is bounded and measurable. Since the weight function describes the properties of the medium in which the Brownian motion moves, relative to the external medium's properties, the weight function can be random and unbounded. In this article, we discuss the possibility to consider Hilbert-space-valued weights. It appears that the existence of Hilbert-valued Dynkin-renormalized self-intersection local time is equivalent to the embedding of the values of Hilbert-valued weight into a Hilbert–Schmidt brick. Using Dorogovtsev's sufficient condition for the embedding of compact sets into a Hilbert–Schmidt brick in terms of an isonormal process, we prove the existence of Hilbert-valued Dynkin-renormalized self-intersection local time. Also using Dynkin's construction we construct the self-intersection local time for the deterministic image of the planar Wiener process.