Brownian motion reflected on Brownian motion

 We study Brownian motion reflected on an ``independent' Brownian path. We prove results on the joint distribution of both processes and the support of the parabolic measure in the space-time domain bounded by a Brownian path. We show that there exist two different natural local times for a Brownian path reflected on a Brownian path. Received: 25 October 2000 / Revised version: 30 March 2001 / Published online: 20 December 2002     

Forward Brownian motion

We consider processes which have the distribution of standard Brownian motion (in the forward direction of time) starting from random points on the trajectory which accumulate at \(-\infty \) . We show that these processes do not have to have the distribution of standard Brownian motion in the backward direction of time, no matter which random time we take as the origin. We study the maximum and minimum rates of growth for these processes in the backward direction. We also address the question of which extra assumptions make one of these processes a two-sided Brownian motion.     

Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion

Any solution of the functional equation

Brownian motion in cones

Summary. We study the asymptotic behavior of Brownian motion and its conditioned process in cones using an infinite series representation of its transition density. A concise probabilistic interpretation of this series in terms of the skew product decomposition of Brownian motion is derived and used to show properties of the transition density. Received: 2 April 1996 / In revised form: 21 December 1996     

Fake exponential Brownian motion

We construct a fake exponential Brownian motion, a continuous martingale different from classical exponential Brownian motion but with the same marginal distributions, thus extending results of Albin and Oleszkiewicz for fake Brownian motions. The ideas extend to other diffusions.     

Hyperbolic branching Brownian motion

Summary. Hyperbolic branching Brownian motion is a branching diffusion process in which individual particles follow independent Brownian paths in the hyperbolic plane ? ² , and undergo binary fission(s) at rate λ > 0. It is shown that there is a phase transition in λ: For λ≦ 1/8 the number of particles in any compact region of ? ² is eventually 0, w.p.1, but for λ > 1/8 the number of particles in any open set grows to ∞ w.p.1. In the subcritical case (λ≦ 1/8) the set Λ of all limit points in ∂? ² (the boundary circle at ∞) of particle trails is a Cantor set, while in the supercritical case (λ > 1/8) the set Λ has full Lebesgue measure. For λ≦ 1/8 it is shown that w.p.1 the Hausdorff dimension of Λ is δ = (1−√1−8 λ)/2. Received: 2 November 1995 / In revised form: 22 October 1996     

Weak convergence to fractional Brownian motion in Brownian scenery

 Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries. In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery. Received: 17 April 2002 / Revised version: 11 October 2002 / Published online: 15 April 2003 Research supported by NSFC (10131040). Mathematics Subject Classification (2002): 60J55, 60J15, 60J65 Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery     

Brownian motion in a quasi-cone

We determine precise logarithmic asymptotics of the probability of a large exit time for Brownian motion in a quasi-cone. This answers a question formally posed by Lifshits and Shi (Bernoulli 8:745–765, 2002), but first studied by Li (Ann Probab 31:1078–1096, 2001).     

Brownian motion in twisted domains

The tail behavior of a Brownian motion's exit time from an unbounded domain depends upon the growth of the ``inner radius' of the domain. In this article we quantify this idea by introducing the notion of a twisted domain in the plane. Roughly speaking, such a domain is generated by a planar curve as follows. As a traveler proceeds out along the curve, the boundary curves of the domain are obtained by moving out units along the unit normal to the curve when the traveler is units away from the origin. The function is called the growth radius. Such domains can be highly nonconvex and asymmetric. We give a detailed account of the case , . When , a twisted domain can reasonably be interpreted as a ``twisted cone.'

     

Random A-permutations and Brownian motion

We consider a random permutation τ _n uniformly distributed over the set of all degree n permutations whose cycle lengths belong to a fixed set A (the so-called A-permutations). Let X _n (t) be the number of cycles of the random permutation τ _n whose lengths are not greater than n ^t , t ∈ [0, 1], and $l(t) = \sum\nolimits_{i \leqslant t,i \in A} {1/i,t > 0} $ . In this paper, we show that the finite-dimensional distributions of the random process $\{ Y_n (t) = (X_n (t) - l(n^t ))/\sqrt {\varrho \ln n} ,t \in [0,1]\} $ converge weakly as n → ∞ to the finite-dimensional distributions of the standard Brownian motion {W(t), t ∈ [0, 1]} in a certain class of sets A of positive asymptotic density ?.     

Quadratic functionals of Brownian motion

Functionals of Brownian motion can be dealt with by realizing them as functionals of white noise. Specifically, for quadratic functionals of Brownian motion, such a realization is a powerful tool to investigate them. There is a one-to-one correspondence between a quadratic functional of white noise and a symmetric L²(R²)-function which is considered as an integral kernel. By using well-known results on the integral operator we can study probabilistic properties of quadratic or certain exponential functionals of white noise. Two examples will illustrate their significance.     

Brownian motion and algorithm complexity

The Brownian motion is shown to be a useful tool in analysing some sorting and tree manipulation algorithms.     

Enumerating graphs and Brownian motion

Constants in the asymptotic formulae of E. M. Wright for the number of labeled connected graphs on n vertices and n − 1 + k edges (k fixed) are shown to be moments of the mean distance from the origin in a certain restricted Brownian motion. © 1997 John Wiley & Sons, Inc.