On the Brownian curve and its circumscribing sphere (original) (raw)

The Annals of Probability, 1998

A fixed 2-dimensional projection of a 3-dimensional Brownian motion is almost surely neighborhood recurrent; is this simultaneously true of all the 2-dimensional projections with probability one? Equivalently: 3-dimensional Brownian motion hits any infinite cylinder with probability one; does it hit all cylinders? This papers shows that the answer is no. Brownian motion in three dimensions avoids random cylinders and in fact avoids bodies of revolution that grow almost as fast as cones.

Nonintersecting Brownian motions on the unit circle

The Annals of Probability, 2016

We consider an ensemble of n nonintersecting Brownian particles on the unit circle with diffusion parameter n −1/2 , which are conditioned to begin at the same point and to return to that point after time T , but otherwise not to intersect. There is a critical value of T which separates the subcritical case, in which it is vanishingly unlikely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper, we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n → ∞, and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode processes in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painlevé II equation of size 2 × 2. The proofs are based on the determinantal structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral.

Mathematical Communications 5(2000), 75-85 75 Intersection properties of Brownian paths

2000

Abstract. This review presents a modern approach to intersec-tions of Brownian paths. It exploits the fundamental link between inter-section properties and percolation processes on trees. More precisely, a Brownians path is intersect-equivalent to certain fractal percolation. It means that the intersection probabilities of Brownian paths can be esti-mated up to constant factors by survival probabilities of certain branching processes.

On conformally invariant subsets of the planar Brownian curve

Eprint Arxiv Math 0105192, 2001

We define and study a family of generalized non-intersection exponents for planar Brownian motions that is indexed by subsets of the complex plane: For each AsubsetCCA\subset\CCAsubsetCC, we define an exponent xi(A)\xi(A)xi(A) that describes the decay of certain non-intersection probabilities. To each of these exponents, we associate a conformally invariant subset of the planar Brownian path, of Hausdorff dimension 2−xi(A)2-\xi(A)2−xi(A). A consequence of this and continuity of xi(A)\xi(A)xi(A) as a function of AAA is the almost sure existence of pivoting points of any sufficiently small angle on a planar Brownian path.

Winding of planar Brownian curves

Journal of Physics A: Mathematical and General, 1990

We compute the joint probability for a closed Brownian curve to wind n times around a prescribed point and to enclose a given algebraic area. An estimate from below of the arithmetic area is obtained. Since the pioneering work of Edwards [l], the study of path integrals in the presence of topological constraints has aroused considerable interest. On the one hand these techniques are of direct relevance for polymer physics while on the other hand they are connected with some rigorous mathematical results. Consider for instance the two-dimensional Brownian motion on the punctured plane P-(0). The problem of finding the asymptotic probability distribution of the total angle e (t) wound at time t around 0 was first addressed by Spitzer [2] who showed that X =: 2e(t)/ln t is distributed according to a Cauchy law for t + +CO. This result was then extended by Pitman and Yor [3] to the case of n prescribed points. This question has also been reexamined by Rudnick and Hu [4] who showed that by removing a disc from the plane, instead of a point, the asymptotic distribution changes drastically from a Cauchy law to an exponential law (which thus leads to finite moments). Recent results of Belisle [ 5 ] on the winding of a discrete random walk indeed confirm that the limiting law has an exponential tail. The winding number distribution was also discussed by Wiegel in the context of polymer entanglements [ 6 ]. An apparently unrelated problem concerns the probability distribution of the area enclosed by a planar Brownian curve. First raised by Levy [7] and solved magisterially by the use of Fourier-Wiener series, this problem was more recently reexamined by Brereton and Butler [8], Khandekar and Wiegel [ 9 ] and Duplantier [lo]. The purpose of this paper is to extend this approach to the case of the joint probability distribution for a closed planar Brownian walk to wind n times around a prescribed point and enclose a given algebraic area (the initial = final point has been left unspecified). Interestingly enough, this quantity is related to the two-body partition function of a gas of particles obeying fractional statistics (anyons). The plan of the paper is as follows: for pedagogical reasons we first rederive Wiegel's results concerning the probability 9 (A) for a closed planar Brownian curve to enclose after a time 7 a given algebraic area A. This quantity can be expressed in terms of the partition function of a charged particle embedded in a constant magnetic field. This partition function diverges as the total area of the plane but an adequate normalisation leads back to the finite P (A). We then consider the probability B,(n)

Reflection and coalescence between independent one-dimensional Brownian paths

Annales de l'Institut Henri Poincare (B) Probability and Statistics, 2000

Take two independent one-dimensional processes as follows: (B t , t ∈ [0, 1]) is a Brownian motion with B 0 = 0, and (β t , t ∈ [0, 1]) has the same law as (B 1−t , t ∈ [0, 1]); in other words, β 1 = 0 and β can be seen as Brownian motion running backwards in time. Define (γ t , t ∈ [0, 1]) as being the function that is obtained by reflecting B on β. Then γ is still a Brownian motion. Similar and more general results (with families of coalescing Brownian motions) are also derived. They enable us to give a precise definition (in terms of reflection) of the joint realization of finite families of coalescing/reflecting Brownian motions.

Plane Brownian motion has strictlyn-multiple points

Israel Journal of Mathematics, 1985

It is shown that, almost surely, for all natural n, there are points which the plane Brownian motion visits exactly n times. A point x is k-multiple (respectively strictly k-multiple) for a map [ if, and only if, Card/-l{x} = > k (resp. = k). Let Z be a plane Brownian motion. (All Brownian motions discussed here are continuous, and defined on R+ = [0,~[.) Dvoretzky, Erd6s and Kakutani proved in [2] that, with probability 1, for all natural n, Z admits n-multiple points. With this at hand, we provide a simple proof to the following THEOREM. Almost surely, [or all natural n, Z admits strictly n-multiple points.* Fix a natural n = 2. We shall see that, almost surely, Z admits at least one strictly n-multiple point. Suppose Z(to) has n-multiple points (which, by [2], is the case for almost all to). So there are n mutually disjoint closed rational subintervals of R÷ whose images under Z(to) have a common point of intersection. (A rational interval is one with rational endpoints.) Since the set of finite sets of rational intervals is only countable, it is enough to show that if I1,...,I, are mutually disjoint ' One of us proved recently ([1]) that if S is a closed subset of R÷ without interior points, then, almost surely, there exist points in the plane whose inverse image under Z is order-similar to S. This, of course, is stronger than our theorem (to obtain the almost sure existence of strictly n-multiple points, take S = {1,..., n}), but its proof (the one in [1], at least) is relatively involved.

What is the probability of intersecting the set of Brownian double points?

The Annals of Probability, 2007

We give potential theoretic estimates for the probability that a set A contains a double point of planar Brownian motion run for unit time. Unlike the probability for A to intersect the range of a Markov process, this cannot be estimated by a capacity of the set A. Instead, we introduce the notion of a capacity with respect to two gauge functions simultaneously.