相关论文: Winding and intersection of Brownian motions

Let $D_N$ be the set of points around which a planar Brownian motion winds at least $N$ times. We prove that the random measure on the plane with density $2 \pi N 1_{D_N}$ with respect to the Lebesgue measure converges almost surely weakly,…

We give asymptotic estimations on the area of the sets of points with large Brownian winding, and study the average winding between a planar Brownian motion and a Poisson point process of large intensity on the plane. This allows us to give…

A Brownian loop is a random walk circuit of infinitely many, suitably infinitesimal, steps. In a plane such a loop may or may not enclose a marked point, the origin, say. If it does so it may wind arbitrarily many times, positive or…

We prove that the occupation measures of Brownian motions conditioned to have large intersections converge weakly, up to spatial shifts, to the measure whose density is the square of an optimizer of the Gagliardo-Nirenberg inequality. We do…

We prove a conjecture of Lalley and Sellke [Ann. Probab. 15 (1987)] asserting that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a double exponential, or Gumbel,…

Consider p independent Brownian motions in R^d, each running up to its first exit time from an open domain B, and their intersection local time l as a measure on B. We give a sharp criterion for the finiteness of exponential moments,…

We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq 2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any measurable…

We study a correlated Brownian motion in two dimensions, which is reflected, stopped or killed in a wedge represented as the intersection of two half spaces. First, we provide explicit density formulas, hinted by the method of images. These…

The Brownian motion of a test particle interacting with a quantum scalar field in the presence of a perfectly reflecting boundary is studied in (1 + 1)-dimensional flat spacetime. Particularly, the expressions for dispersions in velocity…

We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding…

Let \ell be the projected intersection local time of two independent Brownian paths in R^d for d=2,3. We determine the lower tail of the random variable \ell(U), where U is the unit ball. The answer is given in terms of intersection…

In this article we establish some estimates related to the Gaussian densities and to Hermite polynomials in order to obtain an almost sure estimate for each term of the It\^{o}-Wiener expansion of the self-intersection local times of the…

There is a close connection between intersections of Brownian motion paths and percolation on trees. Recently, ideas from probability on trees were an important component of the multifractal analysis of Brownian occupation measure, in joint…

Let B_1,B_2, ... be independent one-dimensional Brownian motions defined over the whole real line such that B_i(0)=0. We consider the nth iterated Brownian motion W_n(t)= B_n(B_{n-1}(...(B_2(B_1(t)))...)). Although the sequences of…

This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz.\spacefactor =1000 Brownian motion, bridge, excursion, meander and double meander; for the Brownian…

Consider the first exit time of one-dimensional Brownian motion $\{B_s\}_{s\geq 0}$ from a random passageway. We discuss a Brownian motion with two time-dependent random boundaries in quenched sense. Let $\{W_s\}_{s\geq 0}$ be an other…

The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $\mathsf{GL}(N;\mathbb{C}),$ in the sense of $\ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution…

Suppose that a sequence of metric measure spaces X_n=(X_n, d_n, m_n) satisfies RCD*(K,N) with Diam(X_n) <D and m_n(X_n)=1. Then Sturm's D-convergence of X_n is equivalent to the weak convergence of the laws of Brownian motions on X_n.

We show that with probability 1, the trace B[0,1] of Brownian motion in space, has positive capacity with respect to exactly the same kernels as the unit square. More precisely, the energy of occupation measure on B[0,1] in the kernel…

It is well known that path probabilities of Brownian motion correspond to the equilibrium configurational probabilities of flexible Gaussian polymers, while those of active Brownian motion correspond to in-extensible semiflexible polymers.…