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We study a model for the entanglement of a two-dimensional reflecting Brownian motion in a bounded region divided into two halves by a wall with three or more small windows. We map the Brownian motion into a Markov Chain on the fundamental groupoid of the region. We quantify entanglement of the path with the length of the appropriate element in this groupoid. Our main results are a law of large numbers and a central limit theorem for this quantity. The constants appearing in the limit theorems are expressed in terms of a coupled system of quadratic equations.
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In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.
Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyis parking problem, alternatively called blocking RSA: at every vertex of the tree a particle (or car) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function.
We study random walks on the giant component of the ErdH{o}s-Renyi random graph ${cal G}(n,p)$ where $p=lambda/n$ for $lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $( u {bf d})^{-1}log n pm (log n)^{1/2+o(1)}$, where $ u$ and ${bf d}$ are the speed of random walk and dimension of harmonic measure on a ${rm Poisson}(lambda)$-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.
In this paper we will study a stiff problem in two-dimensional space and especially its probabilistic counterpart. Roughly speaking, the heat equation with a parameter $varepsilon>0$ is under consideration: [ partial_t u^varepsilon(t,x)=frac{1}{2} abla cdot left(mathbf{A}_varepsilon(x) abla u^varepsilon(t,x) right),quad tgeq 0, xin mathbb{R}^2, ] where $mathbf{A}_varepsilon(x)=text{Id}_2$, the identity matrix, for $x otin Omega_varepsilon:={x=(x_1,x_2)in mathbb{R}^2: |x_2|<varepsilon}$ while $$mathbf{A}_varepsilon(x):=begin{pmatrix} a_varepsilon^- & 0 0 & a^shortmid_varepsilon end{pmatrix}$$ with two positive constants $a^-_varepsilon, a^shortmid_varepsilon$ for $xin Omega_varepsilon$. There exists a diffusion process $X^varepsilon$ on $mathbb{R}^2$ associated to this heat equation in the sense that $u^varepsilon(t,x):=mathbf{E}^xu^varepsilon(0,X_t^varepsilon)$ is its unique weak solution. Note that $Omega_varepsilon$ collapses to the $x_1$-axis, a barrier of zero volume, as $varepsilondownarrow 0$. The main purpose of this paper is to derive all possible limiting process $X$ of $X^varepsilon$ as $varepsilondownarrow 0$. In addition, the limiting flux $u$ of the solution $u^varepsilon$ as $varepsilondownarrow 0 $ and all possible boundary conditions satisfied by $u$ will be also characterized.
We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].
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