Asymmetric cooperative motion in one dimension

Symmetric cooperative motion in one dimension

We explore the relationship between recursive distributional equations and convergence results for finite difference schemes of parabolic partial differential equations (PDEs). We focus on a family of random processes called symmetric cooperative motions, which generalize the symmetric simple random walk and the symmetric hipster random walk introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, arXiv:1909.07367]. We obtain a distributional convergence result for symmetric cooperative motions and, along the way, obtain a novel proof of the Bernoulli central limit theorem. In addi…

高维对称合作运动

Symmetric Cooperative Motion in Higher Dimensions

本文通过将递归分布方程视为多孔介质方程的离散化，证明了高维对称合作运动的分布收敛性，并引入新的比较论证分析离散过程的概率质量函数，进而建立了逼近多孔介质方程ZKB/Barenblatt解的有限差分格式的多维收敛结果。

The motion of the tagged particle in asymmetric exclusion process with long jumps

We prove the law of large numbers and invariance principles for the tagged particle in the asymmetric exclusion process with long jumps when the process starts from its equilibrium measure.

The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Goncalves and Jara and the corresponding uniqueness result of Gubinelli and Perkowski.

Time since maximum of Brownian motion and asymmetric Levy processes

Motivated by recent studies of record statistics in relation to strongly correlated time series, we consider explicitly the drawdown time of a Levy process, which is defined as the time since it last achieved its running maximum when observed over a fixed time period [0,T]. We show that the density function of this drawdown time, in the case of a completely asymmetric jump process, may be factored as a function of $t$ multiplied by a function of T-t. This extends a known result for the case of pure Brownian motion. We state the factors explicitly for the cases of exponential down-jumps with dr…

Diffusion approximation for a simple kinetic model with asymmetric interface

We study a diffusion approximation for a model of stochastic motion of a particle in one spatial dimension. The velocity of the particle is constant but the direction of the motion undergoes random changes with a Poisson clock. Moreover, the particle interacts with an interface in such a way that it can randomly be reflected, transmitted, or killed, and the corresponding probabilities depend on whether the particle arrives at the interface from the left, or right. We prove that the limit process is a minimal Brownian motion, if the probability of killing is positive. In the case of no killing,…