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limit theorems for functionals of Gaussian vectors

Frontiers of Mathematics in China 2017年第4期12卷 821-842页

作者： Hongshuai DAI Guangjun SHEN Lingtao KONG School of Statistics Shandong University of Finance and Economics Jinan 250014 China Department of Mathematics Anhui Normal University Wuhu 241000 China

Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine that the limit of partial sums of functionals of a stationary Gaussian sequence of random vectors is an operator self-similar process.

关键词： Gaussian vector operator self-similar process operator fractional Brownian motion scaling limit

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Some limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks

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Acta Mathematica Sinica,English Series 2007年第6期23卷 997-1012页

作者： Vladimir VATUTIN Jie XIONG Steklov Mathematical Institute Gubkin street 8 119991 Moscow Russia Department of Mathematics University of Tennessee Knoxville TN 37996 1300 USA Department of Mathematics Hebei Normal University Shijiazhuang 050016 P. R. China

We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial （finite） number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n^β particles and consider the scaled process Zt^n（·） = Znt（√n·）, where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/*** addition,if 1/4〈β〈1/2 then n-^（2β-1/2）Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit.

关键词： Renewal equation branching particle system scaling limit

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Hydrodynamic limit for Exclusion Processes

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Communications in Mathematics and Statistics 2018年第4期6卷 417-480页

作者： Tadahisa Funaki Department of Mathematics School of Fundamental Science and EngineeringWaseda University3-4-1 OkuboShinjuku-kuTokyo 169-8555Japan

The exclusion process,sometimes called Kawasaki dynamics or lattice gas model,describes a system of particles moving on a discrete square lattice with an interaction governed by the exclusion rule under which at most one particle can occupy each *** mostly discuss the symmetric and reversible *** weakly asymmetric case recently attracts attention related to KPZ equation;*** and Giacomin(Commun Math Phys 183:571–607,1995)for a simple exclusion case and Gonçalves and Jara(Arch Ration Mech Anal 212:597–644,2014)for an exclusion process with speed change,see also Gonçalves et al.(Ann Probab 43:286–338,2015),Gubinelli and Perkowski(J Am Math Soc 31:427–471,2018).In Sect.1,as a warm-up,we consider a simple exclusion process and discuss its hydrodynamic limit and the corresponding fluctuation limit in a proper space–time *** this model,one can derive a linear heat equation and a stochastic partial differential equation(SPDE)in the limit,*** 2 is devoted to the entropy method originally invented by Guo et al.(Commun Math Phys 118:31–59,1988).We consider the exclusion process with speed change,in which the jump rate of a particle depends on the configuration nearby the *** gives a non-trivial interaction among *** study only the case that the jump rate satisfies the so-called gradient *** hydrodynamic limit,which leads to a nonlinear diffusion equation,follows from the local ergodicity or the local equilibrium of the system,and this is shown by establishing one-block and twoblock *** also discuss the fluctuation limit which follows by showing the so-called Boltzmann–Gibbs *** 3 explains the relative entropy method originally due to Yau(Lett Math Phys 22:63–80,1991).This is a variant of GPV method and gives another proof for the hydrodynamic *** difference between these two methods is as *** N^(d)be the volume of the domain on which the system is defined(typically,d-dim

关键词： Exclusion process Kawasaki dynamics Hydrodynamic limit scaling limit Relative entropy Nonlinear diffusion equation

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