Resume

• 2010

  Doctor of Philosophy (PhD)

  Mathematics

  Duke University

  3.96/4

• 2006

  Bachelor of Science (B.S.)

  Mathematics

  Debate Team

  Peking University

• 1

  In this paper we prove the existence of a nontrivial stationary distribution\nfor a forest model with Grass

  Saplings and Trees

  by comparing with the two\ntype contact process model of Krone and considering the long range limit. Our\nproof shows that if a particle systems has states $\\{0

  2\\}$ and is\nattractive

  then coexistence occurs in the long-range model when the absorbing\nstate $(0

  0)$ is an unstable fixed point of the mean- ?field ODE for $(u_1;\nu_2)$. The result we obtain in this way is asymptotically sharp for Krone's\nmodel

  but the Staver-Levin forest model

  like the quadratic contact process

  \nmay have a nontrivial stationary distribution when $(0

  0)$ is attracting.

  Jian-Guo Liu

  In this paper

  we consider particle systems with interaction and Brownian motion. We prove that when the initial data is from the sampling of Chorin’s method

  i.e.

  the initial vertices are on lattice points hi∈ℝd with mass ρ0(hi)hd

  where ρ0 is some initial density function

  then the regularized empirical measure of the interacting particle system converges in probability to the corresponding mean-field partial differential equation with initial density ρ0

  under the Sobolev norm of L∞(L2)∩L2(H1). Our result is true for all those systems when the interacting function is bounded

  Lipschitz continuous and satisfies certain regular condition. And if we further regularize the interacting particle system

  it also holds for some of the most important systems of which the interacting functions are not Lipschitz continuous. For systems with repulsive Coulomb interaction

  this convergence holds globally on any interval [0

  t]. And for systems with attractive Newton force as interacting function

  we have convergence within the largest existence time of the regular solution of the corresponding Keller–Segel equation.

  Convergence of Stochastic Interacting Particle Systems in Probability under a Sobolev Norm

  Nicolas Lanchier

  Latin American Journal of Probability and Mathematical Statistics

  The stacked contact process is a stochastic model for the spread of an infection within a population of hosts located on the d-dimensional integer lattice. Regardless of whether they are healthy or infected

  hosts give birth and die at the same rate and in accordance to the evolution rules of the neutral multitype contact process. The infection is transmitted both vertically from infected parents to their offspring and horizontally from infected hosts to nearby healthy hosts. The population survives if and only if the common birth rate of healthy and infected hosts exceeds the critical value of the basic contact process. The main purpose of this work is to study the existence of a phase transition between extinction and persistence of the infection in the parameter region where the hosts survive.

  Some rigorous results for the stacked contact process

  Yuan

  Zhang

  Duke University

  Texas A&M University

  National Tsing Hua University

  SAMSI

  Peking University

  UCLA

  National Tsing Hua University

  Assistant Adjunct Professor

  UCLA

  Duke University

  Grader of undergraduate and graduate courses

  calculus lab assistant/ instructor.

  Graduate Teaching Assistant

  Raleigh-Durham

  North Carolina Area

  Peking University

  Visiting Assistant Professor

  Texas A&M University

  Graduate Research Assistant

  Duke University

  Duke University

  Instructor of Math 212

  Summer Session Faculty

  Raleigh-Durham

  North Carolina Area

  Graduate Research Fellow

  Raleigh-Durham

  North Carolina Area

  SAMSI

• Photography

  Stochastic Simulation

  LaTeX

  Matlab

  Probability Theory

  Intrinsic structure study of whale vocalizations

  Robert Calderbank

  Loren Nolte

  Douglas Nowacek

  Wenjing Liao

  Xiaobai Sun

  Intrinsic structure study of whale vocalizations

  The Evolving Voter Model on Thick Graphs

  Anirban Basak

  In 1971

  Schelling introduced a model in which families move if they have too many neighbors of the opposite type. In this paper

  we will consider a metapopulation version of the model in which a city is divided into N neighborhoods

  each of which has L houses. There are ρNL red families and ρNL blue families for some ρ < 1/2. Families are happy if there are ≤ρcL families of the opposite type in their neighborhood and unhappy otherwise. Each family moves to each vacant house at rates that depend on their happiness at their current location and that of their destination. Our main result is that if neighborhoods are large

  then there are critical values ρb < ρd < ρc

  so that for ρ < ρb

  the two types are distributed randomly in equilibrium. When ρ > ρb

  a new segregated equilibrium appears; for ρb < ρ < ρd

  there is bistability

  but when ρ increases past ρd the random state is no longer stable. When ρc is small enough

  the random state will again be the stationary distribution when ρ is close to 1/2. If so

  this is preceded by a region of bistability. \n\n

  Exact solution for a metapopulation version of Schelling's model

  Jian-Guo Liu

  In this paper we develop a new martingale method to show the convergence of the regularized empirical measure of many particle systems in probability under a Sobolev norm to the corresponding mean field PDE. Our method works well for the simple case of Fokker Planck equation and we can estimate a lower bound of the rate of convergence. This method can be generalized to more complicated systems with interactions.

  Convergence of Diffusion-Drift Many Particle Systems in Probability under a Sobolev Norm

  Weak Convergence of a Seasonally Forced Stochastic Epidemic Model

  Alun Lloyd

  In this study

  we extend the results of Kurtz

  to show the weak convergence\nof epidemic processes that include explicit time dependence

  specifically where\nthe transmission parameter

  $\\beta(t)$

  carries a time dependency. We first\nshow that when population size goes to infinity

  the time inhomogeneous process\nconverges weakly to the solution of the mean-field ODE. Our second result is\nthat

  under proper scaling

  the Central Limit type fluctuations converge to a\ndiffusion process.

  Weak Convergence of a Seasonally Forced Stochastic Epidemic Model

  The contact process with fast voting

  Thomas Liggett

  Consider a combination of the contact process and the voter model in which deaths occur at rate 1 per site

  and across each edge between nearest neighbors births occur at rate λ and voting events occur at rate θ. We are interested in the asymptotics as θ→∞ of the critical value λc(θ) for the existence of a nontrivial stationary distribution. In d≥3

  λc(θ)→1/(2dρd) where ρd is the probability a d dimensional simple random walk does not return to its starting point.In d=2

  λc(θ)/log(θ)→1/4π

  while in d=1

  λc(θ)/θ1/2 has lim inf≥1/2√ and lim sup<∞.The lower bound might be the right answer

  but proving this

  or even getting a reasonable upper bound

  seems to be a difficult problem.

  The contact process with fast voting

  Stacy Tantum

  Loren Nolte

  On Marine Mammal Detection Bound