Schedule-at-a-Glance

36 PRIMA 2013 Abstractssamples and investigate asymptotic convergence of optimalvalues and optimal solutions as sample size increases.The analysis provides a unified framework for asymptoticconvergence of some data-driven problems and extendsthe classical asymptotic convergence analysis in stochasticprogramming. The discussion is extended to a stochasticNash equilibrium problem where each player takes arobust action on the basis of their subjective expectedobjective value.Special Session 18ProbabilityThe motion of a tagged particle in the simpleexclusion processDayue Chen & Peng ChenPeking University, Chinadayue@pku.edu.cnThe simple exclusion process is an interacting particlesystem. There is no birth or death of particles. Eachparticle perform an independent random walk. The walkis suspended when a particle jumps to the site of anotherparticle. Therefore a tagged particle behaviors very muchlike a random walk with a fixed rate of slow down. Inthis talk we shall review some limit theorems of a taggedparticle in the simple exclusion and report our progressin this direction.Multivariate normal approximation by Stein’smethod: the concentration inequality approachLouis H. Y. ChenNational University of Singapore, Singaporematchyl@nus.edu.sgIn this talk we will describe Stein’s method for normalapproximation and explain the role of concentration inequalitiesin proving Kolmogorov bounds in one dimension.We will then discuss multivariate extensions of boththe normal approximation and the concentration inequalities.The multivariate concentration inequalities are thenapplied to multivariate normal approximation for independentsummands as well as for locally dependent summands.This talk is based on joint work with Xiao Fang.Stable processes with driftZhen-Qing ChenUniversity of Washington, USAzqchen@uw.eduSuppose that d ≥ 1 and α ∈ (1, 2). Let Y be a rotationallysymmetric α-stable process on R d and b an R d -valued measurable function on R d belonging to a certainKato class of Y . We show that dX t = dY t + b(X t)dt withX 0 = x has a unique weak solution for every x ∈ R d . LetL b = −(−∆) α/2 + b · ∇, which is the infinitesimal generatorof X b . Denote by Cc∞ (R d ) the space of smooth functionson R d with compact support. We further show thatthe martingale problem for (L b , Cc∞(Rd )) has a uniquesolution for each initial value x ∈ R d . Moreover, sharptwo-sided estimates for the transition density function ofX can be obtained as well as that of its subprocess killedupon leaving a C 2 -smooth open set. This talk is basedon a joint work with Longmin Wang.Brownian motion in a heavy tailed PoissonianpotentialRyoki FukushimaKyoto University, Japanryoki@kurims.kyoto-u.ac.jpConsider the d-dimensional Brownian motion in a randompotential defined by attaching a non-negative and polynomiallydecaying potential around Poisson points. Weintroduce a repulsive interaction between the Brownianpath and the Poisson points by weighting the measureby the Feynman-Kac functional. Under the (annealed)weighted measure, it is shown that the Brownian motiontends to localize around the origin and the properly scaledprocess converges in law to a Ornstein-Uhlenbeck process.Martingale problem under non-linear expectationsXin GuoUniversity of California, Berkeley, USAxinguo@berkeley.eduWe consider the martingale problem under the frameworkof nonlinear expectations, analogous to that in a probabilityspace in the seminal paper of Stroock and Varadhan(1969).We first establish an appropriate comparison theorem andthe existence result for the associated state-dependentfully non-linear parabolic PDEs. We then construct theconditional expectation from the viscosity solution of thePDEs, and solve the existence of martingale problems.Under this non-linear expectation space, we further developthe stochastic integral and the Itô’s type formula,which are consistent with Peng’s G-framework. As anapplication, we introduce the notion of weak solution ofSDE under the non-linear expectation.Scaling limits of interacting particle systemsin high dimensions.Mark HolmesUniversity of Auckland, New Zealandm.holmes@auckland.ac.nzA number of (interacting) particle systems at criticality,such as the voter model, the contact process, orientedpercolation, and lattice trees, have been conjectured orproved to behave like the critial branching random walk,when the dimension is high enough so that the interactionis weak. One version of this statement is that appropriatelyrescaled versions of these models converge to super-Brownian motion. We will discuss what is known and notknown about these models in the context of convergenceto super-Brownian motion.Parabolic Littlewood-Paley inequality forφ(−∆)-type operators and applications tostochastic integro-differential equationsPanki KimSeoul National University, Koreapkim@snu.ac.krIn this talk we introduce a parabolic version of theLittlewood-Paley inequality for the operators of the typeφ(−∆), where φ is a Bernstein function. As an application,we construct an L p-theory for the stochastic integrodifferentialequations of the type du = (−φ(−∆)u+f) dt+g dW t. This is a joint work with Ildoo Kim and Kyeong-Hun Kim.Quenched invariance principles for randomwalks and random divergence forms in randommedia with a boundaryTakashi Kumagai