Southern California Probability Symposium
University of California, Irvine
Saturday December 8, 2012
Place:Balboa Island A, Student Center (building 113 on the campus map here).
Parking: Parking is available across the street from the Student Center at Student Center Parking Structure near the intersection of Pereira and W. Peltason Drives. A permit may be purchased at the kiosk upon entering the structure. UC participants may use their UC parking permits. Signs will be posted to direct you to Balboa Island A.
Check here for a list of past Southern California Probability Symposiums.
9:30 - 10:00 Continental breakfast
10:00 - 10:50 Chuck Newman, UCI will speak on "Coarsening Models"
11:00 - 11:50 Alexandra Chronopoulou, UCSB will speak on "Sequential Change Detection for Fractional SDEs"
12:00 - 1:30 Lunch Break
1:30 - 2:20 Tonci Antunovic UCLA will speak on "Tug-of-War and the Infinity Laplace Equation With Neumann Boundary Conditions"
2:30 - 3:20 Quentin Berger, USC will speak on "Influence of a Correlated Disorder in the Polymer Pinning Model"
3:30 - 4:00 Coffee break
4:00 - 4:50 Bruce Driver, USCD will speak on "Feynman Path Integrals in Curved Spaces"
Title: Coarsening Models
Coarsening models are continuous time Markov processes whose states are the assignments of one of two possible values (say +1 or -1) to the vertices of some (usually infinite) graph like Z^d
(with nearest-neighbor edges) or a homogeneous tree. The transition rules (which are
the zero temperature limit of stochastic Ising models) are
that at rate one each vertex updates by adjusting to agree with a strict
majority of its neighbors or in the event of a tie, tosses a fair coin. One is often interested in an initial state in which sites choose values
independently with probability p of being +1. These models have been or can be used to study evolution in time of spatial structure in materials or in voting preferences. Among the questions of interest are, for p = 1/2, whether sites change preference infinitely often and, for p > 1/2,
whether all sites are eventually +1, and how the answers to these
questions depend on the underlying graph. We will review some old
results about Z^d for d <= 2 and recent somewhat unexpected results (jointly with Damron and Sidoravicius) about two dimensional slabs. There are many open problems.
Title: Sequential Change Detection for Fractional SDEs
We will consider the problem of sequentially detecting a change in a stochastic process that satisfies a fractional stochastic differential equation with an arbitrary Hurst index, H. For this class of dynamics, we will establish sufficient conditions for the classical Cumulative Sums (CUSUM) test to be an exact (non-asymptotic) solution to Lorden's
minimax optimal stopping problem. In this way, we will extend well-known optimality properties of CUSUM for diffusion processes. The main techniques for these extensions come from fractional calculus and Malliavin calculus.
Title: Tug-Of-War and the Infinity Laplace Equation with Neumann Boundary Conditions
Tug-of-War is a stochastic zero sum, two player game played by moving a token in a domain until it hits its boundary. At each step Player II pays to Player I a certain value determined by the current position of the token, and the order of moves is determined by fair coin tosses. In a work of Peres, Schramm, Sheffield and Wilson these stochastic games were used to obtain new results on the existence and uniqueness of solutions for certain Infinity Laplace equations with Dirichlet boundary conditions. In this talk we will study the limiting behavior of the game values for Tug-of-War of prescribed horizon, and use it to prove the existence results for the Infinity Laplace equation with vanishing Neumann boundary conditions. This is a joint work with Yuval Peres, Scott Sheffield and Stephanie Somersille.
Title: Influence of a Correlated Disorder in the Polymer Pinning Model
In the study of critical phenomena, the question of the influence of disorder is central. The idea is to compare the phase transition of a disordered system to that of its non-disordered (or homogeneous) version. One can then discuss whether the presence of randomness changes or not the critical properties of the system. This question of relevance/irrelevance of disorder has recently been a great source of investigation in the polymer pinning model, and is now mathematically understood in the case of an IID disorder.
After introducing the pinning model and describing the existing results, we will comment on the influence of spatially correlated disorder in this framework. In particular, we will stress how very strong correlations can have a crucial impact on the critical behavior of the system.
Title: Feynman Path Integrals in Curved Spaces
In a first (perhaps second) course on quantum mechanics one learns to quantize a classical mechanical system in the operator formalism via "canonical quantization." However, when dealing with classical systems with non-flat configuration spaces, canonical quantization may be ambiguous due to problems with "operator orderings." On the other hand at first blush, Feynman's path integral interpretation of quantum mechanics does not seem to suffer from these ambiguities. However, there is no free lunch and the same ambiguities reappear in the Feynman picture when one actually tries to precisely define these path integral expressions. This talk will describe some attempts to mathematically interpret Feynman's picture for quantum mechanical systems in geometric settings. Choices will have to be made and these choices lead to different quantizations of the same classical system. (All terms in this abstract will be explained at least at the level needed for this talk.)