9:50 - 10:40 Jinqiao
Duan,
IPAM
and
Illinois
Institute of Technology, "Random Dynamical Systems with Non-Gaussian
Noises"

10:40 - 11:10 BREAK

11:10 - 12:00 Tom Alberts, Cal Tech, "The Continuum Directed
Random Polymer and the KPZ Universality Class"

12:00 - 2:00 LUNCH

2:00 - 2:50 Allan
Sly,
UC
Berkeley, "Asymptotic Learning on Social Networks"

2:50 - 3:20 BREAK

3:20 - 4:10 Tomoyuki Ichiba, UC Santa Barbara, "On collision of Brownian particles and applications"

4:10 - 5:00 Konstantin Zuev, USC, "Markov Chain Monte Carlo Revolution in Reliability Engineering"

6:00 DINNER

Free parking is available--just tell the parking booth attendant that you are going to the Southern California Probability Symposium.

On Saturdays the only entrance likely to be open is Entrance 5, on the northern edge of campus at Jefferson Blvd. and McClintock Ave. Sometimes Entrance 1 off Exposition Blvd. on the south side of campus is open as well--you can look if you pass by; if the gate is closed it will be readily apparent.

Kaprelian Hall is on the western edge of campus, at the intersection of Vermont Ave. and 36th Place.

Organizers:

Ken Alexander, alexandr (at) usc (dot) edu

Peter Baxendale, baxendal (at) usc (dot) edu

Please email one of the organizers if you would like to attend the dinner.

Thanks to Tom Liggett and Jim Pitman for putting together the following: The history of SCPS

Tomoyuki Ichiba

Title: On collision of Brownian particles and applications

In this talk we examine the colliding behavior of Brownian particles which diffuse on the real line determined by a class of stochastic differential equations. The absence and the presence of triple (or higher order) collisions among the particles are crucial in analysis of local time processes accumulated by these collisions. Especially, this analysis sheds light on some important characteristics (e.g., identification, solvability, time-reversal, invariant distributions) of the stochastic system with piece-wise constant or degenerate coefficients. As case studies, we consider a financial equity market model with rank based characteristics as well as a systemic risk analysis of interbank lending system.

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Tom Alberts

Title: The Continuum Directed Random Polymer and the KPZ Universality Class

The discrete directed polymer model is a well studied example of a Gibbsian disordered system and a random walk in a random environment. The usual goal is to understand how the random environment affects the behavior of the underlying walk and how this behavior varies with a temperature parameter that determines the strength of the environment. At infinite temperature the environment has no effect and the walk is the simple random walk, while at zero temperature the environment dominates and the walk follows a single path along which the environment is largest. For temperatures in between there is a competition between the walk wanting to behave diffusively (like simple random walk) and following a path of highest energy (like last passage percolation).

In this talk I will describe recent joint work with Kostya Khanin and Jeremy Quastel for taking a scaling limit of the directed polymer model to construct a continuous path in a continuum environment. We end up with a one-parameter family of random probability measures (indexed by the temperature parameter) that we call the continuum directed random polymer. As the temperature parameter varies the paths cross over from Brownian motion to what is conjectured to be a continuum limit of last passage percolation. This cross over is an inherent feature of the KPZ universality class, which I will also briefly describe.

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Konstantin Zuev

Title: Markov Chain Monte Carlo Revolution in Reliability Engineering

One of the most important and computationally challenging problems in reliability engineering is to estimate the failure probability for a dynamic system, that is, the probability of unacceptable system performance. The failure probability is usually expressed as an integral over a high-dimensional parameter space that is difficult to evaluate numerically. Over the past decade, the engineering research community has realized the importance of stochastic simulation methods for reliability analysis that are based on Markov chain Monte Carlo (MCMC) algorithms. In this talk, in the spirit of a recent paper by Persi Diaconis with a similar title, I will describe the MCMC revolution that has happened over the last decade in the field of reliability engineering that lies at the boundary of engineering sciences and applied probability.

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Jinqiao Duan

Title: Random Dynamical Systems with Non-Gaussian Noises

Gaussian processes, such as Brownian motion, have been widely used in modeling fluctuations, while some complex phenomena in engineering and science involve non-Gaussian Levy noises. Thus dynamical systems driven by non-Gaussian noises have attracted considerable attention recently. The speaker first reviews dynamical issues for nonlinear systems with non-Gaussian Levy noises, and then presents recent work on the escape probability, bifurcation and random invariant manifolds. Non-Gaussianity of the noises manifests as nonlocality at some ‘macroscopic’ level. The differences in dynamics under Gaussian and non-Gaussian noises are highlighted, theoretically or numerically.

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Allan Sly

Asymptotic Learning on Social Networks

We consider a model of social learning consisting of Bayesian agents connected by a network who are each given an independent signal about an unknown state of the world. The agents proceed to iteratively communicate their beliefs to their neighbours saying which state they believe is more likely and learn from their neighbours beliefs. We study the question of asymptotic learning: do all agents learn the state of the world with probability that approaches one as the number of agents tends to infinity?

Joint work with Elchanan Mossel and Omer Tamuz.