The probability seminar meets at 10am on Thursdays in AP&M 6402 unless specifically indicated otherwise. Please address all inquiries to this year's seminar coordinator: Professor Williams, williams at math dot ucsd dot edu

** FALL 2006 **

Stochastic Modelling Methods for Gene Finding

For an abstract, click here.

Professor Youri Davydov, University of Lille I, France

Ergodic properties of crystallization processes. For an abstract, click here.

Samuel Kou, Harvard University.

Stochastic modeling in single molecule biophysics.

For an abstract, click here.

Title: "Universality Questions in Random Matrix Theory"

For an abstract, click here.

Michael Kinnally, graduate student, UCSD.

On "Qualitative behavior of stochastic delay equations with bounded memory" following M. Scheutzow, Stochastics, 12 (1984), 41-80.

Priscilla Greenwood, Arizona State University

Autonomous stochastic resonance produces epidemic oscillations of fluctuating size.

For an abstract, click here.

Jomy Alappattu, graduate student, UC Berkeley.

Fragmentation and coalescence of conditioned Galton-Watson forests. For an abstract, click here.

Michael Cranston, UC Irvine

Large deviations for parabolic Anderson and other random media models. For an abstract, click here.

San Diego Chapter of the American Statistical Association, One day conference

S. Molchanov, UNC Charlotte, visiting UCI.

On a class of differential-functional equations. For an abstract, click here.

Nathanael Berestycki, University of British Columbia

Random walks, geometry and comparative genomics. For an abstract, click here.

Nathanael Berestycki, University of British Columbia.

Hydrodynamic limits of spatially structured coalescents. For an abstract, click here.

Please note that many talks for Winter are on unusual days and at unusual times.

Ery Arias-Castro, UCSD.

Searching for a Trail of Evidence in a Maze. For an abstract, click here.

"Schramm-Loewner Evolutions on Riemann surfaces"

Information Theory and Its Applications, Workshop, UCSD, If you are planning to attend, please register. The cost is minimal to cover catering. Registration can be done for just the days that you plan to attend.

Professor Frank Kelly, University of Cambridge

Abstract: Variability in the number of simultaneous flows present can have a substantial impact on the perceived performance of packet networks such as the Internet. While the packet level behaviour of a given set of flows is by now well understood, less is known about the stochastic behaviour of the number of flows in progress on different routes through the network. In this talk we describe recent work on Brownian models of networks in heavy traffic.

Joint work with Ruth Williams.

Southern California Probability Symposium/Conference in Honor of Ted Harris, USC.

Modeling Marked Point Processes.

Abstract: New probability models are proposed for the analysis of marked point processes. These models deal with the type of data that arrive or are observed in possibly unequal time intervals such as financial transactions, earthquakes among others. The models treat both the time between event arrivals and the observed marks as stochastic processes. We adopt a class of bivariate distributions to form the bivariate mixture transition distribution(BMTD). In these models the conditional bivariate distribution of the next observation given the past is a mixture of conditional distributions given each one of the last p observations or a selection of past p events. The identifiability of the model is investigated, and EM algorithm is developed to obtain estimates of the model parameters. Simulation and real data examples are used to demonstrate the utility of these models.

"Interest Rate Markets with Stochastic Volatility". For an abstract, click here.

Markov Models on Trees: Reconstruction and Applications

Abstract: Markov models on trees arise naturally in many fields, notably in molecular biology - as models of evolution; in statistical physics - as models of spin systems; and in networking - as models of broadcasting. In this talk, I will discuss various inference problems motivated especially by applications in statistical phylogenetics, i.e. the reconstruction of evolutionary histories of organisms from their molecular sequences. In particular, I will consider the "root reconstruction" problem: how accurately can one guess the value at the root of the tree, given the state at the leaves? I will focus on recent work establishing new conditions for the impossibility of such reconstruction. I will also discuss the related "phylogenetic reconstruction" problem: given enough samples at the leaves, can one reconstruct the tree that generated this data and, if so, how efficiently? I will present a recent result on a sharp transition in the number of samples required to recover the tree topology, using a connection to the root reconstruction problem above. Time permitting, I will describe briefly connections to computational learning theory and network tomography as well. This is joint work with S. Bhamidi, C. Borgs, J. Chayes, C. Daskalakis, E. Mossel, and R. Rajagopal.

"Quasi-invariant measures on path space".

Abstract: "Let $N$ denote a manifold equipped with a finite Borel measure $\gamma$. A vector field $Z$ on $N$ is said to be admissible with respect to $\gamma$ if $Z$ admits an integration by parts formula. The measure $\gamma$ is said to be quasi-invariant under $Z$ if the class of null sets of $\gamma$ is preserved by the flow generated by $Z$. In this talk we study the law $\gamma$ of an elliptic diffusion process with values in a closed compact manifold. We construct a class of admissible vector fields for $\gamma$, show that $\gamma$ is quasi-invariant under these vector fields, and give a formula for the associated family of Radon-Nikodym derivatives $d\gamma_s\over d\gamma$.

Seminar on Stochastic Processes, Fields Institute, Toronto ON.

Jacek Leskow (Nowy Sacz, Poland, visiting UCSD)

"Relative measurability and time series analysis. A non-stochastic perspective."

Abstract: The concept of relative measure was fairly popular among Polish mathematicians of 1930 in Lvov. Steinhaus and Urbanik were working on introducing a relative measure and relative measurability into the area of random variables. Recent work on signal processing and time series has led to re-discovery of the 'old-school' theorems and application to data generated by signals or time-series. Some fundamental work was done by Garnder and continuation of this work was done by Leskow and Napolitano.

A short informal introduction to a nonstochastic approach to time series inference via relative measurability will be presented. Applications to signal forecasting will be presented.

Professor Ron Getoor, UCSD.

Walsh's interior reduite

Abstract: This will be an expository talk. I'll begin by introducing the concepts of reduite (reduced function) and balayage (swept measure) in classical potential theory and their interpretations in terms of Brownian motion. I'll then discuss the extension of these ideas to Markov processes as in Hunt's fundamental memoir. After introducing h-transforms I'll be able to define the interior reduite and discuss some of its properties following Walsh. If time permits I'll give some indications of recent work in this area by Fitzsimmons and myself.

Vladimir Rotar, San Diego State University

On asymptotic proximity of probability distributions and the non-classical invariance principle.

Abstract:

Usually, a limit theorem of Probability Theory is a theorem that concerns convergence of a sequence of distributions P_n to a distribution P. However, there is a number of works where the traditional setup is modified, and the object of study is two sequences of distributions, P_n and Q_n, and the goal consists in establishing conditions implying the convergence

P_n - Q_n ->0 (1)

In particular problems,P_n and Q_n are, as a rule, the distributions of the r.v.'s f(X_1,...,X_n) and f(Y_1,...,Y_n) , where f(.) is a function, and X_1,X_2,... and Y_1,Y_2,... are two sequences of r.v.'s. The aim here is rather to show that different random arguments X_1,...,X_n may generate close distributions of f(X_1,...,X_n) , than to prove that the distribution of f(X_1,...,X_n) is close to some fixed distribution (which, above else, may be not true). Clearly, such a framework is more general than the traditional one. First, as was mentioned, the distributions P_n and Q_n, themselves do not have to converge. Secondly, the sequences P_n and Q_n are not assumed to be tight, and the convergence in (1) covers situations when a part of the probability mass or the whole distributions "move away to infinity'", while the distributions P_n and Q_n, are approaching each other.

We consider a theory on this point, including the very definition of convergence (1), and a particular example of the invariance principle in the general non-classical setup.

Professor Guillaume Bonnet, UC Santa Barbara

"Non-linear SPDEs for Highway Traffic Flows: Theory, and Calibration to Traffic Data"

Abstract: Highway traffic flows are generally modeled by partial differential equations (PDEs). These models are used by traffic engineers for road design, planning or management. However, they often fail to capture important features of empirical traffic flow studies, particularly at small scales. In this talk, I will propose a fairly simple stochastic model for highway traffic flows in the form of a nonlinear stochastic partial differential equation (SPDE) with random coefficients driven by a Poisson random measure. I will discuss the well posedness of the proposed equation as well as the corresponding inverse problem that I will illustrate by its calibration to high resolution traffic data from highway 101 in Los Angeles. I will also present a more sophisticated spde in the form of a system of coupled hyperbolic-parabolic equations.

Professor Mor Harchol-Balter (Computer Science Department, Carnegie Mellon University).

"Analysis of Join-the-Shortest-Queue Routing in Web Server Farms"

ABSTRACT: We present the first analysis of the Join-the-Shortest-Queue (JSQ) routing policy for Web server farms. Web server farms involve a collection of Processor-Sharing (PS) servers, whereas prior analyses of JSQ have always assumed First-Come-First-Serve (FCFS) servers. This work introduces a new technique: Single-Queue-Approximation (SQA), and uses the technique to prove some interesting insensitivity properties for Web server farms.

Joint with: Varun Gupta, Karl Sigman, and Ward Whitt.

James Norris, Cambridge University.

Planar aggregation and the coalescing Brownian flow.