UVA Probability Seminar
Wednesdays, 3:30 - 4:30pm
Kerchof 128
Organizers: Christian Gromoll & Tai Melcher
Mathematics Other Seminars Maps & Directions

Fall 2011

24 Aug Organizational meeting
31 Aug Patrick Driscoll (UVa)
Smooth densities for area-like processes driven by fractional Brownian motion
7 Sep Christian Gromoll
Progress on the dual random intersection graph
14 Sep Tai Melcher
A quasi-invariance result on an abstract Wiener group
28 Sep Sivan Leviyang (Georgetown)
Scaling limits for stochastic models of HIV infection
19 Oct Nicos Georgiou (U Utah)
Large deviations for the log-gamma polymer in 1+1-dimensions
26 Oct Patrick Driscoll
What is Rough Path Analysis? I
2 Nov Patrick Driscoll
What is Rough Path Analysis? II
9 Nov David Sivakoff (Duke)
Contact process on modular random graphs
30 Nov Jessica Zuniga (Duke)
Merging and stability for time inhomogeneous Markov chains
* please note time, date, and/or location change

Abstracts

Smooth densities for area-like processes driven by fractional Brownian motion

Patrick Driscoll (UVa)

The fractional Brownian motions are a family of stochastic processes which resemble Brownian motion in many key ways, yet lack the quality of independence of increments. The focus of this talk will be proving the smoothness of densities for solutions to a particular class of differential equations driven by fractional Brownian motion. Heuristically, these solutions may be thought of analogues to the Levy's area process for a standard Brownian motion.
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A quasi-invariance result on an abstract Wiener group

Tai Melcher

We'll discuss recent results for "subelliptic" heat kernel measures on infinite-dimensional analogues of the Heisenberg group. Certain functional inequalities lead to a proof of quasi-invariance for these measures. I'll try to give basic definitions and to provide some motivation.
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Scaling limits for stochastic models of HIV infection

Sivan Leviyang (Georgetown)

During infection, the HIV population is both attacked and attacks immune system cells. The resultant coupling of HIV and immune system dynamics is often modeled through deterministic ODEs, but in recent years a growing body of work has centered on stochastic models. These models are not well understood and many important mathematical questions remain unanswered. In this talk, I will present a specific system of stochastic ODEs that model HIV evasion of immune system attack. The nature of the ODEs will depend on various population parameters resulting in different scaling limits. I will connect the stochastic models and associated scaling limits to some open biological questions involving HIV.
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Large deviations for the log-gamma polymer in 1+1-dimensions

Nicos Georgiou (U Utah)

A directed polymer in a random environment is, loosely speaking, a directed random walk path coupled to interact with a random environment. The talk is mainly about explicit results for the partition function of an "exactly solvable" polymer model in dimension 1+1. The computational tools that make the calculations tractable are a (very) specific distribution for the weights and a version of Burke's theorem tailored to the polymer setting.

The computational techniques are strong enough to give the law of large numbers, sharp variance bounds and an explicit large deviation principle for the partition function and this results will be discussed during the talk.

If time allows, we will also discuss large deviations for the polymer exit point and path that also apply to more general models.

This is joint work with Timo Seppalainen.
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Contact process on modular random graphs

David Sivakoff (Duke)

We studied the contact process (or SIS epidemic) on a pair of dense networks with sparse connections between them. I will give an intuitive derivation for the distribution of the time at which the contact process jumps from one part of the network to the other, and outline the proof.
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Merging and stability for time inhomogeneous Markov chains

Jessica Zuniga (Duke)

In this talk we will discuss the quantitative analysis concerning the asymptotic behavior of time inhomogeneous finite Markov chains. A time inhomogeneous Markov chain is said to be merging if it asymptotically forgets where it started. We are interested in obtaining bounds on the merging time of an inhomogeneous chain, which is analogous to the study of mixing times for time homogeneous chains.

To study this behavior, we develop singular value techniques in the context of time inhomogeneous chains and introduce the notion of
c-stability, which can be viewed as a generalization of the case when a time inhomogeneous chain admits an invariant measure. We describe some examples where these techniques yield quantitative results for time inhomogeneous chains. This is joint work with Laurent
Saloff-Coste.
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Mathematics Other Seminars Maps & Directions