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Probability Seminar
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Mondays, 2:00 - 2:50pm
Kerchof 326

Organizers: Christian Gromoll & Tai Melcher
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Spring 2007

17 Jan* Adrian P. C. Lim (Cornell) -- (joint with Mathematical Physics seminar, 3:30pm in Kerchof 317)
Path integral quantization
5 Feb Matt Cecil (U Conn)
Taylor map on complex path groups
19 Feb Anja Sturm (U Del)
Voter models with heterozygosity selection
12 Mar Joe Blitzstein (Harvard)
Random network models: Sampling and estimation
19 Mar Bert Zwart (Georgia Tech)
Bandwidth sharing networks in overload
2 Apr Weining Kang (Carnegie Mellon)
Characterizations of the invariant measure for a class of reflected diffusions via the extended Skorokhod map
* please note time, date, and/or location changes

Abstracts

Path integral quantization (pdf)
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Adrian P. C. Lim (Cornell)
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A typical path integral on a manifold M, is an informal expression of the form

(1)shim 1/Z \int_{s \in H(M)} f(s) exp(-E(s)) Ds

where H(M) is a space of paths in M with energy E(s) < \infty, f is a real valued function on H(M), Ds is a "Lebesgue measure," and Z is a normalization constant. The use of path integrals for "quantizing" classical mechanical systems (whose classical energy is E) started with Feynman in [2] with very early beginnings being traced back to Dirac [1]. In this talk, I will give several rigorous definitions to Equation (1), by reviewing work done by Driver and Andersson and recently by me. The idea is to approximate H(M) by finite dimensional subspaces consisting of broken geodesics and then to pass to the limit of finer and finer approximations.

[1] P. A. M. Dirac, Physikalische Zeitschrift der Sowjetunion 3 (1933), 64.

[2] R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948), 367--387.
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Taylor map on complex path groups (pdf)
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Matt Cecil (U Conn)
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The classical Taylor expansion can be interpreted as describing a unitary map from holomorphic functions on C^d which are square integrable with respect to a suitable Gaussian measure to the set of its "derivatives at the origin," interpreted as the symmetric tensor algebra (Fock space) along with a suitable norm. This map is related to the Fock space representation of classical quantum mechanics. Analogous maps have been shown to exist when C^d is replaced by a Hilbert space or by a non-commutative Lie group G. In this talk, I will review these known cases and the challenges they present. The goal will be to describe a Taylor map which takes functions on the space of paths on a Lie group which are square integrable with respect to a heat kernel measure to subspace of the dual of the universal enveloping algebra, a setting in which the base group is both non-commutative and infinite dimensional.
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Voter models with heterozygosity selection (pdf)
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Anja Sturm (U Del)
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In this talk we consider variations of the usual voter model which favor types that are locally less common. This may be understood as a selective advantage for rare types leading to more variation in the particle population. The voter models with selection considered here are dual to certain systems of branching annihilating random walks that are parity preserving. Coexistence of types in one model is related to survival of particles in the other. We consider conditions for the existence of homogeneous invariant laws in which types coexist as well as convergence to these laws.

This is joint work with Jan Swart (UTIA Prague).
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Random network models: Sampling and estimation (pdf)
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Joe Blitzstein (Harvard)
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The usual Erdos-Renyi model for random graphs is mathematically beautiful, but is rarely useful as a model for real-world networks. Thus, many more-complicated models are being explored for such applications as social, biological, and information networks. We will describe a method for converting combinatorial theorems into efficient sampling schemes, and compare sequential importance sampling techniques to MCMC techniques. For the widely-studied case of exponential random graph models, we discuss challenges and approaches for parameter estimation. Parts based on joint work with Sourav Chatterjee and Persi Diaconis.
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Bandwidth sharing networks in overload (pdf)
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Bert Zwart (Georgia Tech)
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Processor Sharing (PS) queues were originally introduced to analyze the performance of time-sharing in computer networks. Nowadays, PS queues are one of the most popular congestion models for TCP traffic on the Internet. From a methodological perspective, PS is a challenging service discipline since a measure-valued state descriptor is necessary to analyze the system.

This talk is focused on such systems in overload. We look at a single queue and consider the scenario where customers leave impatient as their waiting time grows too large. We propose a fluid (functional LLN) approximation of this system and investigate the fixed point of this approximation. This leads to several qualitative insights into the dynamics of PS models with impatience. We show that the impact of impatience on the performance of the system can be quite substantial and propose an admission control scheme to reduce its effect.

We will also consider the extension of some of these results to bandwidth sharing networks. The main challenge there is to prove uniqueness of solutions of certain fixed point equations.
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Characterizations of the invariant measure for a class of reflected diffusions via the extended Skorokhod map (pdf)
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Weining Kang (Carnegie Mellon)
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In this talk we focus on a class of reflected diffusions in polyhedrons defined via the extended Skorokhod map. We show that the reflected diffusion has certain boundary property, i.e., the pushing process in the reflected diffusion charges zero on the amount time the reflected diffusion spends at the boundary away from a "bad" set V where two or more faces meet. Further, under suitable conditions, the reflected diffusion has an unique invariant measure and such an invariant measure satisfies a basic adjoint relationship.
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