UVA Probability Seminar
Wednesdays, 4:30 - 5:30pm
Kerchof 317
 Organizers: Christian Gromoll, Tai Melcher, & Leonid Petrov
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# Spring 2016

27 Jan Jing Dong (Northwestern) $\varepsilon$-strong simulation for multidimensional stochastic differential equations via rough path analysis Amber Puha (CSU San Marcos) Analysis of processor sharing queues via relative entropy Todd Kemp (UC San Diego) Lie groups, Lie algebras, and heat flow Todd Kemp* (UC San Diego) Analysis Focus Year colloquium, meets 4-5p Random matrices, heat flow, and Lie groups Pascal Moyal* (Université de Technologie de Compiègne) Analysis Focus Year colloquium, meets 3-4p in Clark 101 Large-graph properties of uniform random graphs using measure-valued processes Pascal Moyal* (Université de Technologie de Compiègne) Analysis Focus Year colloquium, meets 4-5pm in Clark 101 Stability of the general stochastic matching model Guillaume Barraquand (Columbia) Fluctuations of the first particle in exclusion processes Ruth Williams (UC San Diego) Criticality and Adaptivity in Enzymatic Networks
* please note time, date, and/or location change

# Abstracts

 $\varepsilon$-strong simulation for multidimensional stochastic differential equations via rough path analysis Jing Dong (Northwestern) Consider a multidimensional diffusion process, $X$. Let $\varepsilon>0$ be a deterministic user defined tolerance error parameter. We develop a systematic way to construct a probability space, supporting both $X$ and a fully simulatable piecewise constant process $X_\varepsilon$, such that $X_\varepsilon$ is within $\varepsilon$ distance from $X$ under the uniform metric on compact time intervals with probability one. Our construction requires a detailed study of continuity estimates of the Itô map using Lyons' theory of rough paths. We approximate the underlying Brownian paths, jointly with the Lévy areas, with a deterministic error bound in the underlying rough path metric. top of page Analysis of processor sharing queues via relative entropy Amber Puha (CSU San Marcos) In this talk, we discuss a new approach to studying the asymptotic behavior of fluid model solutions for critically loaded processor sharing queues. For this, we introduce a notion of relative entropy associated with measure-valued fluid model solutions. In contrast to the approach used in Puha and Williams (2004), which does not readily generalize to networks of processor sharing queues, we expect the approach developed in this paper to be more robust. Indeed, we anticipate that similar notions involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions for stochastic networks operating under various protocols naturally described by measure-valued processes.top of page Lie groups, Lie algebras, and heat flow Todd Kemp (UC San Diego) Using two familiar examples, the unitary group $\mathrm{U}(N)$ and the general linear group $\mathrm{GL}(N)$, I will illustrate the main analytic and geometric constructs associated with Lie groups and their Lie algebras. We will touch on vector fields, PDEs, and representation theory. The goal is to understand what it means to talk about the {\em heat kernel} on a Lie group. Time permitting, we will talk about the diffusion processes (i.e. Brownian motion) that actually transport heat in these geometric examples. This lecture is aimed at graduate students and anyone interested, without any necessary background in differential geometry, Lie theory, or stochastic analysis.top of page Random matrices, heat flow, and Lie groups Todd Kemp (UC San Diego) Random matrix theory studies the behavior of the eigenvalues (or singular values) of random matrices as the dimension grows. Initiated by Wigner in the 1950s, there is now a rich and well-developed theory of the universal behavior of such random eigenvalues in models that are natural generalizations of the Gaussian case. In this talk, I will discuss a generalization of these kinds of results in a new direction. A Gaussian random matrix can be thought of as an instance of Brownian motion on a Lie algebra; this opens the door to studying the eigenvalues (and singular values) of Brownian motion on Lie groups. I will present recent progress understanding the asymptotic spectral distribution of Brownian motion on unitary groups and general linear groups. The tools needed include probability theory, combinatorics, and representation theory.top of page Large-graph properties of uniform random graphs using measure-valued processes Pascal Moyal* (Université de Technologie de Compiègne) We address large-graph asymptotic properties of random graphs, identified via their degree distributions. In all cases, the graphs are constructed by the uniform random pairing procedure, leading to the well-known configuration model. These construction algorithms follow a Markov dynamics, allowing us to represent the state of the partially constructed random graph by a point measure-valued Markov process, until completion. We apply this procedure in two different contexts: (1) for describing the spread of an epidemics on a randomly connected population and (2) to construct independent sets of the graphs, corresponding for instance, in the CSMA context, to the largest class of contemporarily active agents. top of page Stability of the general stochastic matching model Pascal Moyal* (Université de Technologie de Compiègne) Consider a stochastic system in which, to each node of a graph G is associated a point process (seen as represeting arrivals of 'items'). Upon arrival, any item associated to node k is either (i) matched with an item present in the system and corresponding to a node j, such that j and k share an edge in G or (ii) is put in line, waiting for a future match. A matching policy is specified, which allows to chose the item to be matched with in the case where several items are available for a match. We analyze the stability of this system: we first exhibit a general necessary stability condition, that is reminiscent of the condition of existence of a perfect matching on a bipartite graph (Hall's Theorem). We then investigate the structual properties of the graph G, rendering the necessary consition also sufficient (i.e. stability is obtained for any matching algorithm). We also discuss our ongoing investigations on the connection of this model with the construction of a maximal matching on a random graph. (based on joint works with J. Mairesse, and O. Perry). top of page Fluctuations of the first particle in exclusion processes Guillaume Barraquand* (Columbia) We will consider exclusion processes on the one-dimensional integer lattice, starting from a densely packed configuration (step initial condition). We address the non-universal behaviour of the first particle. We will review some known results (TASEP, ASEP), and then focus on two different dynamics. -The multi-particle asymmetric diffusion model, a Bethe ansatz solvable process introduced by Sasamoto and Wadati. For these dynamics, the first particle fluctuates on the $t^{1/3}$ scale with Tracy-Widom GUE fluctuations. -The Facilitated TASEP, an exclusion process where each particle jumps to the right by one at rate 1 only when the right neighbouring site is empty, and the left neighbouring site is occupied. For these dynamics, the first particle fluctuates on the $t^{1/3}$ scale with - surprisingly - Tracy-Widom GSE fluctuations. In the latter case, we will sketch the main lines of our proofs. The exact solvability comes from a coupling with a model of Last Passage Percolation on a half-quadrant, that we analyse using the formalism of Pfaffian Schur processes. Joint works with Jinho Baik, Ivan Corwin and Toufic Suidan. top of page Criticality and Adaptivity in Enzymatic Networks Ruth Williams (UC San Diego) The contrast between stochasticity of biochemical networks and regularity of cellular behavior suggests that biological networks generate robust behavior from noisy constituents. Identifying the mechanisms that confer this ability on biological networks is essential to understanding cells. Here we use stochastic queueing models to investigate one potential mechanism. In living cells, enzymes perform the critical function of acting as catalysts to ensure that important reactions occur at rates fast enough to sustain life. We show how competition among different molecular species for the attention of a limited pool of shared enzymes in enzymatic networks can produce strong correlations between the different species when these systems are poised near a critical state where the substrate input flux is equal to the maximum processing capacity. We then consider the enzymatic networks with adaptation, where the limiting resource is produced in proportion to the demand for it. In this setting, we show that strong correlations are robustly producedacross a broad range of system parameters.This adaptive queueing motif suggests a natural control mechanism for producing strong correlations in biological systems. Based on joint work with Paul Steiner, Jeff Hasty and Lev Tsimring.top of page
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