Oliver Tough : The Fleming-Viot Particle System with McKean-Vlasov dynamics
- Probability,Uploaded Videos ( 1332 Views )Quasi-Stationary Distributions (QSDs) describe the long-time behaviour of killed Markov processes. The Fleming-Viot particle system provides a particle representation for the QSD of a Markov process killed upon contact with the boundary of its domain. Whereas previous work has dealt with killed Markov processes, we consider killed McKean-Vlasov processes. We show that the Fleming-Viot particle system with McKean-Vlasov dynamics provides a particle representation for the corresponding QSDs. Joint work with James Nolen.
David Aldous: Probability Seminar
- Probability,Uploaded Videos ( 649 Views )David Aldous, Probability Seminar Sept 30, 2021 TITLE: Can one prove existence of an infectiousness threshold (for a pandemic) in very general models of disease spread? ABSTRACT: Intuitively, in any kind of disease transmission model with an infectiousness parameter, there should exist a critical value of the parameter separating a very likely from a very unlikely resulting pandemic. But even formulating a general conjecture is challenging. In the most simplistic model (SI) of transmission, one can prove this for an essentially arbitrary large weighted contact network. The proof for SI depends on a simple lemma concerning hitting times for increasing set-valued Markov processes. Can one extend to SIR or SIS models over similarly general networks, where the lemma is no longer applicable?
Ruth Williams : Control of Stochastic Processing Networks
- Probability ( 222 Views )Stochastic processing networks (SPNs) are a significant generalization of conventional queueing networks that allow for flexible scheduling through dynamic sequencing and alternate routing. SPNs arise naturally in a variety of applications in operations management and their control and analysis present challenging mathematical problems. One approach to these problems, via approximate diffusion control problems, has been outlined by J. M. Harrison. Various aspects of this approach have been developed mathematically, including a reduction in dimension of the diffusion control problem. However, other aspects have been less explored, especially, solution of the diffusion control problem, derivation of policies by interpretating such solutions, and limit theorems that establish optimality of such policies in a suitable asymptotic sense. In this talk, for a concrete class of networks called parallel server systems which arise in service network and computer science applications, we explore previously undeveloped aspects of Harrison's scheme and illustrate the use of the approach in obtaining simple control policies that are nearly optimal. Identification of a graphical structure for the network, an invariance principle and properties of local times of reflecting Brownian motion, will feature in our analysis. The talk will conclude with a summary of the current status and description of open problems associated with the further development of control of stochastic processing networks. This talk will draw on aspects of joint work with M. Bramson, M. Reiman, W. Kang and V. Pesic.
Laurie Field : Relating variants of SLE using the Brownian loop measure
- Probability ( 205 Views )In this talk I will discuss a framework for transforming one variant of the SchrammLoewner evolution (SLE) into another. The main tool in this approach is the Brownian loop measure. A simple case is to relate the reversal of radial SLE to whole-plane SLE, which looks the same locally. Writing the formula one might naïvely expect fails, because the loop measure term is infinite. In joint work with Greg Lawler, we show that there is a finite normalized version of the loop measure term, and that with this change, the naïve formula relating the two SLEs becomes correct.
Yu Gu : Scaling limits of random fluctuations in homogenization of divergence form operators
- Probability ( 182 Views )Recently, quantitative stochastic homogenization of operators in divergence form has witnessed important progress. Our goal is to go beyond the error bound to analyze statistical fluctuations around the homogenized limit. We prove a pointwise two-scale expansion and a large scale central limit theorem for the solution. The approach is probabilistic. The main ingredients include the Kipnis-Varadhan method applied to symmetric diffusion in random environment, a quantitative martingale central limit theorem, the Helffer-Sjostrand covariance representation and Stein's method. This is joint work with Jean-Christophe Mourrat.
Jan Rychtar : The evolution of cooperation kin selection and greenbeard genes
- Probability ( 164 Views )Abstract:One way to convince ourselves that no cooperation can evolve among defectors is via a simple yet one of the most famous games in all of game theory - the Prisoners dilemma (PD) game. The players of this game adopt one of the two strategies: a) a cooperator who pays a cost so that another individual can receive a benefit, or b) a defector who can receive benefits, but it has no cost as it does not deal out any benefits at all. As seen from this formulation, no rational individual would opt to be a cooperator. Yet, we can see cooperation everywhere around us and thus (assuming defectors were here first) there must exist at least one mechanism for its evolution. Nowak (2006, 2012) discusses several of such mechanisms, including the kin selection by which cooperation can spread if the benefits go primarily to genetic relatives. In this talk we will introduce a simple PD-like asymmetric matrix game and show how Hamiltons rule can easily be recovered. We will also introduce a simple PD-like symmetric matrix game to model the evolution of cooperation via greenbeard mechanism, which can be seen as a special case of kin selection.
Kavita Ramanan : Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs
- Probability ( 160 Views )Many applications can be modeled as a large system of homogeneous interacting particles on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle.
Krishna Athreya : Preferential attachment random graphs with general weight function
- Probability ( 156 Views )Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth rates for the growth of the degree for each vertex with n and the behavior of the empirical distribution of the degrees. In this talk we will consider three cases: the weight function w(.) is superlinear, linear, and sublinear. Using recently obtained limit theorems for the growth rates of a pure birth continuous time Markov chains and an embedding of the discrete time graph sequence in a sequence of continuous time pure birth Markov chains, we establish a number of results for all the three cases. We show that the much discussed power law growth of the degrees and the power law decay of the limiting degree distribution hold only in the linear case, i.e., when w(.) is linear
Mokshay Madiman : A Shannon-McMillan-Breiman theorem for log-concave measures and applications in convex geometry
- Probability ( 132 Views )Our primary goal is to describe a strong quantitative form of the Shannon-McMillan-Breiman theorem for log-concave probability measures on linear spaces, even in the absence of stationarity. The main technical result is a concentration of measure inequality for the ``information content'' of certain random vectors. We will also briefly discuss implications. In particular, by combining this concentration result with ideas from information theory and convex geometry, we obtain a reverse entropy power inequality for convex measures that generalizes the reverse Brunn-Minkowski inequality of V. Milman. Along the way, we also develop a new information-theoretic formulation of Bourgain's hyperplane conjecture, as well as some Gaussian comparison inequalities for the entropy of log-concave probability measures. This is joint work with Sergey Bobkov (Minnesota).
Shankar Bhamidi : Two philosophies for random graphs and networks: Local weak convergence and scaling limits
- Probability ( 110 Views )The last few years have witnessed an explosion in the number of mathematical models for random graphs and networks, as well as models for dynamics on these network models. In this context I would like to exhibit the power of two well known philosophies in attacking problems in random graphs and networks: First, local weak convergence: The idea of local neighborhoods of probabilistic discrete structures (such as random graphs) converging to the local neighborhood of limiting infinite objects has been known for a long time in the probability community and has proved to be remarkably effective in proving convergence results in many different situations. Here we shall give a wide range of examples of the above methodology. In particular, we shall show how the above methodology can be used to tackle problems of flows through random networks, where we have a random network with nodes communicating via least cost paths to other nodes. We shall show in some models on the completely connected network how the above methodology allows us to prove the convergence of the empirical distribution of edge flows, exhibiting how macroscopic order emerges from microscopic rules. Also, we shall show how for a wide variety of random trees (uniform random trees, preferential attachment trees arising from a wide variety of attachment schemes, models of trees from Statistical Physics etc) the above methodology shows the convergence of the spectral distribution of the adjacency matrix of theses trees to a limiting non random distribution function. Second, scaling limits: For the analysis of critical random graphs, one often finds that properly associated walks corresponding to the exploration of the graph encode a wide array of information (including the size of the maximal components). In this context we shall extend work of Aldous on Erdos-Renyi critical random graphs to the context of inhomogeneous random graph models. If time permits we shall describe the connection between these models and the multiplicative coalescent, arising from models of coagulation in the physical sciences.
Ivan Matic : Deterministic Walks in Random Environments
- Probability ( 107 Views )A deterministic walk in a random environment can be understood as a general finite-range dependent random walk that starts repeating the loop once it reaches a site it has visited before. Such process lacks the Markov property. We will talk about the exponential decay of the probabilities that the walk will reach sites located far away from the origin.
Ahmed Bou-Rabee : Homogenization with critical disorder
- Probability ( 0 Views )Homogenization is the approximation of a complex, “disordered” system by a simpler, “ordered” one. Picture a walker on a grid. In each step, the walker chooses to walk along a neighboring edge with equal probability. At large scales, the walker approximates Brownian motion. But what if some edges are more likely to be traversed than others? I will discuss recent advances in the theory of quantitative homogenization which make it possible to analyze random walk with drift and other models in probability. Joint work with Scott Armstrong and Tuomo Kuusi.