Large scale dynamics and geometry in stochastic systems
Abstract
The proposed project considers the large scale behavior of types of stochastic processes, which connect microscopic views to macroscopic laws. Such processes include those used in the modeling of traffic and fluids, whereas others are employed in the modeling of wireless or sensor networks and associated geometric and stochastic optimizations. In one set of problems, the space-time behaviors of mass density, their fluctuations, and interface evolutions in interacting systems of random walks are studied. In another set of problems, relations between the geometry of paths in discrete, random networks with those in the continuum are considered. Still another set of problems, focuses on limit behavior of time-inhomogenous Markov processes with different scales in their transition rates. More specifically, questions will include the following: (1) Derivation of a `nonlinear martingale characterization of the stochastic partial differential equations which govern the fluctuations of the mass in nonequilibrium and multi-type weakly-asymmetric particle systems. (2) Development of a novel probabilistic `Gamma convergence approach to understand how well geodesic or certain optimal paths formed in random point process graphs approximate those in the continuum. (3) Understanding a new class of `stick-breaking processes, with potential use in nonparametric Bayesian statistics, as a limit of time-inhomogeneous Markov chains which form simplified models of simulated annealing. These questions are designed in part to understand fundamental concerns in basic settings. Moreover, the approaches envisioned themselves will be of interest, robust, and of use in other settings. In terms of significance with respect to the Army Research Office (ARO) and broader scientific effects, the project aims to understand important dynamical and geometric features of the large scale structure of fluids and networks in complex models. The first part of the proposal develops scaling limit connections between concerns in the areas of `stochastic partial differential equations , `measure-valued stochastic processes , and `interacting stochastic particle systems . The second part introduces a new probabilistic framework to estimate geometric objects of interest in (sensor) networks formed from data, and might be viewed through the lenses of of `statistical testing and validation of network models and `geometric methods for statistical inference . The third part develops the limit theory of time-inhomogeneous processes, where the possible transitions depends both on the spatial location and the time, and might be seen in the context of `modeling at intermediate timescales . Connections in this work to `stick-breaking processes may also be viewed in terms of `Bayesian and nonparametric statistics . In terms of education, facets of the proposal, are good questions on which to train young researchers; several are working with the PI on this project. In terms of dissemination, proposed also is to disseminate results in talks at conferences and other venues, by posting to the arXiv, and by keeping a useful webpage. In terms of promoting research interactions, the PI is an organizer for the long-running `Frontier Probability Days , a national conference on probability and applications, the next iteration planned for 2020.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Apr 01, 2019
- Source ID
- W911NF1810311
Entities
People
- Sunder Sethuraman
Organizations
- Army Contracting Command
- United States Army
- University of Arizona