Kavita Ramanan - Interacting Stochastic Processes on Sparse Random Graphs

Explore a 50-minute lecture on interacting stochastic processes on sparse random graphs, delivered by Kavita Ramanan for the International Mathematical Union. Delve into recent progress in characterizing hydrodynamic limits and marginal dynamics of stochastic processes on sparse random interaction graphs. Compare their behavior with classically studied mean-field limits arising from complete interaction graphs. Examine applications in statistical physics, neuroscience, biology, and engineering. Learn about local weak convergence of graphs, process convergence results, and global empirical measure convergence. Investigate marginal dynamics on various graph structures, including infinite d-regular trees and unimodular Galton-Watson trees. Discover analogous convergence results for jump processes and explore Markovian approximations for detecting phase transitions and analyzing transient behavior.

Intro
Interacting Stochastic Processes
A Prototype Examples Pairwise Interacting Diffusions
Global Empirical Measure Process
Key Questions
Outline of the Rest of the Talk
Classical Mean-Field Results for Interacting Diffusions
Summary of the Classical Case
Challenges in the Sparse Regime
Local weak convergence of graphs
Local convergence of marked graphs
Examples of local weak convergence of deterministic graphs
Modes of Local Convergence for Random Graph Sequences
Other Examples of Local weak convergence of random graphs
A More General Class of Interacting Diffusions
1. Process Convergence Results
Global Empirical Measure Convergence Results
2. Global Empirical Measure Convergence
Marginal Dynamics on the Line
Key Properties of the Marginal Dynamics/Local Equations
Elements of the Proof: 1. A Filtering Lemma
Elements of the Proof: 2. A Markov Random Field Property
Summary: Beyond Mean-Field Limits
Infinite d-regular trees
Unimodular Galton-Watson trees
Marginal Dynamics on Galton Watson Trees
Interacting Jump Process Dynamics
Analogous Convergence Results Assumption
Convergence Results for Jump Processes (contd.)
Marginal Dynamics for Jump Processes on A-Regular Trees
Markovian Approximations to the Local Equations
Detecting Phase Transitions via Markov Approximations
Markovian Approximations for Transient Behavior
Acknowledgment for Numerical Simulations