Directed polymers are a statistical mechanics model for random growth.  Their partition functions are solutions to a discrete stochastic heat equation.  This talk will discuss their logarithmic derivatives, which are solutions to a discrete stochastic Burgers equation.  Of interest is the success or failure of the  "one force-one solution principle" for this equation.  I will…

We give an explicit description of the jointly invariant measures for the KPZ equation. These are couplings of Brownian motions with drift, and can be extended to a process defined for all drift parameters simultaneously. We term this process the KPZ horizon (KPZH). As a corollary of this description, we resolve a recent conjecture of…

In this talk, I will present recent results on game theoretical formulation of optimal debt management problems in an infinite time horizon with exponential discount, modeled as a noncooperative interaction between a borrower and a pool of risk-neutral lenders. Here, the yearly income of the borrower is governed by a stochastic process and bankruptcy instantly occurs…

Models for networks that evolve and change over time are ubiquitous in a host of domains including modeling social networks, understanding the evolution of systems in proteomics, the study of the growth and spread of epidemics etc. This talk will give a brief summary of three recent findings in this area where stochastic approximation techniques…

 Models for networks that evolve and change over time are ubiquitous in a host of domains including modeling social networks, understanding the evolution of systems in proteomics, the study of the growth and spread of epidemics etc. This talk will give a brief summary of three recent findings in this area where stochastic approximation techniques…

The goal of this talk is to price American-type financial contracts in the presence of Knightian uncertainty. In particular, instead of choosing a particular probabilistic model to represent the price process of some underlying asset (on which an American option is written), we first restrict our attention to the whole class of models that are…

Consider a dense Erdös-Rényi random graph with parameters n and p, with p fixed in (0,1). Let H_n(p) be the hitting time between two distinct vertices: run simple random walk from a vertex w, wait until it hits another vertex v, and average over the random walk. What does the distribution of H_n(p) look like? Heuristic…

We investigate the long-time behavior and stationary fluctuations of an infinite-dimensional rank-based diffusion process, called the Atlas model, where particles move as independent Brownian motions, with the lowest ranked particle at any time getting a unit upward drift. The associated process of gaps between successive ranked particles possesses an uncountable collection of invariant measures. We…

Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and…

We consider a class of disordered mean-field combinatorial optimization problems, focusing on the Gibbs measure, where the inverse temperature does not vary with the size of the graph and the edge weights are sampled from a general distribution. We prove Central Limit Theorems for the log-partition function, the weight of a typical configuration, and the…

The Fisher-KPP equation was introduced in 1937 to model the spread of an advantageous gene through a spatially distributed population. Remarkably precise information on the traveling front has been obtained via a connection with branching Brownian motion, beginning with works of McKean and Bramson in the 70s. I will discuss an extension of this probabilistic…