The study of how fluctuations arise in a stochastic process - e.g., how a queue overflows or how a financial market crashes - is fundamental for predicting and controlling their behavior. In physics, fluctuations are also known to be connected to the response of systems to perturbations and external forces.

The idea of this project is to see how a Markov process 'builds up' fluctuations in time by conditioning (in a probabilistic sense) that process on observing a given fluctuation and by describing this conditioning as a new Markov process, called the effective or fluctuation process.

• Applications: Brownian motion, random walks, nonequilibrium systems, financial time series, etc.
• Keywords: Brownian bridges, Doob transforms, fluctuation paths, pathways or dynamics, large deviation or rare event conditioning
• Subject areas: Theoretical physics, statistical physics, stochastic processes

This an application of the previous project focused on using conditioning on rare events to find network characteristics of random graphs, such as connected components, typical regions, central nodes, etc. A related project is to study escape-type problems of random walks evolving on random graphs in the limit of large graphs.

• Applications: All sorts of graphs arising in physical and manmade applications
• Keywords: Random graphs, complex networks, random walks on graphs
• Subject areas: Random walks, graphs theory, numerical simulations

Large deviations are difficult to probe numerically as they are, by definition, rarely seen in direct simulations (think of a meteorite hitting earth). The aim of this project is to develop efficient numerical methods, based on Monte-Carlo simulations, to 'accelerate' the sampling of large deviations or rare events in general in Markov processes. This is an active area in physics, statistics, engineering, and financial mathematics.

• Applications: Fluctuations and response of equilibrium and nonequilibrium systems, any other area/topic where rare events are critical
• Keywords: Monte Carlo algorithms, importance sampling, splitting and cloing methods, control theory
• Subject areas: Stochastic processes, numerical simulations, coding (Python, Matlab, C, C++)

Dynamical phase transitions are sudden changes that arise in the fluctuations of stochastic processes or, more precisely, in the mechanisms that create fluctuations. The goal of this project is to study dynamical phase transitions in Markov processes and, especially, in stochastic differential equations. Analogies with phas transitions arising in physics, e.g., in solids and liquids at a certain critical temperature, are established via large deviation theory.

• Applications: Statistical physics, forecasting
• Keywords: Phase transitions in large deviations
• Subject areas: Statistical physics, Markov processes, large deviations

Markov processes jumping at random from one state to another (depending on their current state) are used in physics to model equilibrium and nonequilibrium systems perturbed by noise. Recently, there has been a lot of interest on a new class of Markov processes, called reset processes, evolving stochastically for some time and then jumping to a fixed 'reset' point. The goal of this project is to develop the theory of these processes and, here again, their large deviation properies.

• Applications: Nonequilibrium systems, population dynamics, random searches, etc.
• Keywords: Reset processes, processes with catastrophes or killings, absorbing processes
• Subject areas: Stochastic processes, probability theory, statistical physics