Asymptotics and ergodicity of hypoelliptic random processes

Randomness has been used to model numerous phenomena in physics, biology, finance etc. Starting with the classical example of Brownian motion used to describe the motion of particles subject to thermal fluctuations and later to model the value of stock prices over time, stochastic techniques have found many applications. For example, randomness is a key ingredient in algorithms used to analyze large data. One of the major questions in such an analysis is understanding if and how a random system converges to an equilibrium. The research in this award will study such convergence depending on the models used. The project includes training graduate students, introducing undergraduate students to research, while the results will be disseminated through publications and presentations at conferences. The project concerns problems combining probability, analysis, geometry. One of the directions of research is to study limits laws such as small deviations, laws of iterated logarithm and large deviations for hypoelliptic diffusions and random walks. These questions are closely related to the Cameron-Martin-Girsanov type quasi-invariance in hypoelliptic settings, applications to functional inequalities, and smoothness of probability laws in hypoelliptic and singular settings. The methods include diverse probabilistic techniques such as coupling and Dirichlet forms. In particular, research concerns large and small deviations, the Onsager–Machlup functional which can be viewed as an analog of the Lagrangian of a dynamical system, and convergence to equilibrium of a large particle system with singular potentials. Both degeneracy (lack of ellipticity) and high dimensions have to be dealt with new techniques coming from different fields such as probability, ergodic theory and sub-Riemannian geometry. While many of these settings arise naturally in applications, their mathematical analysis is not easy. In addition to the theoretical significance of such questions, some answers have practical uses. For example, the rate of convergence to the equilibrium, its dependence on the number of particles and other parameters, or an explicit form of the rate function have many applications. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.