Probabilistic Methods in Analysis, Geometry, and Beyond

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This project develops several research directions combining probability, geometry and analysis, with many problems motivated by physics. The main object of study are random systems with a certain level of degeneracy similar to a constrained movement. Such degenerate diffusions in high or infinite dimensions have many applications in different fields including quantum field theory (QFT), turbulence, chemical dynamics, and large data environments. While they are useful in modelling many of such phenomena, degenerate and high-dimensional nature of the setting pose mathematical challenges. A number of the projects are for graduate and possibly undergraduate students. In addition, the results will be used for mentoring and educational activities at the local middle school and at the undergraduate level.One of the directions of research is to study Cameron-Martin-Girsanov type quasi-invariance in hypoelliptic settings, and its applications to functional inequalities, smoothness of probability laws in subelliptic and singular settings. This is closely related to asymptotic behavior of such diffusions, namely, large 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 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.