Li works in Probability and Stochastic Analysis, with special interests in stochastic differential and stochastic partial differential equations, their geometry and dynamics. She works on stochastic processes on geometric spaces, multi-scale systems, rough paths, and limit theorems.
Here is a list of topics she works on:
Stochastic processes and Stochastic Differential Equations on manifolds,
Interplay of proeprties of stochastic processes and that of manifolds,
Geometry of diffusion operators,
Sum of Squares of Vector Fields : sub-elliptic, Hoermander’s conditions,
Multi-scale stochastic dynamics on geometric spaces,
Complexity reduction with geometric invariance and conservation laws,
Ergodic properties of stochastic processes,
Multi-Scale stochastic equations driven by fractional Brownian Motions (fBM)– in this new theory one incorporates techniques from and inspired by the rough path theory as well as stochastic dynamical systems.
Theory of fluctuation of stochastic partial differential equations, estimates on the heat kernel and kernels of SDEs driven by fBMs.
Heat Kernels Estimates (gradient and Hessian),
Stochastic Flows;
Analysis on Path Spaces over manifolds: Sobolev calculus, Hodge DeRham theory; Infinite Dimensional Laplacian, Poincare inequality, Logarithmic Sobolev Inequalities;
Special Processes: Strict local martingales, Stochastic Bridges.