Xue-Mei Li specializes in Probability and Stochastic Analysis, with a focus on stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs), their geometry and dynamics. Her work spans stochastic processes on geometric spaces, multi-scale systems, rough paths, and limit theorems.
Her research topics include:
- Stochastic processes and SDEs on manifolds
- Interplay between stochastic processes and manifold properties
- Geometry of diffusion operators
- Sum of squares of vector fields: sub-elliptic operators, Hörmander’s conditions
- Multi-scale stochastic dynamics on geometric spaces
- Complexity reduction via geometric invariance and conservation laws
- Ergodic properties of stochastic processes
- Multi-scale stochastic equations driven by fractional Brownian motion (fBM), incorporating techniques from rough path theory and stochastic dynamical systems
- Fluctuation theory of SPDEs, including heat kernel estimates and SDEs driven by fBM
- Heat kernel estimates: gradient and Hessian bounds
- Stochastic flows
- Analysis on path spaces over manifolds: Sobolev calculus, Hodge–De Rham theory, infinite-dimensional Laplacians, Poincaré and logarithmic Sobolev inequalities
- Special processes: strict local martingales, stochastic bridges.