## Overview

Geometric Aspects of stochastic processes and Stochastic processes on geometric spaces; Heat Kernels Estimates (gradient and Hessian) for Parabolic Schroedinger Equations; Stochastic Dynamical Systems (large and short time asymptotics), Stochastic Differential Equations, Stochastic Flows; Limit theorems, Law of Large number and Ergodicity of Stochastic Processes on Geometric Spaces; Sum of Squares of Vector Fields (elliptic, sub-elliptic or with Hoermanderâ€™s conditions); Geometry of sub-elliptic operators; Stochastic Averaging, Singular Perturbation, Perturbation to Conservation Laws, Complexity Reduction and Multi-Scale Analysis ;MeanFfields Equations; Malliavin calculus, Infinite dimensional Sobolev calculus and Hodge theory on path and loop spaces; Spectral Gap for the Infinite Dimensional Laplacian and Logarithmic Sobolev Inequalities for Probability Measures on path and Loop Spaces; Differential forms and harmonic maps, Hodge DeRham theory; Martingales, local martingales, and semi-martingales. Special processes: Brownian bridges and reflected Brownian motions.

## Guest Lectures

Plenary lecture @ 8th International Congress of Chinese Mathematicians, 2019

Lecture course @ THE 12TH CHINESE SUMMER SCHOOL for Ph.D. students of China, July 2007.

Lecture course @ Osaka University, Japan, August 2003

Lecture course @JYVASKYLA SUMMER SCHOOL, Finland, August 2013

Lecture course @ Partial Differential Equations Probability Thematic Trimester; Toulouse, May 2014

Lecture course @ The University of Campinas, Brazil, August 2008

lecture course @ DIMITSANA SUMMER SCHOOL on “Stochastic Differential Geometry and Applications in Finance”, Greece, July 2005,, Greece

Lectures @ The Indo-UK school on SPDE and applications, Bangalore, December 2016