## 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