Stochastic Analysis and its Applications

Abstract：

The first half of the course is devoted to explaining fundamental concepts, terms, facts and tools in probability theory and stochastic analysis. Then, in the second half, we pick up some topics in stochastic partial differential equations as applications of stochastic analysis.

Contents：

(1) Foundations of Probability Theory (5 lectures)

Probability space, Dynkin's pi-lambda theorem, Convergence of random variables, Independence, Conditional probability, Strong law of large numbers, Kolmogorov's inequality, Convergence in law, Central limit theorem

(2) Foundations of Stochastic Analysis (9 lectures)

Discrete and continuous time martingales, Brownian motion, Stochastic integrals, Ito's formula, Stochastic differential equations, Relation to PDEs

(3) Applications of Stochastic Analysis (14 lectures)

Stochastic partial differential equations, Random interfaces, (Stochastic) Motion by mean curvature, Stochastic Allen-Cahn equation, Time-dependent Ginzburg-Landau equation, Other topics.