NOC:Probabilistic Methods in PDE

Lecture 1 - Prerequisite Measure Theory - Part 1

Lecture 2 - Prerequisite Measure Theory - Part 2

Lecture 3 - Prerequisite Measure Theory - Part 3

Lecture 4 - Random variable

Lecture 5 - Stochastic Process

Lecture 6 - Conditional Expectation

Lecture 7 - Preliminary for Stochastic Integration - Part 1

Lecture 8 - Preliminary for Stochastic Integration - Part 2

Lecture 9 - Definition and properties of Stochastic Integration - Part 1

Lecture 10 - Definition and properties of Stochastic Integration - Part 2

Lecture 11 - Further properties of Stochastic Integration

Lecture 12 - Extension of stochastic integral

Lecture 13 - change of variable formula and proof - Part 1

Lecture 14 - change of variable formula and proof - Part 2

Lecture 15 - Brownian motion as the building block

Lecture 16 - Brownian motion and its martingale property - Part 1

Lecture 17 - Brownian motion and its martingale property - Part 2

Lecture 18 - Application of Ito’s rule on Ito process

Lecture 19 - Harmonic function and its properties

Lecture 20 - Maximum principle of harmonic function

Lecture 21 - Dirichlet Problem and bounded solution

Lecture 22 - Example of a Dirichlet problem

Lecture 23 - Regular points at the boundary

Lecture 24 - Zarembas cone condition for regularity

Lecture 25 - Summary of the Zaremba's cone condition

Lecture 26 - Continuity of candidate solution at regular points - Part 1

Lecture 27 - Continuity of candidate solution at regular points - Part 2

Lecture 28 - Summary of bounded solution to the Dirichlet Problem

Lecture 29 - Stochastic representation of bounded solution to a heat equation - Part 1

Lecture 30 - Stochastic representation of bounded solution to a heat equation - Part 2

Lecture 31 - Uniqueness of solution to the heat equation 

Lecture 32 - Remark on Tychonoff's Theorem

Lecture 33 - Widder’s result and its extension on heat equation

Lecture 34 - Solution to the mixed initial boundary value problem 

Lecture 35 - The Feynman-Kac formula 

Lecture 36 - Kac’s theorem on the stochastic representation of solution to a second-order linear ODE - Part 1

Lecture 37 - Kac’s theorem on the stochastic representation of solution to a second-order linear ODE - Part 2

Lecture 38 - Geometric Brownian motion

Lecture 39 - A system of stochastic differential equations in application

Lecture 40 - Brownian bridge

Lecture 41 - Simulation of stochastic differential equations

Lecture 42 - Stochastic differential equations: Uniqueness

Lecture 43 - Stochastic differential equations: Existence - Part 1

Lecture 44 - Stochastic differential equations: Existence - Part 2

Lecture 45 - Stochastic differential equations: Existence - Part 3

Lecture 46 - Stochastic differential equations: Weak solution

Lecture 47 - Functional Stochastic Differential Equations

Lecture 48 - Statement of Dirichlet and Cauchy problems with variable coefficients elliptic operators

Lecture 49 - Cauchy Problem with variable coefficients: Feynman-Kac formula - Part 1

Lecture 50 - Cauchy Problem with variable coefficients: Feynman-Kac formula - Part 2

Lecture 51 - Semigroup of bounded linear operators on Banach space - Part 1

Lecture 52 - Semigroup of bounded linear operators on Banach space - Part 2

Lecture 53 - Growth property of C0 semigroup

Lecture 54 - Unique semigroup generated by a bounded linear operator

Lecture 55 - Homogeneous initial value problem

Lecture 56 - Mild solution to homogeneous initial value problem

Lecture 57 - Mild solution to inhomogeneous initial value problem

Lecture 58 - Sufficient condition for existence of classical solution of IVP

Lecture 59 - Tutorial on Resolvant operator

Lecture 60 - Feynman-Kac formula and the formula of variations of constants

Lecture 61 - Non-autonomous evolution problem and mild/generalized solution

Lecture 62 - Sufficient condition for existence of an evolution system

Lecture 63 - Y-valued solution

Lecture 64 - mild/generalized solution to Semi-linear Evolution Problem

Lecture 65 - Existence of classical solution - Part 1

Lecture 66 - Existence of classical solution - Part 2

Lecture 67 - Conclusion video