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