Probabilistic Methods in PDE

Overview

Syllabus 12

1. week-01
   • Prerequisite Measure Theory - Part 01
   • Prerequisite Measure Theory - Part 02
   • Prerequisite Measure Theory - Part 03
   • Random variable
   • Stochastic Process
   • Conditional Expectation
2. week-02
   • Preliminary for Stochastic Integration - Part 01
   • Preliminary for Stochastic Integration - Part 02
   • Definition and properties of Stochastic Integration - Part 01
   • Definition and properties of Stochastic Integration - Part 02
   • Further properties of Stochastic Integration
3. week-03
   • Extension of stochastic integral
   • change of variable formula and proof - Part 1
   • change of variable formula and proof - Part 2
   • Brownian motion as the building block
   • Brownian motion and its martingale property - Part 1
   • Brownian motion and its martingale property - Part 2
4. week-04
   • Application of Ito’s rule on Ito process
   • Harmonic function and its properties
   • Maximum principle of harmonic function
   • Dirichlet Problem and bounded solution
   • Example of a Dirichlet problem
   • Regular points at the boundary
   • Zarembas cone condition for regularity
5. week-05
   • Summary of the Zaremba's cone condition
   • Continuity of candidate solution at regular points - Part 1
   • Continuity of candidate solution at regular points - Part 2
   • Summary of bounded solution to the Dirichlet Problem
   • Stochastic representation of bounded solution to a heat equation - Part 1
   • Stochastic representation of bounded solution to a heat equation - Part 2
6. week-06
   • Uniqueness of solution to the heat equationÂ
   • Remark on Tychonoff's Theorem
   • Widder’s result and its extension on heat equation
   • Solution to the mixed initial boundary value problemÂ
   • The Feynman-Kac formulaÂ
7. week-07
   • Kac’s theorem on the stochastic representation of solution to a second-order linear ODE - Part 1
   • Kac’s theorem on the stochastic representation of solution to a second-order linear ODE - Part 2
   • Geometric Brownian motion
   • A system of stochastic differential equations in application
   • Brownian bridge
8. week-08
   • Simulation of stochastic differential equations
   • Stochastic differential equations: Uniqueness
   • Stochastic differential equations: Existence part 1
   • Stochastic differential equations: Existence part 2
   • Stochastic differential equations: Existence part 3
   • Stochastic differential equations: Weak solution
9. week-09
   • Functional Stochastic Differential Equations
   • Statement of Dirichlet and Cauchy problems with variable coefficients elliptic operators
   • Cauchy Problem with variable coefficients: Feynman-Kac formula: Part 1
   • Cauchy Problem with variable coefficients: Feynman-Kac formula: Part 2
   • Semigroup of bounded linear operators on Banach space - Part 1
10. week-10
    • mod10lec52-Semigroup of bounded linear operators on Banach space part 2
    • mod10lec53-Growth property of C0 semigroup
    • mod10lec54-Unique semigroup generated by a bounded linear operator
    • mod10lec55-Homogeneous initial value problem
    • mod10lec56-Mild solution to homogeneous initial value problem
11. week-11
    • mod11lec57-Mild solution to inhomogeneous initial value problem
    • mod11lec58-Sufficient condition for existence of classical solution of iIVP
    • mod11lec59-Tutorial on Resolvant operator
    • mod11lec60-Feynman-Kac formula and the formula of variations of constants
    • mod11lec61-Non-autonomous evolution problem and mild/generalized solution
12. week-12
    • mod12lec62-Sufficient condition for existence of an evolution system
    • mod12lec63-Y-valued solution
    • mod12lec64-mild/generalized solution to Semi-linear Evolution Problem
    • mod12lec65-Existence of classical solution Part 1
    • mod12lec66-Existence of classical solution Part 2
    • mod12lec67-Conclusion

Advantages and disadvantages

Some points apply to every course of this kind; see how we rank.

Taught by

• Prof. Anindya GoswamiIISER Pune