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Course11110613467 lectures

Probabilistic Methods in PDE

Prof. Anindya Goswami · IISER Pune

Start with lecture 1 At NPTEL

Lectures

week-01

  1. Prerequisite Measure Theory - Part 01 Video
  2. Prerequisite Measure Theory - Part 02 Video
  3. Prerequisite Measure Theory - Part 03 Video
  4. Random variable Video
  5. Stochastic Process Video
  6. Conditional Expectation Video

week-02

  1. Preliminary for Stochastic Integration - Part 01 Video
  2. Preliminary for Stochastic Integration - Part 02 Video
  3. Definition and properties of Stochastic Integration - Part 01 Video
  4. Definition and properties of Stochastic Integration - Part 02 Video
  5. Further properties of Stochastic Integration Video

week-03

  1. Extension of stochastic integral Video
  2. change of variable formula and proof - Part 1 Video
  3. change of variable formula and proof - Part 2 Video
  4. Brownian motion as the building block Video
  5. Brownian motion and its martingale property - Part 1 Video
  6. Brownian motion and its martingale property - Part 2 Video

week-04

  1. Application of Ito’s rule on Ito process Video
  2. Harmonic function and its properties Video
  3. Maximum principle of harmonic function Video
  4. Dirichlet Problem and bounded solution Video
  5. Example of a Dirichlet problem Video
  6. Regular points at the boundary Video
  7. Zarembas cone condition for regularity Video

week-05

  1. Summary of the Zaremba's cone condition Video
  2. Continuity of candidate solution at regular points - Part 1 Video
  3. Continuity of candidate solution at regular points - Part 2 Video
  4. Summary of bounded solution to the Dirichlet Problem Video
  5. Stochastic representation of bounded solution to a heat equation - Part 1 Video
  6. Stochastic representation of bounded solution to a heat equation - Part 2 Video

week-06

  1. Uniqueness of solution to the heat equation Video
  2. Remark on Tychonoff's Theorem Video
  3. Widder’s result and its extension on heat equation Video
  4. Solution to the mixed initial boundary value problem Video
  5. The Feynman-Kac formula Video

week-07

  1. Kac’s theorem on the stochastic representation of solution to a second-order linear ODE - Part 1 Video
  2. Kac’s theorem on the stochastic representation of solution to a second-order linear ODE - Part 2 Video
  3. Geometric Brownian motion Video
  4. A system of stochastic differential equations in application Video
  5. Brownian bridge Video

week-08

  1. Simulation of stochastic differential equations Video
  2. Stochastic differential equations: Uniqueness Video
  3. Stochastic differential equations: Existence part 1 Video
  4. Stochastic differential equations: Existence part 2 Video
  5. Stochastic differential equations: Existence part 3 Video
  6. Stochastic differential equations: Weak solution Video

week-09

  1. Functional Stochastic Differential Equations Video
  2. Statement of Dirichlet and Cauchy problems with variable coefficients elliptic operators Video
  3. Cauchy Problem with variable coefficients: Feynman-Kac formula: Part 1 Video
  4. Cauchy Problem with variable coefficients: Feynman-Kac formula: Part 2 Video
  5. Semigroup of bounded linear operators on Banach space - Part 1 Video

week-10

  1. mod10lec52-Semigroup of bounded linear operators on Banach space part 2 Video
  2. mod10lec53-Growth property of C0 semigroup Video
  3. mod10lec54-Unique semigroup generated by a bounded linear operator Video
  4. mod10lec55-Homogeneous initial value problem Video
  5. mod10lec56-Mild solution to homogeneous initial value problem Video

week-11

  1. mod11lec57-Mild solution to inhomogeneous initial value problem Video
  2. mod11lec58-Sufficient condition for existence of classical solution of iIVP Video
  3. mod11lec59-Tutorial on Resolvant operator Video
  4. mod11lec60-Feynman-Kac formula and the formula of variations of constants Video
  5. mod11lec61-Non-autonomous evolution problem and mild/generalized solution Video

week-12

  1. mod12lec62-Sufficient condition for existence of an evolution system Video
  2. mod12lec63-Y-valued solution Video
  3. mod12lec64-mild/generalized solution to Semi-linear Evolution Problem Video
  4. mod12lec65-Existence of classical solution Part 1 Video
  5. mod12lec66-Existence of classical solution Part 2 Video
  6. mod12lec67-Conclusion Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISER Pune
Department
Mathematics
Teacher
Prof. Anindya Goswami
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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