Learn · Probabilistic Methods in PDE
Lecture 24 of 67week-04
Zarembas cone condition for regularity
Prof. Anindya Goswami · IISER Pune
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All the lectures
- Prerequisite Measure Theory - Part 01
- Prerequisite Measure Theory - Part 02
- Prerequisite Measure Theory - Part 03
- Random variable
- Stochastic Process
- Conditional Expectation
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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