Probabilistic Methods in PDE
Introduction: The probabilistic method in PDE is equally used in Pure and Applied Mathematics research. This is regarded as a very powerful tool by the researchers working on the theory of differential equations. However, as the topic demands expertise on both PDE and probability theory, an initiative to teach this as a structured course is vastly absent globally, including in India. There is hardly any lecture note or an online course accessible for the mathematics students. There is no dedicated book for mathematics students that focuses on this topic and assembles all important aspects suitable for an introduction to this topic. Young researchers like Ph.D. students or junior postdoctoral fellows, who aspire to learn this subject, resort on several different books and study by themselves, which often consumes a considerable amount of their productive time.
To change the present scenario and to boost up research on this very powerful and vibrant topic, this introductory course has been designed. I have offered this once informally at IISER Pune and then officially at Justus-Liebig University, Giessen, Germany for research students in 2019. This course, although an advanced one, attracts students with a background of PDE, Probability Theory, Mathematical Finance, or Mathematical Physics. This course allows a researcher to confidently take up an original research problem in the related field.
The students are suggested to take up this course only after learning some advanced topics in Stochastic processes. An example of an advanced stochastic process course can be found here. This would be considered as a prerequisite. Nevertheless, all the relevant terms and definitions would be clearly mentioned and all the relevant results would be recalled. Only a few of these prerequisite results would be proved in the class and the rest would not be proved for the sake of time limitations.
This course content is mainly based on two different books, one on stochastic calculus and another on semigroup theory. Many theorems would be proved in the lectures with greater details than the reference books. The lectures can be found in this YouTube playlist, sponsored by NPTEL.
Reference:
Ioannis Karatzas and Steven Shreve, Brownian Motion and Stochastic Calculus, GTM, Springer-Verlag New York, 1998.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York, 1983
Teaching Assistants:
Mr. Ravishankar Kopildev Yadav
Dr. Somnath Pradhan
Lectures Lessons/Topics (Week-wise description)
Mathematical formulation of stochastic processes
Brief review of L2 theory of stochastic integration
Ito’s formula
Probabilistic method in Dirichlet problem
Further topics of Dirichlet problem and Probabilistic method in the heat equation
Further topics of Probabilistic method in the heat equation
Tychonoff's uniqueness theorem for the heat equation video31 video32
Widder’s result and its extension on the positive solution to the heat equation video33
The solution to the mixed initial boundary value problem video34
Feynman Kac formula: Stochastic representation of the solution to the Cauchy problem for the backward heat equation with continuous terminal data, potential function, and Lagrangian video35
Application of Feynman Kac formula and Ito's formula
Kac’s theorem on the stochastic representation of solution to a second-order linear ODE and a detailed proof video36 video37
Tutorial: Geometric Brownian motion(GBM), GBM as a solution to a Stochastic differential equation video38
System of stochastic differential equations in application to financial economics video39
Brownian Bridge video40
Stochastic differential equations
PDE with variable coefficient elliptic operators
Functional Stochastic Differential Equations
Elliptic and uniformly elliptic operators, Operator associated with weak solution to SDE and FSDE video47
Statement of Dirichlet problems with variable coefficients elliptic operator video48
Cauchy Problem with variable coefficients: Feynman-Kac formula video49 video50
Semigroup of bounded linear operators, strongly continuous and uniformly continuous semigroup, Infinitesimal generator
Infinitesimal generator of a finite state continuous time Markov chain is bounded video51
Feynman Kac formula and its abstraction with Semi-group theory
Infinitesimal generator of a uniformly continuous semigroup is bounded and vise versa.
Infinitesimal generator of the semigroup of translation in a Euclidean space is unbounded, a first order linear PDE. video52
Growth property of C0 semigroup video53
Unique semigroup generated by a bounded linear operator. video54
Properties of C0 semigroup and its application in solving homogeneous initial value problem. video55
Mild solution to hIVP
Hille-Yoshida theorem, Yoshida approximation video56
Mild solution to linear evolution problems
Inhomogeneous initial value problem, the formula of variations of constants, sufficient condition for existence of classical solution video57 video58
Tutorial on Resolvant operator video59
The relation between the Feynman-Kac formula and the formula of variations of constants video60
homogeneous non-autonomous evolution problems and solution operator, evolution system video61
inhomogeneous non-autonomous evolution problem and mild/generalized solution video61
Mild solution to semilinear evolution problem
An extension of Hille-Yoshida theorem and its application to solve inhomogeneous non-autonomous evolution problem video62, video63
Semilinear evolution problem with Lipschitz additive term and its mild solution, the integral equation of the mild solution.
Existence and uniqueness of mild solution to (sEP). video64
Regularity of mild solution, a sufficient condition for the existence of a classical solution, a detailed proof. video65, video66
Conclusion video67
The slides used in the video recordings have been revised and improved with occasionally more written clarifications and also by eliminating typographical errors. Those slides are saved as presentationxx.pdf and numbered sequentially and can be viewed or downloaded from the following file cabinet. The presentations are not numbered according to the video lecture number. These are rather arranged topics-wise. With the help of TAs and some other students, printable Lecture notes from the presentation files are being prepared. The video numbers corresponding to each note are to be mentioned. Some notes may include additional corrections, examples or clarifications.
presentation22.pdf - Non-autonomous Evolution Problem
presentation21.pdf - Feynman-Kac Formula vs Formula of variations of constants
presentation20.pdf - Inhomogeneous Cauchy problem
presentation19.pdf - Classical & Mild solution to homogeneous IVP, Hille-Yosida Theorem, Yosida Approximation.
presentation18.pdf - Definition, examples and growth of C_0 semigroups
presentation17.pdf - Cauchy Problem with variable coefficients: Proof of Feynman-Kac formula
presentation16.pdf - Operator associated with Weak Solution to FSDE. Statement of Feynman-Kac formula
presentation15.pdf - Weak Solution and Operator associated to that
presentation14.pdf - Existence and uniqueness of SDE solution
presentation13A.pdf - Tutorial: GBM, Mean reverting process, and Brownian bridge
presentation13.pdf - A second order linear ODE
presentation12.pdf - The Feynman-Kac formula
presentation11.pdf - Nonnegative solution. Solution to the mixed initial boundary value problem
presentation10.pdf - Uniqueness of solution to the heat equation
presentation09.pdf - Existence of solution to the Heat equation and its smoothness. (unbounded domain)
presentation08.pdf - Dirichlet Problem: part 2 - Continuity of candidate solution at regular boundary points
presentation07.pdf - Dirichlet problem: part 1 - Bounded solution
presentation06.pdf - Harmonic function and mean value property, Maximum Principle
presentation05.pdf - Brownian motion and its quadratic variation
presentation04.pdf - Stochastic integral w.r.t. continuous local martingale
presentation03.pdf - L2 theory of stochastic integration and its properties.
presentation02.pdf - Preliminary notations for defining Stochastic Integration
presentation01.pdf - Mathematical formulation of stochastic processes
presentation00.pdf - Prerequisite concepts from Measure Theory
Lecture04.pdf - video12, video13, video14, video15
Lecture03.pdf - video09 video10 video11
Lecture02.pdf - video07 video08
Lecture01.pdf - video04 video05 video06
Lecture00.pdf - video01 video02 video03