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
1. Prerequisite concepts from Measure Theory video01 video02 video03
2. Kolmogorov’s model of probability, random variable, conditional expectation video04 video06
3. stochastic processes, filtration, measurability of processes video05
Brief review of L2 theory of stochastic integration
1. Doob-Meyer decomposition,
2. square integrable continuous martingale (SCM) and its quadratic variations video07
3. stochastic integral of simple processes w.r.t. SCMs video08
4. L2 theory of stochastic integration video09 video10
5. properties of stochastic integral. video11
Ito’s formula

1. local martingale, semi-martingale
2. stochastic integral w.r.t. continuous local martingale video12
3. Ito’s rule and its detailed proof video13 video14
4. Martingale representation Theorem, Time change for martingales video15
5. Brownian motion and its quadratic variation. video16 video17

Probabilistic method in Dirichlet problem

1. application of Ito’s rule on Ito process video18
2. harmonic function and mean value property, maximum principle video19 video20
3. stochastic representation of the bounded solution to a Dirichlet problem video21 video22 video23
4. Zaremba's cone condition and regular point of boundary video24

Further topics of Dirichlet problem and Probabilistic method in the heat equation

1. Continuity of candidate solution at regular boundary points with detailed proofs video25 video26 video27 video28
2. stochastic representation of a solution to a heat equation video29
3. Sufficient condition on unbounded initial data for the existence of a classical solution to the heat equation video30

Further topics of Probabilistic method in the heat equation
1. Tychonoff's uniqueness theorem for the heat equation video31 video32
2. Widder’s result and its extension on the positive solution to the heat equation video33
3. The solution to the mixed initial boundary value problem video34
4. 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
1. Kac’s theorem on the stochastic representation of solution to a second-order linear ODE and a detailed proof video36 video37
2. Tutorial: Geometric Brownian motion(GBM), GBM as a solution to a Stochastic differential equation video38
3. System of stochastic differential equations in application to financial economics video39
4. Brownian Bridge video40
Stochastic differential equations
1. Simulation of stochastic differential equations video41
2. Existence and uniqueness result of stochastic differential equations: A detailed proof. video42, video43, video44, video45
3. Weak solution of stochastic differential equation video46
PDE with variable coefficient elliptic operators
1. Functional Stochastic Differential Equations
2. Elliptic and uniformly elliptic operators, Operator associated with weak solution to SDE and FSDE video47
3. Statement of Dirichlet problems with variable coefficients elliptic operator video48
4. Cauchy Problem with variable coefficients: Feynman-Kac formula video49 video50
5. Semigroup of bounded linear operators, strongly continuous and uniformly continuous semigroup, Infinitesimal generator
6. Infinitesimal generator of a finite state continuous time Markov chain is bounded video51
Feynman Kac formula and its abstraction with Semi-group theory
1. Infinitesimal generator of a uniformly continuous semigroup is bounded and vise versa.
2. Infinitesimal generator of the semigroup of translation in a Euclidean space is unbounded, a first order linear PDE. video52
3. Growth property of C0 semigroup video53
4. Unique semigroup generated by a bounded linear operator. video54
5. Properties of C0 semigroup and its application in solving homogeneous initial value problem. video55
6. Mild solution to hIVP
7. Hille-Yoshida theorem, Yoshida approximation video56
Mild solution to linear evolution problems
1. Inhomogeneous initial value problem, the formula of variations of constants, sufficient condition for existence of classical solution video57 video58
2. Tutorial on Resolvant operator video59
3. The relation between the Feynman-Kac formula and the formula of variations of constants video60
4. homogeneous non-autonomous evolution problems and solution operator, evolution system video61
5. inhomogeneous non-autonomous evolution problem and mild/generalized solution video61
Mild solution to semilinear evolution problem
1. An extension of Hille-Yoshida theorem and its application to solve inhomogeneous non-autonomous evolution problem video62, video63
2. Semilinear evolution problem with Lipschitz additive term and its mild solution, the integral equation of the mild solution.
3. Existence and uniqueness of mild solution to (sEP). video64
4. Regularity of mild solution, a sufficient condition for the existence of a classical solution, a detailed proof. video65, video66
5. 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.

presentation23.pdf - Semi-linear Evolution Problem

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