About the course:

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 the topic as a structured course is vastly absent globally, including in India. There is hardly any lecture note or a course accessible for the mathematics students. There is no book for mathematics students which focuses on this topic and assembles all important aspects suitable for an introduction to this topic. Young researchers like PhD students or junior postdoctoral fellows, who aspire to learn the topic, resorts on several different books and study by themselves, which often consumes considerable amount of their productive time.

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 the topic as a structured course is vastly absent globally, including in India. There is hardly any lecture note or a course accessible for the mathematics students. There is no book for mathematics students which focuses on this topic and assembles all important aspects suitable for an introduction to this topic. Young researchers like PhD students or junior postdoctoral fellows, who aspire to learn the topic, resorts on several different books and study by themselves, which often consumes considerable amount of their productive time.

To change the present discouraging scenario and to boost up research on this very powerful and vibrant topic, I have designed this introductory course. I have offered this once informally at IISER Pune and then officially at Justus-Liebig University, Giessen, Germany for research students. 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.

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.

INTENDED AUDIENCE : Doctoral students or researchers in the area of partial differential equations or stochastic processes who wish to learn the probabilistic techniques in PDE for solving research problems either in pure or applied mathematics such as Mathematical Finance or Mathematical Physics.

PREREQUISITES : Appropriate for students with MSc in Pure Mathematics with specialization in Analysis and/or Probability Theory. Prerequisite courses: Measure Theory, Functional Analysis, Probability Theory, Stochastic Processes,

INDUSTRY SUPPORT : This course develops tools to solve deterministic Evolution Problem arising in physical scenarios with random noise. The evolution problems include, for example, heat equation and option price equations. Therefore, the cutting edge R&D sectors of Finance industry should value this course.

Week 1 : Mathematical formulation of stochastic processes

Week 2 : Brief review of L2 theory of stochastic integration

Week 3 : Ito’s formula

Week 4 : Probabilistic method in Dirichlet problem

Week 5 : Further topics of Dirichlet problem and Probabilistic method in heat equation

Week 6 : Further topics of Probabilistic method in heat equation

Week 7 : Feynman Kac formula

Week 8 : Stochastic differential equations

Week 9 : PDE with general elliptic operators

Week 10 : Feynman Kac formula and its abstraction with semigroup theory

Week 11 : Mild solution to linear evolution problems

Week 12 : Mild solution to semilinear evolution problem

1. Ioannis Karatzas and Steven Shreve, Brownian Motion and Stochastic Calculus, GTM, Springer-Verlag New York, 1998.

2. A Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York, 1983.

Anindya Goswami received his Bachelor's degree in Mathematics from St. Xavier's College, Calcutta in 2002. Later in the same year, he joined the Integrated Ph.D. program in the Department of Mathematics in Indian Institute of Science, Bangalore. Following the completion of MS degree in 2005, he received the SPM fellowship as part of the National Award for best performance in National Eligibility Test in Mathematical Sciences. He was bestowed with the Doctorate degree from the Department of Mathematics, IISc in the year 2008. The following three years, he carried out postdoctoral research in the University of Twente, Netherlands; INRIA- Rennes, France; and Technion- Israel Institute of Technology, Israel respectively. He joined IISER Pune as an Assistant Professor in fall, 2011. Since then, he has offered a variety of graduate and undergraduate courses- Multivariable Calculus, Point-set Topology, Measure Theory, Functional Analysis, Numerical Analysis, Measure Theoretic Probability Theory, Stochastic Processes, Mathematical Finance, to name a few. He was reappointed at the same department as an Associate Professor in spring, 2018. His current research interest comprises of Non-cooperative Stochastic Dynamic Game, Stochastic Control, Mathematical Finance, and Queuing Network.

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams:**30 April 2023** Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

**CRITERIA TO GET A CERTIFICATE**

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

**YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.**

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IISER Pune. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams:

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IISER Pune. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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