Probabilistic Methods in PDE Course | Prof. Anindya Goswami | NPTEL
Course Details
| Exam Registration | 9 |
|---|---|
| Course Status | Ongoing |
| Course Type | Elective |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 18 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Unlocking the Power of Probability in Partial Differential Equations
The world of mathematical analysis is witnessing a profound synthesis. The deterministic realm of Partial Differential Equations (PDEs) is increasingly being illuminated by the stochastic insights of probability theory. This fusion, known as Probabilistic Methods in PDE, has emerged as a cornerstone of modern research in both pure and applied mathematics. To address a critical gap in structured learning for this advanced topic, Prof. Anindya Goswami of IISER Pune offers a comprehensive 12-week postgraduate course.
Why This Course is a Game-Changer
Despite its power, the interdisciplinary nature of probabilistic PDEs—demanding deep expertise in both PDE theory and measure-theoretic probability—has left a significant void in formal education. Globally, and particularly in India, there is a notable absence of structured courses, consolidated lecture notes, or dedicated textbooks for mathematics students. Aspiring researchers, such as PhD students and postdoctoral fellows, are often forced to piece together knowledge from disparate sources, a time-consuming and inefficient process.
This course, meticulously designed by Prof. Goswami, aims to change that. It provides a rigorous, self-contained introduction, empowering researchers to confidently tackle original problems in fields ranging from mathematical physics to quantitative finance.
Meet the Instructor: Prof. Anindya Goswami
Prof. Anindya Goswami brings a wealth of academic excellence and international research experience to this course. A recipient of the prestigious SPM fellowship for topping the National Eligibility Test in Mathematical Sciences, he earned his PhD from the Indian Institute of Science, Bangalore. His postdoctoral journey took him to leading institutions in the Netherlands, France, and Israel. Since joining IISER Pune in 2011 (and being promoted to Associate Professor in 2018), he has built a strong teaching and research profile in stochastic analysis, control, game theory, and mathematical finance.
Course Overview and Objectives
This advanced course serves as a bridge, connecting the theories of stochastic processes and differential equations. The primary objective is to equip students with the tools to solve deterministic evolution problems—like the heat equation or Black-Scholes option pricing model—that are driven or influenced by random noise.
Intended Audience: Doctoral students and researchers in PDEs or stochastic processes who wish to employ probabilistic techniques in pure or applied mathematics research.
Prerequisites and Foundation
The course is designed for students with an MSc in Pure Mathematics, specializing in Analysis and/or Probability Theory. A solid foundation is required in:
- Measure Theory and Functional Analysis
- Measure-Theoretic Probability Theory & Stochastic Processes
- Ordinary and Partial Differential Equations
- Calculus, Metric Spaces, and Point-Set Topology
To help students prepare, Prof. Goswami recommends several foundational NPTEL courses, ensuring the material is accessible despite its advanced nature. The course itself is self-contained, with necessary basics revisited throughout the lectures.
Detailed 12-Week Course Layout
| Week | Topics Covered |
|---|---|
| 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 |
Primary Reference Books
The course content synthesizes material from two seminal texts:
- Brownian Motion and Stochastic Calculus by Ioannis Karatzas and Steven Shreve (Springer GTM).
- Semigroups of Linear Operators and Applications to Partial Differential Equations by A. Pazy (Springer).
Prof. Goswami’s lectures will often provide more detailed proofs and explanations than found in these references, offering unique pedagogical value.
Industry Relevance and Applications
The skills developed in this course have direct, high-value applications in cutting-edge industries. The ability to model and solve evolution problems with random perturbations is crucial in:
- Mathematical Finance: For derivative pricing, risk management, and developing sophisticated financial models.
- Mathematical Physics: For understanding diffusion processes, quantum mechanics, and statistical physics.
- R&D Sectors: Any field involving dynamic systems subject to uncertainty, from engineering to data science.
This course is more than an academic pursuit; it is training in a powerful analytical framework that bridges deterministic models and the stochastic nature of real-world phenomena. For any serious researcher looking to master the probabilistic approach to PDEs, this structured journey led by Prof. Anindya Goswami is an invaluable opportunity.
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