Feynman-Kac formula: path integral formulation
Goal: Understand one of the most fundamental formulas related to the concept of path integral formulation (which is Feynman-Kac formula) and be familiar with its applications.
Study Team: Marcel Majocha, Adrian Mirecki, Aniszia Vajda, Alex Muresan
Supervision: Marcin Napiórkowski
Application: Quantum Physics, QFT
Perspective: Immerse into the constructive world of quantum field theory
Math related to the topic: probability theory, real analysis (in particular measure theory)
Logical scheme:
- Basics of measure and probability theory
- Introduction to stochastic processes (random walk, Brownian motion etc.)
- Diffusion models
- …
Bonus: Historical note
References:
- Kac, M. On distributions of certain Wiener functionals, Transactions of the American Mathematical Society, vol. 65, no. 1, 1949, pp. 1–13
- Moral, Pierre Del. Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer, 2004.
- Pavliotis, Grigorios A. Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations, Springer, 2014.
- Kolmogorov, A. N., and S. V. Fomin, Measure, Lebesgue Integrals and Hilbert Space, Academic Press, 1962.
- Glimm and Jaffe, Quantum physics: a functional integral point of view
(list will expand during the project)
Summer project
Fundamentals of probability theory and stochastic processes
Responsible: Aniszia Vajda
Scheme:
a. Basics of measure theory and integration
b. Probability spaces and related concepts
c. Radom variables. Law of large numbers; independence and the central limit theorem
d. Discrete and continuous stochastic processes
e. Markov processes
f. Application to Brownian motion.
Main references:
Klenke, Probability theory: A comprehensive course
Grimmett and Stirzacker, Probability and random processes
More references here.