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19242101 Aufbaumodul: Stochastics IV "Malliavin calculus and applications"

Summer Term 2023

lecture and exercise by Prof. Dr. Nicolas Perkowski

Time and place

  • Lecture: Thursdays, 10:00 -12:00, SR 009, Arnimallee 6. Starting April 27.

  • Exercise Session: Thursdays, 12:00 -14:00, SR 009, Arnimallee 6.

  • Final Exam:  to be announced in due course

Prerequisits: Stochastics I — III.

Assessment

To receive credits fo the course you need to

  • actively participate in the exercise session 
  • work on and successfully solve the weekly exercises 
  • pass the final exam (see above)   

Exercises 

Problem sets will be put online every Thursday and can be found under Assignements in the Whiteboard portal. The solutions will be discussed in the tutorial.

Course Overview/ Content:

I will give an introduction to Malliavin calculus, a fundamental tool in modern stochastic analysis that allows differentiation of random variables defined on a Gaussian probability space, such as Wiener space. It serves as an infinite-dimensional generalization of analytical concepts like Fourier transforms and Sobolev spaces. The calculus enables the development of an „analysis on Wiener space“ and it provides a framework for studying properties of stochastic processes. We will also discuss applications of Malliavin calculus, including:

  • existence of transition densities for stochastic differential equations under Hörmander’s condition („the noise allows movement in each direction from each point“)
  • hypercontractivity („all moments of polynomials of Gaussians are comparable“, this is crucial for the pathwise approach to SPDEs)
  • the fourth moment theorem (a universality result akin to the central limit theorem, but it applies in very different situations than the CLT and strikingly we only need to check convergence of the second and fourth moment to get weak convergence)
  • construction of two-dimensional Euclidean quantum field theories.

We will start with a crash course on (finite-dimensional) distributions, Fourier transforms and Sobolev spaces. Then we will mostly follow Hairer’s lecture notes.

References

  • Hairer’s lecture notes and further literature:
  • D. Nualart. (2006): The Malliavin calculus and related topics. Springer.
  • Janson, S. (1997): Gaussian Hilbert spaces. Cambridge University Press.