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Discrete Stochastic Analysis

This is a TCC course and will run on Mondays 10-12. Notes (in progress) can be found here Lecture Notes (last edited 21/10/19)

Synopsis: Part of the classical stochastic analysis is devoted to the analysis of the so-called Wiener chaos, which expresses L2 random variables as a series expansion of iterated Wiener-Ito integrals. Theories like Malliavin calculus, hypercontractivity, Wick normalisation etc. play a significant role in the analysis of these expansions and associated gaussian spaces.

From the point of view of statistical mechanics of disordered systems or theoretical computer science and boolean functions, one is motivated to look at discrete analogues of Wiener chaos and develop tools that will allow to analyse these discrete structures. Furthermore, one is interested in scaling limits, which amounts to establishing convergence of the discrete structure to the continuum objects.

We use the term ``Discrete Stochastic Analysis'' to describe a set of tools that fall into this framework.
The topics we will expose in these lectures cover

  • general Lindeberg principles
  • convergence of multilinear polynomials of random variables to Wiener chaoses
  • Hoeffding decomposition
  • the Fourth Moment Theorem
  • discrete versions of elements of Malliavin calculus
  • Stein's method
  • discrete versions of general functional (such as Poincare) inequalities
  • applications

Time permitting, we will also aim to present the basics (in a discrete setting) of one of the recent breakthrough theories on singular SPDEs, the theory of Paracontrolled Distributions of Gubinnelli, Imkeller and Perkowski.