Algebraic and combinatorial aspects
in stochastic calculus
(3+3 hours) Computations involving classical iterated stochastic integrals or rough paths rely (often only implicitly) on shuffle and quasi-shuffle Hopf algebras (in the geometric, or Stratonovich, respectively Itô calculus framework). Recently, similar phenomena have been unveiled in the context of moments-cumulants calculus in classical and free probability theory. In this series of lectures we outline the basic ideas and techniques underpinning algebraic and combinatorial aspects of stochastic differential equations as well as moments-cumulants relations in probability theory. A particular emphasis will be payed on the interplay between Hopf algebra structures (and generalizations thereof) on permutations, surjections, set partitions and rooted trees. This abstract approach will be complemented by studying concrete problems relevant to more advanced applications.
(5 hours) We will present second order asymptotic results for random variables living in a fixed Wiener chaos of a random measure, by stressing their deep connections with Rota and Wallstrom's combinatorial theory of stochastic integration. We will also illustrate three remarkable geometric applications: to random geometric graphs, to nodal sets of random waves, and to real polarization problems.
(5 hours) Free Probability Theory has a combinatorial description
which shows the analogy with classical probability theory quite
clearly; compared to the latter the lattice of all partitions of a set
is replaced by the lattice of non-crossing partitions. There exists
also a free analogue of a Brownian motion and correspondingly a free
stochastic calculus. Again, main properties of these free stochastic
integrals (like Itô formulas or martingale polynomials) are closely
related with properties of non-crossing partitions.
I will introduce the basics about free probability, in particular, the key notion of a free cumulant, and will then concentrate on the main ideas and results around free stochastic calculus, mostly from the combinatorial perspective.
Rough paths, regularity structures and renormalisation [Slides]
(5 hours) We want to present the algebraic and analytic aspects of the theories of rough paths and of regularity structures and their links with renormalisation and Hopf algebras.