On Krylov-Veretennikov expansion for stochastic semi-groups and coalescing stochastic flows

Andrey Dorogovtsev, Institute of Mathematics, National Academy of Sciences of Ukraine. Part of the Probability, Stochastic Modelling and Financial Mathematics seminar.

This lecture is devoted to chaotic representation of non-classical multiplicative functionals. We begin with the semi-group of strong random operators in Skorokhod sense. Since the strong random operator can be non bounded, such semi-groups can demonstrate a very complicated behavior and have an interesting structure.

We discuss the corresponding examples and a variant of Trotter formula for Arratia flow. Despite of above mentioned complexities it occurs that when the stochastic semi-group is the multiplicative functional from the cylindrical Wiener process it has a chaotic representation which is the generalization of the Krylov-Veretennikov expansion. In the second part of the lecture we discuss the chaotic representations for the functionals from Arratia flow of coalescing Brownian particles on the real line. Here coalescence makes the picture essentially non Gaussian.

Despite of this we are able to find the chaotic representation in terms of multiple integrals with respect to Brownian motions from the flow. Such integrals contain some correction terms corresponding to the coalescing phenomena. Presented results leads to deeper understanding of the structure of complicated multiplicative systems and noise which they generate.

The lecture is partially based on the joint work with Georgiy Riabov and Nikolai Vovchanskiy.