Second quantisation for skew convolution products of measures in Banach spaces

TL;DR

This paper explores the second quantisation framework for skew convolution products of measures in Banach spaces, linking measure theory with operator theory and chaos expansions, especially for infinitely divisible measures.

Contribution

It introduces a second quantisation approach for skew convolution measures in Banach spaces, extending the functorial procedure to this context, with detailed analysis for Gaussian and Poisson cases.

Findings

Skew convolution measures can be lifted to operators via second quantisation.

The second quantisation process is explicitly characterized using chaos expansions.

The approach applies to infinitely divisible measures, including Gaussian and Poisson types.

Abstract

We study measures in Banach space which arise as the skew convolution product of two other measures where the convolution is deformed by a skew map. This is the structure that underlies both the theory of Mehler semigroups and operator self-decomposable measures. We show how that given such a set-up the skew map can be lifted to an operator that acts at the level of function spaces and demonstrate that this is an example of the well known functorial procedure of second quantisation. We give particular emphasis to the case where the product measure is infinitely divisible and study the second quantisation process in some detail using chaos expansions when this is either Gaussian or is generated by a Poisson random measure.