Second quantisation for skew convolution products of infinitely divisible measures

TL;DR

This paper extends second quantisation techniques to skew convolution products of infinitely divisible measures on Banach spaces, unifying Gaussian and Poissonian cases through a representation of transition operators.

Contribution

It introduces a framework for representing transition operators as second quantisations in the context of skew convolution products of infinitely divisible measures.

Findings

Unified representation for Gaussian and Poissonian cases

Extension of second quantisation to skew convolution products

Representation of transition operators as second quantisations

Abstract

Suppose $\lambda_1$ and $\lambda_2$ are infinitely divisible Radon measures on real Banach spaces $E_1$ and $E_2$, respectively and let $T:E_{1} \rightarrow E_{2}$ be a Borel measurable mapping so that $T(\lambda_1) * \rho = \lambda_2 $ for some Radon probability measure $\rho$ on $E_{2}$. Extending previous results for the Gaussian and the Poissonian case, we study the problem of representing the `transition operator' $P_{T}:L^{p}(E_{2}, \lambda_{2}) \rightarrow L^{p}(E_{1}, \lambda_{1})$ given by $$ P_{T}f(x) = \int_{E_{2}}f(T(x) + y)d\rho(y) %% d\rho(y) instead of \rho(dy) in order to unify notations $$ as the second quantisation of a contraction operator acting between suitably chosen `reproducing kernel Hilbert spaces' associated with $\lambda_1$ and $\lambda_2$.