Second quantisation for skew convolution products of infinitely divisible measures

Abstract

Suppose ${\lambda }_1$ and ${\lambda }_2$ are infinitely divisible Radon measures on real Banach spaces $E_1$ and $E_2$, respectively and let $T:E_1\to E_2$ be a Borel measurable mapping so that $T({\lambda }_1)\ast \rho ={\lambda }_2$ 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' $P_T:L^p(E_2,{\lambda }_2)\to L^p(E_1,{\lambda }_1)$ given by $$P_{T}f(x)={\int }_{E_2}f(T(x)+y)d\rho (y)$$ as the second quantisation of a contraction operator acting between suitably chosen `reproducing kernel Hilbert spaces' associated with ${\lambda }_1$ and ${\lambda }_2$.

Some typos have been corrected. To appear in IDAQP