  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 $\lambda_2$.
