  This article, which is substantially motivated by the previous joint work
with J. McKay [8], establishes the analytic analogues of the relations we found
free probability has with Witt vectors. Therefore, we first present a novel
analytic derivation of an exponential map which relates the free additive
convolution on $\mathbb{R}$ with the free multiplicative convolution on either
the unit circle or $\mathbb{R}_+^*$, for compactly supported, freely infinitely
divisible probability measures. We then introduce several new operations on
these measures, which gives rise to more extended classes of operations. Then
we consider the relation with classical infinite divisibility and the geometry
of the spaces involved. Finally, we discuss the general structure, using the
language of operads and algebraic theories, of the various operations we
defined give rise to.
