16 research outputs found
On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution
Let M denote the space of Borel probability measures on the real line. For
every nonnegative t we consider the transformation
defined for any given element in M by taking succesively the the (1+t) power
with respect to free additive convolution and then the 1/(1+t) power with
respect to Boolean convolution of the given element. We show that the family of
maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of
composition and that, quite surprisingly, every is a homomorphism
for the operation of free multiplicative convolution.
We prove that for t=1 the transformation coincides with the
canonical bijection discovered by Bercovici and
Pata in their study of the relations between infinite divisibility in free and
in Boolean probability. Here M_{inf-div} stands for the set of probability
distributions in M which are infinitely divisible with respect to free additive
convolution. As a consequence, we have that is infinitely
divisible with respect to free additive convolution for any for every in
M and every t greater than or equal to one.
On the other hand we put into evidence a relation between the transformations
and the free Brownian motion; indeed, Theorem 4 of the paper
gives an interpretation of the transformations as a way of
re-casting the free Brownian motion, where the resulting process becomes
multiplicative with respect to free multiplicative convolution, and always
reaches infinite divisibility with respect to free additive convolution by the
time t=1.Comment: 30 pages, minor changes; to appear in Indiana University Mathematics
Journa
Convolution powers in the operator-valued framework
We consider the framework of an operator-valued noncommutative probability
space over a unital C*-algebra B. We show how for a B-valued distribution \mu
one can define convolution powers with respect to free additive convolution and
with respect to Boolean convolution, where the exponent considered in the power
is a suitably chosen linear map \eta from B to B, instead of being a
non-negative real number. More precisely, the Boolean convolution power is
defined whenever \eta is completely positive, while the free additive
convolution power is defined whenever \eta - 1 is completely positive (where 1
stands for the identity map on B).
In connection to these convolution powers we define an evolution semigroup
related to the Boolean Bercovici-Pata bijection. We prove several properties of
this semigroup, including its connection to the B-valued free Brownian motion.
We also obtain two results on the operator-valued analytic function theory
related to the free additive convolution powers with exponent \eta. One of the
results concerns analytic subordination for B-valued Cauchy-Stieltjes
transforms. The other gives a B-valued version of the inviscid Burgers
equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued
free Brownian motion.Comment: 33 pages, no figure
Free Infinite Divisibility for Ultrasphericals
We prove that the integral powers of the semicircular distribution are freely
infinitely divisible. As a byproduct we get another proof of the free infnite
divisibility of the classical Gaussian distribution.Comment: 10 page
The normal distribution is -infinitely divisible
We prove that the classical normal distribution is infinitely divisible with
respect to the free additive convolution. We study the Voiculescu transform
first by giving a survey of its combinatorial implications and then
analytically, including a proof of free infinite divisibility. In fact we prove
that a subfamily Askey-Wimp-Kerov distributions are freely infinitely
divisible, of which the normal distribution is a special case. At the time of
this writing this is only the third example known to us of a nontrivial
distribution that is infinitely divisible with respect to both classical and
free convolution, the others being the Cauchy distribution and the free
1/2-stable distribution.Comment: AMS LaTeX, 29 pages, using tikz and 3 eps figures; new proof
including infinite divisibility of certain Askey-Wilson-Kerov distibution