117 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
Free probability of type B: analytic interpretation and applications
In this paper we give an analytic interpretation of free convolution of type
B, introduced by Biane, Goodman and Nica, and provide a new formula for its
computation. This formula allows us to show that free additive convolution of
type B is essentially a re-casting of conditionally free convolution. We put in
evidence several aspects of this operation, the most significant being its
apparition as an 'intertwiner' between derivation and free convolution of type
A. We also show connections between several limit theorems in type A and type B
free probability. Moreover, we show that the analytical picture fits very well
with the idea of considering type B random variables as infinitesimal
deformations to ordinary non-commutative random variables.Comment: 28 page
Regularization by free additive convolution, square and rectangular cases
The free convolution is the binary operation on the set of probability
measures on the real line which allows to deduce, from the individual spectral
distributions, the spectral distribution of a sum of independent unitarily
invariant square random matrices or of a sum of free operators in a non
commutative probability space. In the same way, the rectangular free
convolution allows to deduce, from the individual singular distributions, the
singular distribution of a sum of independent unitarily invariant rectangular
random matrices. In this paper, we consider the regularization properties of
these free convolutions on the whole real line. More specifically, we try to
find continuous semigroups of probability measures such that
is the Dirac mass at zero and such that for all positive and all
probability measure , the free convolution of with (or, in
the rectangular context, the rectangular free convolution of with
) is absolutely continuous with respect to the Lebesgue measure, with a
positive analytic density on the whole real line. In the square case, we prove
that in semigroups satisfying this property, no measure can have a finite
second moment, and we give a sufficient condition on semigroups to satisfy this
property, with examples. In the rectangular case, we prove that in most cases,
for in a continuous rectangular-convolution-semigroup, the rectangular
convolution of with either has an atom at the origin or doesn't put
any mass in a neighborhood of the origin, thus the expected property does not
hold. However, we give sufficient conditions for analyticity of the density of
the rectangular convolution of with except on a negligible set of
points, as well as existence and continuity of a density everywhere.Comment: 43 pages, to appear in Complex Analysis and Operator Theor
Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup
Latex, 20 pages, version extended to study Julia-Caratheodory derivatives for the functions involved.International audienceIn his article ''On the free convolution with a semicircular distribution," Biane found very useful characterizations of the boundary values of the imaginary part of the Cauchy-Stieltjes transform of the free additive convolution of a probability measure on the real line with a Wigner (semicircular) distribution. Biane's methods were recently extended by Huang to measures which belong to the partial free convolution semigroups introduced by Nica and Speicher. This note further extends some of Biane's methods and results to free convolution powers of operator-valued distributions and to free convolutions with operator-valued semicirculars. In addition, it investigates properties of the Julia-Caratheodory derivative of the subordination functions associated to such semigroups, extending certain results from the article "Partially Defined Semigroups Relative to Multiplicative Free Convolution" by Bercovici and the author (reference [7] in the text)
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