61 research outputs found
Two-state free Brownian motions
In a two-state free probability space , we define an
algebraic two-state free Brownian motion to be a process with two-state freely
independent increments whose two-state free cumulant generating function is
quadratic. Note that a priori, the distribution of the process with respect to
the second state is arbitrary. We show, however, that if is a von
Neumann algebra, the states are normal, and is faithful,
then there is only a one-parameter family of such processes. Moreover, with the
exception of the actual free Brownian motion (corresponding to ),
these processes only exist for finite time.Comment: 21 page
Free stochastic measures via noncrossing partitions II
We show that for stochastic measures with freely independent increments, the
partition-dependent stochastic measures of math.OA/9903084 can be expressed
purely in terms of the higher stochastic measures and the higher diagonal
measures of the original.Comment: 15 pages, AMS-LaTeX2e. A serious revision following the suggestions
by the refere
Orthogonal polynomials with a resolvent-type generating function
The subject of this paper are polynomials in multiple non-commuting
variables. For polynomials of this type orthogonal with respect to a state, we
prove a Favard-type recursion relation. On the other hand, free Sheffer
polynomials are a polynomial family in non-commuting variables with a
resolvent-type generating function. Among such families, we describe the ones
that are orthogonal. Their recursion relations have a more special form; the
best way to describe them is in terms of the free cumulant generating function
of the state of orthogonality, which turns out to satisfy a type of
second-order difference equation. If the difference equation is in fact first
order, and the state is tracial, we show that the state is necessarily a
rotation of a free product state. We also describe interesting examples of
non-tracial infinitely divisible states with orthogonal free Sheffer
polynomials.Comment: 19 pages; minor improvement
Linearization coefficients for orthogonal polynomials using stochastic processes
Given a basis for a polynomial ring, the coefficients in the expansion of a
product of some of its elements in terms of this basis are called linearization
coefficients. These coefficients have combinatorial significance for many
classical families of orthogonal polynomials. Starting with a stochastic
process and using the stochastic measures machinery introduced by Rota and
Wallstrom, we calculate and give an interpretation of linearization
coefficients for a number of polynomial families. The processes involved may
have independent, freely independent or q-independent increments. The use of
noncommutative stochastic processes extends the range of applications
significantly, allowing us to treat Hermite, Charlier, Chebyshev, free Charlier
and Rogers and continuous big q-Hermite polynomials. We also show that the
q-Poisson process is a Markov process.Comment: Published at http://dx.doi.org/10.1214/009117904000000757 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Monic non-commutative orthogonal polynomials
Among all states on the algebra of non-commutative polynomials, we
characterize the ones that have monic orthogonal polynomials. The
characterizations involve recursion relations, Hankel-type determinants, and a
representation as a joint distribution of operators on a Fock space.Comment: 10 page
Appell polynomials and their relatives
This paper summarizes some known results about Appell polynomials and
investigates their various analogs. The primary of these are the free Appell
polynomials. In the multivariate case, they can be considered as natural
analogs of the Appell polynomials among polynomials in non-commuting variables.
They also fit well into the framework of free probability. For the free Appell
polynomials, a number of combinatorial and "diagram" formulas are proven, such
as the formulas for their linearization coefficients. An explicit formula for
their generating function is obtained. These polynomials are also martingales
for free Levy processes. For more general free Sheffer families, a necessary
condition for pseudo-orthogonality is given. Another family investigated are
the Kailath-Segall polynomials. These are multivariate polynomials, which share
with the Appell polynomials nice combinatorial properties, but are always
orthogonal. Their origins lie in the Fock space representations, or in the
theory of multiple stochastic integrals. Diagram formulas are proven for these
polynomials as well, even in the q-deformed case.Comment: 45 pages, 2 postscript figure
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