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
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