316 research outputs found
Wiener ( Ornstein-Uhlenbeck processes. A generalization of known processes
We collect, scattered through literature, as well as we prove some new
properties of two Markov processes that in many ways resemble Wiener and
Ornstein--Uhlenbeck processes. Although processes considered in this paper were
defined either in non-commutative probability context or through quadratic
harnesses we define them once more as so to say 'continuous time '
generalization of a simple, symmetric, discrete time process satisfying simple
conditions imposed on the form of its first two conditional moments. The finite
dimensional distributions of the first one (say X=(X_{t})_{t\geq0} called
q-Wiener) depends on one parameter q\in(-1,1] and of the second one (say
Y=(Y_{t})_{t\inR} called ({\alpha},q)- Ornstein--Uhlenbeck) on two parameters
({\alpha},q)\in(0,\infty)\times(-1,1]. The first one resembles Wiener process
in the sense that for q=1 it is Wiener process but also that for |q|<1 and
\foralln\geq1: t^{n/2}H_{n}(X_{t}/\surdt|q), where (H_{n})_{n\geq0} are the so
called q-Hermite polynomials, are martingales. It does not have however neither
independent increments not allows continuous sample path modification. The
second one resembles Ornstein--Uhlenbeck process. For q=1 it is a classical OU
process. For |q|<1 it is also stationary with correlation function equal to
exp(-{\alpha}|t-s|) and has many properties resembling those of its classical
version. We think that these process are fascinating objects to study posing
many interesting, open questions.Comment: 25 page
Fock space associated to Coxeter group of type B
In this article we construct a generalized Gaussian process coming from
Coxeter groups of type B. It is given by creation and annihilation operators on
an -Fock space, which satisfy the commutation relation where are
elements of a complex Hilbert space with a self-adjoint involution
and is the number operator with respect to the grading on
the -Fock space. We give an estimate of the norms of creation
operators. We show that the distribution of the operators
with respect to the vacuum expectation
becomes a generalized Gaussian distribution, in the sense that all mixed
moments can be calculated from the second moments with the help of a
combinatorial formula related with set partitions. Our generalized Gaussian
distribution associates the orthogonal polynomials called the
-Meixner-Pollaczek polynomials, yielding the -Hermite polynomials when
and free Meixner polynomials when .Comment: 22 pages, 6 figure
Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
Using the determinantal formula of Biane, Bougerol, and O'Connell, we give
multitime joint probability densities to the noncolliding Brownian motion with
drift, where the number of particles is finite. We study a special case such
that the initial positions of particles are equidistant with a period and
the values of drift coefficients are well-ordered with a scale . We
show that, at each time , the single-time probability density of particle
system is exactly transformed to the biorthogonal Stieltjes-Wigert matrix model
in the Chern-Simons theory introduced by Dolivet and Tierz. Here one-parameter
extensions (-extensions) of the Stieltjes-Wigert polynomials, which are
themselves -extensions of the Hermite polynomials, play an essential role.
The two parameters and of the process combined with time are
mapped to the parameters and of the biorthogonal polynomials. By
the transformation of normalization factor of our probability density, the
partition function of the Chern-Simons matrix model is readily obtained. We
study the determinantal structure of the matrix model and prove that, at each
time , the present noncolliding Brownian motion with drift is a
determinantal point process, in the sense that any correlation function is
given by a determinant governed by a single integral kernel called the
correlation kernel. Using the obtained correlation kernel, we study time
evolution of the noncolliding Brownian motion with drift.Comment: v2: REVTeX4, 34 pages, 4 figures, minor corrections made for
publication in J. Math. Phy
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