959 research outputs found
Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions
This paper reviews known results which connect Riemann's integral
representations of his zeta function, involving Jacobi's theta function and its
derivatives, to some particular probability laws governing sums of independent
exponential variables. These laws are related to one-dimensional Brownian
motion and to higher dimensional Bessel processes. We present some
characterizations of these probability laws, and some approximations of
Riemann's zeta function which are related to these laws.Comment: LaTeX; 40 pages; review pape
Time-fractional equations with reaction terms: fundamental solutions and asymptotics
We analyze the fundamental solution of a time-fractional problem,
establishing existence and uniqueness in an appropriate functional space.
We also focus on the one-dimensional spatial setting in the case in which the
time-fractional exponent is equal to, or larger than, . In this
situation, we prove that the speed of invasion of the fundamental solution is
at least `almost of square root type', namely it is larger than~ for
any given~ and~
Quantile clocks
Quantile clocks are defined as convolutions of subordinators , with
quantile functions of positive random variables. We show that quantile clocks
can be chosen to be strictly increasing and continuous and discuss their
practical modeling advantages as business activity times in models for asset
prices. We show that the marginal distributions of a quantile clock, at each
fixed time, equate with the marginal distribution of a single subordinator.
Moreover, we show that there are many quantile clocks where one can specify
, such that their marginal distributions have a desired law in the class of
generalized -self decomposable distributions, and in particular the class of
self-decomposable distributions. The development of these results involves
elements of distribution theory for specific classes of infinitely divisible
random variables and also decompositions of a gamma subordinator, that is of
independent interest. As applications, we construct many price models that have
continuous trajectories, exhibit volatility clustering and have marginal
distributions that are equivalent to those of quite general exponential
L\'{e}vy price models. In particular, we provide explicit details for
continuous processes whose marginals equate with the popular VG, CGMY and NIG
price models. We also show how to perfectly sample the marginal distributions
of more general classes of convoluted subordinators when is in a sub-class
of generalized gamma convolutions, which is relevant for pricing of European
style options.Comment: Published in at http://dx.doi.org/10.1214/10-AAP752 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Period functions for Maass wave forms. I
Recall that a Maass wave form on the full modular group Gamma=PSL(2,Z) is a
smooth gamma-invariant function u from the upper half-plane H = {x+iy | y>0} to
C which is small as y \to \infty and satisfies Delta u = lambda u for some
lambda \in C, where Delta = y^2(d^2/dx^2 + d^2/dy^2) is the hyperbolic
Laplacian. These functions give a basis for L_2 on the modular surface Gamma\H,
with the usual trigonometric waveforms on the torus R^2/Z^2, which are also
(for this surface) both the Fourier building blocks for L_2 and eigenfunctions
of the Laplacian. Although therefore very basic objects, Maass forms
nevertheless still remain mysteriously elusive fifty years after their
discovery; in particular, no explicit construction exists for any of these
functions for the full modular group. The basic information about them (e.g.
their existence and the density of the eigenvalues) comes mostly from the
Selberg trace formula: the rest is conjectural with support from extensive
numerical computations.Comment: 68 pages, published versio
Adiabatic limits of eta and zeta functions of elliptic operators
We extend the calculus of adiabatic pseudo-differential operators to study
the adiabatic limit behavior of the eta and zeta functions of a differential
operator , constructed from an elliptic family of operators indexed by
. We show that the regularized values and
are smooth functions of at , and we identify
their values at with the holonomy of the determinant bundle, respectively
with a residue trace. For invertible families of operators, the functions
and are shown to extend smoothly to
for all values of . After normalizing with a Gamma factor, the zeta
function satisfies in the adiabatic limit an identity reminiscent of the
Riemann zeta function, while the eta function converges to the volume of the
Bismut-Freed meromorphic family of connection 1-forms.Comment: 32 pages, final versio
Clifford algebras, Fourier transforms and quantum mechanics
In this review, an overview is given of several recent generalizations of the
Fourier transform, related to either the Lie algebra sl_2 or the Lie
superalgebra osp(1|2). In the former case, one obtains scalar generalizations
of the Fourier transform, including the fractional Fourier transform, the Dunkl
transform, the radially deformed Fourier transform and the super Fourier
transform. In the latter case, one has to use the framework of Clifford
analysis and arrives at the Clifford-Fourier transform and the radially
deformed hypercomplex Fourier transform. A detailed exposition of all these
transforms is given, with emphasis on aspects such as eigenfunctions and
spectrum of the transform, characterization of the integral kernel and
connection with various special functions.Comment: Review paper, 39 pages, to appear in Math. Methods. Appl. Sc
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