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, $\frac12$. 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~$ct^\beta$ for any given~$c>0$ and~$\beta\in\left(0,\frac12\right)$

Quantile clocks

Quantile clocks are defined as convolutions of subordinators $L$, 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 $L$, such that their marginal distributions have a desired law in the class of generalized $s$-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 $L$ 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 $\delta$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values ${\eta}(\delta_t,0)$ and $t{\zeta}(\delta_t,0)$ are smooth functions of $t$ at $t=0$, and we identify their values at $t=0$ with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions ${\eta}(\delta_t,s)$ and $t{\zeta}(\delta_t,s)$ are shown to extend smoothly to $t=0$ for all values of $s$. 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