Given the maximum process (St) = (max 0≤r≤t Xr) associated with a diffusion ((Xt),Px), and a continuous function g satisfying g(s) < s, we show how to compute the expectation of the Azéma-Yor stopping time τg = inf{ t > 0 : Xt ≤ g(St) } as a function of x. The method of proof is based upon verifying that the expectation solves a differential equation with two boundary conditions. The third ‘mis...

We study theoretically and numerically third-harmonic generation (THG) from model geometries (interfaces, slabs, periodic media) illuminated by Bessel beams produced by focusing an annular intensity profile. Bessel beams exhibit a phase and intensity distribution near focus different from Gaussian beams, resulting in distinct THG phase matching properties and coherent scattering directions. Exc...

We present a method for constructing new families of solvable one-dimensional diffusions with linear drift and nonlinear diffusion coefficient functions, whose transition densities are obtainable in analytically closed-form. Our approach is based on the so-called diffusion canonical transformation method that allows us to uncover new multiparameter diffusions that are mapped onto various simple...

Non-diffracting Bessel beams, including zero-order and high-order Bessel Beams which carry orbital angular momentum (OAM), enable a variety of important applications in optical micromanipulation, sub-diffraction imaging, high speed photonics/quantum communication, etc. The commonly used ways to create Bessel beams, including an axicon or a digital hologram written to a spatial light modulator (...

It is shown that the tail probability of a Bessel process is the distributio function of a random time which is related to first and last passage times of Bessel processes. AMS Classification 60 J 25, 60 J 60, 60 E 15, 60 G 44, 60 H 10.

Delest, M. In this paper, we show that the generating function for skew Ferrers diagrams according to their width and area is the quotient of new basic Bessel functions. Nous montrons dans cet article que la fonction g&ntratrice des diagrammes de Ferrers gauches selon les paramktres ptrimttre et aire s'exprime en fonction du quotient des q analogues de deux fonctions de Bessel.

We demonstrate the existence of vector Helmholtz-Gauss (vHzG) and vector Laplace-Gauss beams that constitute two general families of localized vector beam solutions of the Maxwell equations in the paraxial approximation. The electromagnetic components are determined starting from the scalar solutions of the two-dimensional Helmholtz and Laplace equations, respectively. Special cases of the vHzG...

In a previous study, we discussed the expansion of two-parameter functions in a double series of Chebyshev polynomials, and, in particular, we presented coefficients for the evaluation of the modified Bessel function (2z I iifne'K,(z) to 20 decimals for all z ^ 5 and all v, 0 g v g 1. In the present study, we give similar coefficients for the evaluation of ge~'z-»l,(z) to at least 20 decimals w...

We consider the exact path sampling of the squared Bessel process and some other continuous-time Markov processes, such as the CIR model, constant elasticity of variance diffusion model, and hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale transformation, and/or change of measure. All these diffusions are broadly us...