Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

TL;DR

This paper investigates the moments of the maximum height of a system of two non-colliding Bessel bridges, expressing them through double Dirichlet series and Jacobi theta functions, with implications for vicious walker models.

Contribution

It introduces a novel expression for moments of non-colliding Bessel bridges using double Dirichlet series and theta functions, linking stochastic processes with special functions.

Findings

Moments of the maximum height are expressed via double Dirichlet series.

Asymptotic behavior of 2-watermelon height moments is determined.

Connections established between Bessel bridges, vicious walkers, and special functions.

Abstract

It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in long-time asymptotics of moments of height of 2-watermelons are completely determined. For the height of 2-watermelons with a wall, the average value was recently studied by Fulmek by a method of enumerative combinatorics.