System of Complex Brownian Motions Associated with the O'Connell Process

TL;DR

This paper studies a generalized Brownian motion process called the O'Connell process, providing a new representation using complex Brownian motions and connecting it to noncolliding Brownian motion in a limiting case.

Contribution

It rewrites the Fredholm determinant kernel for the O'Connell process into a form similar to determinantal processes and introduces a complex Brownian motion representation.

Findings

Reformulation of the kernel to a determinantal-like form

Representation of the process via complex Brownian motions

Recovery of noncolliding Brownian motion in the limit a→0

Abstract

The O'Connell process is a softened version (a geometric lifting with a parameter $a>0$) of the noncolliding Brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length $a$. This process is not determinantal. Under a special entrance law, however, Borodin and Corwin gave a Fredholm determinant expression for the expectation of an observable, which is a softening of an indicator of a particle position. We rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and show, if the number of particles is $N$, the rank of the matrix of the Fredholm determinant is $N$. Then we give a representation for the quantity by using an $N$-particle system of complex Brownian motions (CBMs). The complex function, which gives the determinantal expression to the weight of CBM paths, is not entire, but in the combinatorial limit $a \to 0$ it becomes an entire function providing conformal martingales and the CBM representation for the noncolliding Brownian motion is recovered.