Determinantal process starting from an orthogonal symmetry is a Pfaffian process

TL;DR

This paper demonstrates that noncolliding Brownian motion and squared Bessel processes with orthogonal symmetric initial configurations are Pfaffian processes, providing explicit correlation kernels and extending the class of known Pfaffian point processes.

Contribution

It proves that certain noncolliding stochastic processes with orthogonal symmetry are Pfaffian, with explicit kernels, expanding the understanding of their correlation structures.

Findings

Noncolliding BM and BESQ processes with orthogonal symmetry are Pfaffian.

Explicit 2x2 skew-symmetric correlation kernels are derived.

Connections between noncolliding processes and dilatations of time-reversed processes are established.

Abstract

When the number of particles $N$ is finite, the noncolliding Brownian motion (BM) and the noncolliding squared Bessel process with index $\nu > -1$ (BESQ$^{(\nu)}$) are determinantal processes for arbitrary fixed initial configurations. In the present paper we prove that, if initial configurations are distributed with orthogonal symmetry, they are Pfaffian processes in the sense that any multitime correlation functions are expressed by Pfaffians. The $2 \times 2$ skew-symmetric matrix-valued correlation kernels of the Pfaffians processes are explicitly obtained by the equivalence between the noncolliding BM and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding BM with finite duration in which all particles start from the origin, $N \delta_0$, and by the equivalence between the noncolliding BESQ$^{(\nu)}$ and that of the noncolliding squared generalized meander starting from $N \delta_0$.