Reciprocal Time Relation of Noncolliding Brownian Motion with Drift

TL;DR

This paper establishes a reciprocal time relation for noncolliding Brownian motion with drift, linking processes with and without drift through a dilatation, and explores their determinantal properties for finite and infinite particles.

Contribution

It proves a novel reciprocal time relation connecting drifted and non-drifted noncolliding Brownian motions, extending understanding of their determinantal structures.

Findings

Proved equivalence between dilated drifted process and non-drifted process at reciprocal time.

Analyzed determinantal properties for finite and infinite particle systems.

Extended the theory to include processes starting from degenerate initial conditions.

Abstract

We consider an $N$-particle system of noncolliding Brownian motion starting from $x_1 \leq x_2 \leq ... \leq x_N$ with drift coefficients $\nu_j, 1 \leq j \leq N$ satisfying $\nu_1 \leq \nu_2 \leq ... \leq \nu_N$. When all of the initial points are degenerated to be zero, $x_j=0, 1 \leq j \leq N$, the equivalence is proved between a dilatation with factor $1/t$ of this drifted process and the noncolliding Brownian motion starting from $\nu_1 \leq \nu_2 \leq ... \leq \nu_N$ without drift observed at reciprocal time $1/t$, for arbitrary $t > 0$. Using this reciprocal time relation, we study the determinantal property of the noncolliding Brownian motion with drift having finite and infinite numbers of particles.