The logarithmic derivative for point processes with equivalent Palm measures

TL;DR

This paper explicitly computes the logarithmic derivative for a broad class of determinantal point processes on the real line, including those from random matrix theory, using their quasi-invariance properties.

Contribution

It provides a new explicit formula for the logarithmic derivative of determinantal processes with integrable kernels, extending the understanding of their structure.

Findings

Explicit formulas for logarithmic derivatives of determinantal processes.

Includes classical random matrix processes and de Branges space processes.

Utilizes quasi-invariance to derive the results.

Abstract

The logarithmic derivative of a point process plays a key role in the general approach, due to the third author, to constructing diffusions preserving a given point process. In this paper we explicitly compute the logarithmic derivative for determinantal processes on $\mathbb{R}$ with integrable kernels, a large class that includes all the classical processes of random matrix theory as well as processes associated with de Branges spaces. The argument uses the quasi-invariance of our processes established by the first author.