Permanental processes with kernels that are not equivalent to a symmetric matrix

TL;DR

This paper investigates conditions under which certain non-symmetric kernels of alpha-permanental processes, constructed from symmetric kernels and excessive functions, can be symmetrized or asymptotically symmetrized.

Contribution

It provides new criteria to determine when these non-symmetric kernels are (asymptotically) symmetrizable, expanding understanding of permanental processes with non-equivalent kernels.

Findings

Conditions for symmetrizability of kernels are established.

Criteria for asymptotic symmetrizability are derived.

Results apply to kernels formed from symmetric potential densities and excessive functions.

Abstract

Kernels of $\alpha$-permanental processes of the form \[ v(x,y)=u(x,y)+f(y),\qquad x,y\in S, \] in which $u(x,y)$ is symmetric, and $f$ is an excessive function for the Borel right process with potential densities $u(x,y)$, are considered. Conditions are given that determine whether $\{v(x,y);x,y\in S\}$ is symmetrizable or asymptotically symmetrizable.