Sample path properties of permanental processes

TL;DR

This paper investigates the sample path properties of alpha-permanental processes by relating them to subgaussian processes, enabling the derivation of continuity moduli and asymptotic behaviors with applications to processes with various kernels.

Contribution

The paper establishes that the square root of alpha-permanental processes are subgaussian, extending known sample path properties to a broader class of processes with diverse kernels.

Findings

Derived local and uniform moduli of continuity.

Analyzed behavior at infinity of permanental processes.

Provided examples with non-symmetric and modified kernels.

Abstract

Let $X_{\alpha}=\{X_{\alpha}(t),t\in T\}$, $\alpha>0$, be an $\alpha$-permanental process with kernel $u(s,t)$. We show that $X^{1/2}_{\alpha}$ is a subgaussian process with respect to the metric $\sigma (s,t)= (u(s,s)+u(t,t)-2(u(s,t)u(t,s))^{1/2})^{1/2}$. This allows us to use the vast literature on sample path properties of subgaussian processes to extend these properties to $\alpha$-permanental processes. Local and uniform moduli of continuity are obtained as well as the behavior of the processes at infinity. Examples are given of permanental processes with kernels that are the potential density of transient L\'evy processes that are not necessarily symmetric, or with kernels of the form $ \hat u(x,y)= u(x,y)+f(y)$, where $u$ is the potential density of a symmetric transient Borel right process and $f$ is an excessive function for the process.