Sample path properties of permanental processes

Abstract

Let $X_{\alpha }=\{X_{\alpha }(t),t\in{\cal T} \}$, $\alpha >0$, 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 \[ \sigma (s,t)= (u(s,s)+u(t,t)-2(u(s,t)u(t,s))^{1/2})^{1/2} .\nonumber \] 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 Lévy processes that are not necessarily symmetric, or with kernels of the form \[ \widetilde{u} (x,y)= u(x,y)+f(y),\nonumber \] where $u$ is the potential density of a symmetric transient Borel right process and $f$ is an excessive function for the process.