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Let $X_{\alpha}=\{X_{\alpha}(t),t\in T\}$, $\alpha>0$, be an $\alpha$-permanental process with kernel $u(s,t)$. We show that when $\alpha\le 1/2$, $X^{1/2}_{\alpha}$ is a subgaussian process with respect to to the metric \[ \sigma (s,t)= (u(s,s)+u(t,t)-2(u(s,t)u(t,s))^{1/2})^{1/2} .\] This is also the case when $\alpha>1/2$ and, in addition, \[ \inf_{ t\in T} u(t,t) >0\qquad\mbox{and} \qquad \ \sup_{ t\in T} u(t,t) <\infty\label{1.10kk}.\] 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, and with kernels of the form \[ v(x,y)= u(x,y)+f(y),\] where $u$ is the potential density of a symmetric transient Borel right process and $f$ is an excessive function for the process.

Sample path properties of permanental processes