Necessary and sufficient conditions for the continuity of permanental processes associated with transient Lévy processes

Abstract

Let $u^{\beta}(x,y)$ be the $\beta$-potential density of a transient Lévy process $\overline Y$ and $X_{\alpha}=\{X_{\alpha,x}, x\in R \}$ be the $\alpha$-permanental process determined by $u^{\beta}(x,y)$. Let $\overline L=\{\overline L_{t}^{x}, (t,x),\in R_{+}\times R \}$ be the local time process of $\overline Y$ and let $G=\{G_{x}, x\in R\}$ be the stationary mean zero Gaussian process with covariance $u^{\beta}(x,y)+ u^{\beta}(y,x)$. Then the processes $X_{\alpha}$, $\overline L$ and $G$ are either all continuous almost surely or all unbounded almost surely. Therefore, the well known necessary and sufficient condition for the continuity of $\overline L$ and $G$ is also a necessary and sufficient condition for the continuity of $X_{\alpha}$.