Electronic Communications in Probability

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

Michael Marcus and Jay Rosen

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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  $X_{\alpha}$.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 57, 6 pp.

Dates
Accepted: 4 August 2015
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320984

Digital Object Identifier
doi:10.1214/ECP.v20-4183

Mathematical Reviews number (MathSciNet)
MR3384115

Zentralblatt MATH identifier
1325.60072

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G99: None of the above, but in this section 60G17: Sample path properties

Keywords
permanental processes path properties Lévy processes

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Marcus, Michael; Rosen, Jay. Necessary and sufficient conditions for the continuity of permanental processes associated with transient Lévy processes. Electron. Commun. Probab. 20 (2015), paper no. 57, 6 pp. doi:10.1214/ECP.v20-4183. https://projecteuclid.org/euclid.ecp/1465320984


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References

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