## The Annals of Probability

### A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes

#### Abstract

We provide a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the sufficient condition for continuity which is also known to be necessary. Using an isomorphism theorem of Eisenbaum and Kaspi which relates Markov local times and permanental processes, we obtain a general sufficient condition for the joint continuity of local times.

#### Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 671-698.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750938

Digital Object Identifier
doi:10.1214/12-AOP744

Mathematical Reviews number (MathSciNet)
MR3077522

Zentralblatt MATH identifier
1329.60273

#### Citation

Marcus, Michael B.; Rosen, Jay. A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes. Ann. Probab. 41 (2013), no. 2, 671--698. doi:10.1214/12-AOP744. https://projecteuclid.org/euclid.aop/1362750938

#### References

• [1] Barlow, M. T. (1988). Necessary and sufficient conditions for the continuity of local time of Lévy processes. Ann. Probab. 16 1389–1427.
• [2] Cohn, D. L. (1972). Measurable choice of limit points and the existence of separable and measurable processes. Z. Wahrsch. Verw. Gebiete 22 161–165.
• [3] Dellacherie, C. and Meyer, P. A. (1978). Probabilities et Potential. Hermann, Paris.
• [4] Dynkin, E. B. (1984). Local times and quantum fields. In Seminar on Stochastic Processes, 1983 (Gainesville, Fla., 1983). Progr. Probab. Statist. 7 69–83. Birkhäuser, Boston, MA.
• [5] Dynkin, E. B. (1984). Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55 344–376.
• [6] Eisenbaum, N. and Kaspi, H. (2007). On the continuity of local times of Borel right Markov processes. Ann. Probab. 35 915–934.
• [7] Eisenbaum, N. and Kaspi, H. (2009). On permanental processes. Stochastic Process. Appl. 119 1401–1415.
• [8] Fernique, X. (1997). Fonctions Aléatoires Gaussiennes, Vecteurs Aléatoires Gaussiens. Univ. Montréal Centre de Recherches Mathématiques, Montreal, QC.
• [9] Garsia, A. M., Rodemich, E. and Rumsey, H. Jr. (1970/1971). A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 565–578.
• [10] Heinkel, B. (1977). Mesures majorantes et théorème de la limite centrale dans $C(S)$. Z. Wahrsch. Verw. Gebiete 38 339–351.
• [11] Kwapień, S. and Rosiński, J. (2004). Sample Hölder continuity of stochastic processes and majorizing measures. In Seminar on Stochastic Analysis, Random Fields and Applications IV. Progress in Probability 58 155–163. Birkhäuser, Basel.
• [12] Loewy, R. (1986). Principal minors and diagonal similarity of matrices. Linear Algebra Appl. 78 23–64.
• [13] Marcus, M. B. and Rosen, J. (1992). Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Probab. 20 1603–1684.
• [14] Marcus, M. B. and Rosen, J. (2006). Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge Univ. Press, Cambridge.
• [15] Marcus, M. B. and Rosen, J. (2009). An almost sure limit theorem for Wick powers of Gaussian differences quotients. In High Dimensional Probability V: The Luminy Volume. Inst. Math. Stat. Collect. 5 258–272. IMS, Beachwood, OH.
• [16] Preston, C. (1970/1971). Banach spaces arising from some integral inequalities. Indiana Univ. Math. J. 20 997–1015.
• [17] Preston, C. (1972). Continuity properties of some Gaussian processes. Ann. Math. Statist. 43 285–292.
• [18] Vere-Jones, D. (1997). Alpha-permanents and their applications to multivariate gamma, negative binomial and ordinary binomial distributions. New Zealand J. Math. 26 125–149.