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Intersection local times for interlacements. (English) Zbl 1307.60104
Summary: We define renormalized intersection local times for random interlacements of Lévy processes in \(\mathbb R^d\) and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials.

MSC:
60J55 Local time and additive functionals
60G51 Processes with independent increments; Lévy processes
60J40 Right processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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[1] Arcones, M.; Giné, E., On decoupling, series expansion, and tail behavior of chaos processes, J. Theoret. Probab., 6, 101-122, (1993) · Zbl 0785.60023
[2] Bass, R. F.; Khoshnevisan, D., Intersection local times and Tanaka formulas, Ann. Inst. H. Poincaré, 29, 419-451, (1993) · Zbl 0798.60072
[3] Bass, R. F.; Rosen, J., An almost sure invariance principle for the range of planar random walks, Ann. Probab., 33, 1856-1887, (2005) · Zbl 1085.60018
[4] Dellacherie, C.; Maisonneuve, B.; Meyer, P.-A., Probabilities et potential, (1992), Hermann Paris, Chapitres XVII a XXIV
[5] Dynkin, E. B., Regularized self-intersection local times of planar Brownian motion, Ann. Probab., 16, 58-74, (1988) · Zbl 0641.60085
[6] Eisenbaum, N.; Kaspi, H.; Marcus, M.; Rosen, J.; Shi, Zhan, A ray-knight theorem for symmetric Markov processes, Ann. Probab., 28, 1781-1796, (2000) · Zbl 1044.60064
[7] Gradshteyn, I.; Ryzhik, I., Table of integrals, series and products, (1980), Academic Press Oxford · Zbl 0521.33001
[8] Kingman, J. F.C., (Poisson Processes, Oxford Studies in Probability, (2002), Clarendon Press Oxford)
[9] Le Gall, J.-F., Propriétés d’intersection des marches aléatoires, I. convergence vers le temps local d’intersection, Comm. Math. Phys., 104, 471-507, (1986) · Zbl 0609.60078
[10] Le Gall, J.-F., Wiener sausage and self intersection local times, J. Funct. Anal., 88, 299-341, (1990) · Zbl 0697.60081
[11] Le Jan, Y.; Marcus, M. B.; Rosen, J., Intersection local times, loop soups and permanental Wick powers · Zbl 1368.60080
[12] Marcus, M. B.; Rosen, J., Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes, Ann. Probab., 24, 1130-1177, (1996) · Zbl 0862.60065
[13] Marcus, M. B.; Rosen, J., Renormalized self-intersection local times and Wick power chaos processes, Mem. Amer. Math. Soc., 675, (1999) · Zbl 1230.60005
[14] Marcus, M. B.; Rosen, J., Markov processes, Gaussian processes and local times, (2006), Cambridge University Press New York · Zbl 1129.60002
[15] Rosen, J., Joint continuity of renormalized intersection local times, Ann. Inst. H. Poincaré, 32, 671-700, (1996) · Zbl 0867.60049
[16] Rosen, J., Joint continuity and a Doob-Meyer type decomposition for renormalized intersection local times, Ann. Inst. H. Poincaré, 35, 143-176, (1999) · Zbl 0922.60072
[17] Sznitman, A.-S., An isomorphism theorem for random interlacements, Electron. Comm. Probab., 17, 9, 1-9, (2012) · Zbl 1247.60135
[18] Sznitman, A.-S., Topics in occupation times and Gaussian free fields, (Zurich Lectures in Advanced Mathematics, (2012), EMS Zurich) · Zbl 1246.60003
[19] A.-S. Sznitman, On scaling limits and Brownian interlacements. · Zbl 1303.60022
[20] Varadhan, S. R.S., Appendix to Euclidean quantum field theory by K. symanzyk, (Jost, R., Local Quantum Theory, (1969), Academic Press Reading, MA)
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