On harmonic functions of symmetric Lévy processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, p. 214-235

We consider some classes of Lévy processes for which the estimate of Krylov and Safonov (as in (Potential Anal. 17 (2002) 375-388)) fails and thus it is not possible to use the standard iteration technique to obtain a-priori Hölder continuity estimates of harmonic functions. Despite the failure of this method, we obtain some a-priori regularity estimates of harmonic functions for these processes. Moreover, we extend results from (Probab. Theory Related Fields 135 (2006) 547-575) and obtain asymptotic behavior of the Green function and the Lévy density for a large class of subordinate Brownian motions, where the Laplace exponent of the corresponding subordinator is a slowly varying function.

On considère des classes de processus de Lévy pour lesquels les estimations de Krylov et Safonov (comme dans (Potential Anal. 17 (2002) 375-388)) ne sont pas verifiées donc il n'est pas possible d'utiliser la technique standard d'itération pour obtenir a priori des estimations de continuité Hölder pour des fonctions harmoniques. Bien qu'il soit impossible d'appliquer cette méthode, on obtient des estimations a priori de régularité de fonctions harmoniques pour ces processus. De plus, on étend les résultats de (Probab. Theory Related Fields 135 (2006) 547-575) et on obtient les comportements asymptotiques de la fonction de Green et de la densité de Lévy pour une grande classe de mouvements browniens subordonnés, où l'exposant de Laplace du subordinateur correspondant est une fonction à variation lente.

DOI : https://doi.org/10.1214/12-AIHP508
Classification:  60J45,  60J75,  60J25
Keywords: geometric stable process, Green function, harmonic function, Lévy process, modulus of continuity, subordinator, subordinate brownian motion
@article{AIHPB_2014__50_1_214_0,
     author = {Mimica, Ante},
     title = {On harmonic functions of symmetric L\'evy processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     pages = {214-235},
     doi = {10.1214/12-AIHP508},
     zbl = {1298.60054},
     mrnumber = {3161529},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_1_214_0}
}
Mimica, Ante. On harmonic functions of symmetric Lévy processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 214-235. doi : 10.1214/12-AIHP508. http://www.numdam.org/item/AIHPB_2014__50_1_214_0/

[1] M. T. Barlow, A. Grigor'Yan and T. Kumagai. Heat kernel upper bounds for jump processes and the first exit time. J. Reine Angew. Math. 626 (2009) 135-157. | MR 2492992 | Zbl 1158.60039

[2] R. F. Bass and M. Kassmann. Harnack inequalities for non-local operators of variable order. Trans. Amer. Math. Soc. 357 (2005) 837-850. | MR 2095633 | Zbl 1052.60060

[3] R. F. Bass and D. Levin. Harnack inequalities for jump processes. Potential Anal. 17 (2002) 375-388. | MR 1918242 | Zbl 0997.60089

[4] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR 1406564 | Zbl 0938.60005

[5] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1987. | MR 898871 | Zbl 0667.26003

[6] K. Bogdan and P. Sztonyk. Harnack's inequality for stable Lévy processes. Potential Anal. 22 (2005) 133-150. | MR 2137058 | Zbl 1081.60055

[7] Z.-Q. Chen and T. Kumagai. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl. 108 (2003) 27-62. | MR 2008600 | Zbl 1075.60556

[8] Z.-Q. Chen and T. Kumagai. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields 140 (2008) 277-317. | MR 2357678 | Zbl 1131.60076

[9] N. Ikeda and S. Watanabe. On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2 (1962) 79-95. | MR 142153 | Zbl 0118.13401

[10] M. Kassmann and A. Mimica. Analysis of jump processes with nondegenerate jumping kernels. Preprint, 2011. | MR 3003366 | Zbl 1259.60100

[11] P. Kim and R. Song. Potential theory of truncated stable processes. Math. Z. 256 (2007) 139-173. | MR 2282263 | Zbl 1115.60073

[12] P. W. Millar. First passage distributions of processes with independent increments. Ann. Probab. 3 (1975) 215-233. | MR 368177 | Zbl 0318.60063

[13] A. Mimica. Harnack inequalities for some Lévy processes. Potential Anal. 32 (2010) 275-303. | MR 2595368 | Zbl 1202.60126

[14] A. Mimica. Harnack inequality and Hölder regularity estimates for a Lévy process with small jumps of high intensity. J. Theoret. Probab. 26(2) (2013) 329-348. | MR 3055806 | Zbl 1279.60100

[15] A. Mimica. Heat kernel estimates for symmetric jump processes with small jumps of high intensity. Potential Anal. 36 (2012) 203-222. | MR 2886459 | Zbl 1239.60077

[16] M. Rao, R. Song and Z. Vondraček. Green function estimates and Harnack inequality for subordinate Brownian motions. Potential Anal. 25 (2006) 1-27. | MR 2238934 | Zbl 1107.60042

[17] R. L. Schilling, R. Song and Z. Vondraček. Bernstein Functions: Theory and Applications. de Gruyter, Berlin, 2010. | MR 2598208 | Zbl 1257.33001

[18] H. Šikić, R. Song and Z. Vondraček. Potential theory of geometric stable processes. Probab. Theory Related Fields 135 (2006) 547-575. | Zbl 1099.60051

[19] L. Silvestre. Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55 (2006) 1155-1174. | MR 2244602 | Zbl 1101.45004

[20] R. Song and Z. Vondraček. Harnack inequalities for some classes of Markov processes. Math. Z. 246 (2004) 177-202. | MR 2031452 | Zbl 1052.60064

[21] P. Sztonyk. On harmonic measure for Lévy processes. Probab. Math. Statist. 20 (2000) 383-390. | MR 1825650 | Zbl 0991.60067

[22] P. Sztonyk. Regularity of harmonic functions for anisotropic fractional Laplacians. Math. Nachr. 283 (2010) 289-311. | MR 2604123 | Zbl 1194.47044