On harmonic functions of symmetric Lévy processes

Mimica, Ante

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 214-235.

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.

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.

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

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