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.
Mots clés : 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}, pages = {214--235}, publisher = {Gauthier-Villars}, volume = {50}, number = {1}, year = {2014}, doi = {10.1214/12-AIHP508}, mrnumber = {3161529}, zbl = {1298.60054}, language = {en}, url = {http://www.numdam.org/articles/10.1214/12-AIHP508/} }
TY - JOUR AU - Mimica, Ante TI - On harmonic functions of symmetric Lévy processes JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2014 SP - 214 EP - 235 VL - 50 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/12-AIHP508/ DO - 10.1214/12-AIHP508 LA - en ID - AIHPB_2014__50_1_214_0 ER -
Mimica, Ante. On harmonic functions of symmetric Lévy processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 214-235. doi : 10.1214/12-AIHP508. http://www.numdam.org/articles/10.1214/12-AIHP508/
[1] Heat kernel upper bounds for jump processes and the first exit time. J. Reine Angew. Math. 626 (2009) 135-157. | MR | Zbl
, and .[2] Harnack inequalities for non-local operators of variable order. Trans. Amer. Math. Soc. 357 (2005) 837-850. | MR | Zbl
and .[3] Harnack inequalities for jump processes. Potential Anal. 17 (2002) 375-388. | MR | Zbl
and .[4] Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl
.[5] Regular Variation. Cambridge Univ. Press, Cambridge, 1987. | MR | Zbl
, and .[6] Harnack's inequality for stable Lévy processes. Potential Anal. 22 (2005) 133-150. | MR | Zbl
and .[7] Heat kernel estimates for stable-like processes on -sets. Stochastic Process. Appl. 108 (2003) 27-62. | MR | Zbl
and .[8] Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields 140 (2008) 277-317. | MR | Zbl
and .[9] 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
and .[10] Analysis of jump processes with nondegenerate jumping kernels. Preprint, 2011. | MR | Zbl
and .[11] Potential theory of truncated stable processes. Math. Z. 256 (2007) 139-173. | MR | Zbl
and .[12] First passage distributions of processes with independent increments. Ann. Probab. 3 (1975) 215-233. | MR | Zbl
.[13] Harnack inequalities for some Lévy processes. Potential Anal. 32 (2010) 275-303. | MR | Zbl
.[14] 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] Heat kernel estimates for symmetric jump processes with small jumps of high intensity. Potential Anal. 36 (2012) 203-222. | MR | Zbl
.[16] Green function estimates and Harnack inequality for subordinate Brownian motions. Potential Anal. 25 (2006) 1-27. | MR | Zbl
, and .[17] Bernstein Functions: Theory and Applications. de Gruyter, Berlin, 2010. | MR | Zbl
, and .[18] Potential theory of geometric stable processes. Probab. Theory Related Fields 135 (2006) 547-575. | Zbl
, and .[19] Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55 (2006) 1155-1174. | MR | Zbl
.[20] Harnack inequalities for some classes of Markov processes. Math. Z. 246 (2004) 177-202. | MR | Zbl
and .[21] On harmonic measure for Lévy processes. Probab. Math. Statist. 20 (2000) 383-390. | MR | Zbl
.[22] Regularity of harmonic functions for anisotropic fractional Laplacians. Math. Nachr. 283 (2010) 289-311. | MR | Zbl
.Cité par Sources :