Jump processes, $L$-harmonic functions, continuity estimates and the Feller property

Husseini, Ryad; Kassmann, Moritz

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1099-1115.

Soit $\nu ={\left\{\nu (x,·)\right\}}_{x\in {ℝ}^d}$ une famille de mesures de Lévy, ce travail étudie la régularité de fonctions harmoniques et la propriété de Feller du processus de saut correspondant. Le but principal est d'établir des estimations de continuité pour les fonctions harmoniques sous des conditions faibles sur la famille $\nu$. À la différence des contributions précédentes cette méthode couvre des cas où les bornes inférieures de la probabilité d'atteindre de petits ensembles dégénèrent.

Given a family of Lévy measures $\nu ={\left\{\nu (x,·)\right\}}_{x\in {ℝ}^d}$, the present work deals with the regularity of harmonic functions and the Feller property of corresponding jump processes. The main aim is to establish continuity estimates for harmonic functions under weak assumptions on the family $\nu$. Different from previous contributions the method covers cases where lower bounds on the probability of hitting small sets degenerate.

[1] M. T. Barlow, R. F. Bass, Z.-Q. Chen and M. Kassmann. Non-local Dirichlet form and symmetric jump processes. Trans. Amer. Math. Soc. 361 (2009) 1963-1999. | MR | Zbl

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

[3] R. F. Bass and M. Kassmann. Hölder continuity of harmonic functions with respect to operators of variable orders. Comm. Partial Differential Equations 30 (2005) 1249-1259. | MR | Zbl

[4] R. F. Bass and D. A. Levin. Harnack inequalities for jump processes. Potential Anal. 17(4) (2002) 375-388. | MR | Zbl

[5] R. F. Bass and D. A. Levin. Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc. 354(7) (2002) 2933-2953. | MR | Zbl

[6] R. F. Bass, M. Kassmann and T. Kumagai. Symmetric jump processes: Localization, heat kernels, and convergence. Ann. Inst. H. Poincaré. To appear, 2009.

[7] Z.-Q. Chen. Symmetric jump processes and their heat kernel estimates. Sci. China Ser. A 52(7) (2009) 1423-1445. | MR

[8] Z.-Q. Chen and T. Kumagai. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl. 108(1) (2003) 27-62. | MR | Zbl

[9] E. De Giorgi. Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957) 25-43. | MR | Zbl

[10] S. N. Ethier and T. G. Kurtz. Markov Processes. Wiley, New York, 1986. | MR | Zbl

[11] M. Fukushima, Y. Ōshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin, 1994. | MR | Zbl

[12] W. Hoh. The martingale problem for a class of pseudo-differential operators. Math. Ann. 300(1) (1994) 121-147. | MR | Zbl

[13] W. Hoh. A symbolic calculus for pseudo-differential operators generating Feller semigroups. Osaka J. Math. 35(4) (1998) 789-820. | MR | Zbl

[14] R. Husseini and M. Kassmann. Markov chain approximations for symmetric jump processes. Potential Anal. 27(4) (2007) 353-380. | MR | Zbl

[15] N. Jacob. Feller semigroups, Dirichlet forms, and pseudodifferential operators. Forum Math. 4(5) (1992) 433-446. | MR | Zbl

[16] N. Jacob. A class of Feller semigroups generated by pseudo-differential operators. Math. Z. 215(1) (1994) 151-166. | MR | Zbl

[17] N. Jacob. Pseudo Differential Operators and Markov Processes. Vol. III. Imperial College Press, London, 2005. | MR | Zbl

[18] M. Kassmann. A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differential Equations 34(1) (2009) 1-21. | MR | Zbl

[19] T. Komatsu. Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms. Osaka J. Math. 25(3) (1988) 697-728. | MR | Zbl

[20] T. Komatsu. Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32(4) (1995) 833-860. | MR | Zbl

[21] N. V. Krylov and M. V. Safonov. An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk SSSR 245(1) (1979) 18-20. | MR | Zbl

[22] E. M. Landis. Second Order Equations of Elliptic and Parabolic Type. Amer. Math. Soc., Providence, RI, 1998. | MR | Zbl

[23] J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958) 931-954. | MR | Zbl

[24] K.-I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge, 1999. | MR | Zbl

[25] L. Silvestre. Hölder estimates for solutions of integro differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3) (2006) 1155-1174. | MR | Zbl

[26] R. Schilling and T. Uemura. Dirichlet forms generated by pseudo differential operators: On the Feller property of the associated stochastic process. Tohoku Math. J. 59 (2007) 401-422. | MR | Zbl

[27] R. Song and Z. Vondraček. Harnack inequality for some classes of Markov processes. Math. Z. 246(1, 2) (2004) 177-202. | MR | Zbl