Abstract
We consider a class of pure jump Markov processes in R d whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.
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References
- 1.
Athreya, S., Barlow, M.T., Bass, R.F., and Perkins, E.A.: Degenerate stochastic differential equations and super-Markov chains, Probab. Theory Related Fields, to appear.
- 2.
Barlow, M.T., Bass, R.F., and Gui, C.: 'The Liouville property and a conjecture of De Georgi', Comm. Pure Appl. Math. 53 (2000), 1007-1038.
- 3.
Bass, R.F.: 'Uniqueness in law for pure jump Markov processes', Probab. Theory Related Fields 79 (1988), 271-287.
- 4.
Chen, Z.-Q. and Song, R.: 'Estimates on Green functions and Poisson kernels for symmetric stable processes', Math. Ann. 312 (1998), 465-501.
- 5.
Hoh, W.: 'Pseudodifferential operators with negative definite symbols and the martingale problem', Stochastics Stochastics Rep. 55 (1995), 225-252.
- 6.
Kaussmann, M.: Harnack-Ungleichungen für nichtlokale Differentialoperatoren und Dirichletformen, Ph.D. dissertation, Univ.-Bonn, 2001.
- 7.
Komatsu, T.: 'On the martingale problem for generators of stable processes with perturbations', Osaka J. Math. 21 (1984), 113-132.
- 8.
Tomisaki, M.: 'Some estimates for solutions of equations related to non-local Dirichlet forms', Rep. Fac. Sci. Engrg. Saga Univ. Math. 6 (1978), 1-7.
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Bass, R.F., Levin, D.A. Harnack Inequalities for Jump Processes. Potential Analysis 17, 375–388 (2002). https://doi.org/10.1023/A:1016378210944
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- Harnack inequality
- jump processes
- stable processes
- Lévy systems
- integral equations