Non-local dirichlet forms and symmetric jump processes

Transactions of the American Mathematical Society

Volumn 361, Issue 4, 2009, Pages 1963-1999

  Barlow, Martin T ^a   Bass, Richard F ^b   Chen, Zhen Qing ^c   Kassmann, Moritz ^d  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b UNIVERSITY OF CONNECTICUT   (United States)

^c UNIVERSITY OF WASHINGTON   (United States)

^d UNIVERSITY OF BONN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 62549135472     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-08-04544-3     Document Type: Article

References (43)

1. M. T. Barlow and R. F. Bass. Transition densities for Brownian motion on the Sierpiński carpet. Probab. Theory Rel. Fields, 91 (1992), 307-330. MR1151799 (93k:60203)
2. 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., to appear.
3. R. F. Bass. Uniqueness in law for pure jump processes. Probab. Theory Rel. Fields, 79 (1988), 271-287. MR958291 (89h:60118)
4. R. F. Bass. Probabilistic Techniques in Analysis. Probability and its Applications (New York). Springer-Verlag, New York, 1995. MR1329542 (96e:60001)
5. R. F. Bass. Diffusions and Elliptic Operators. Probability and its Applications (New York). Springer-Verlag, New York, 1997. MR1483890 (99h:60136)
6. R. F. Bass. On Aronson's upper bounds for heat kernels. Bull. London Math. Soc., 34 (2002), 415-419. MR1897420 (2003c:35054)
7. R. Bass and M. Kassmann. Harnack inequalities for non-local operators of variable order. Trans. Amer. Math. Soc., 357 (2005), 837-850. MR2095633 (2005i:60104) (Pubitemid 40251554)
8. R. Bass and M. Kassmann. Ḧolder continuity of harmonic functions with respect to operators of variable order. Comm. Part. Diff. Eq., 30 (2005), 1249-1259. MR2180302 (2006i:31005) (Pubitemid 41623978)
9. R. F. Bass and D. A. Levin. Harnack inequalities for jump processes. Potential Anal., 17 (2002), 375-388. MR1918242 (2003e:60194)
10. R. F. Bass and D. A. Levin. Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc., 354 (2002), 2933-2953. MR1895210 (2002m:60132)
11. R. M. Blumenthal and R. K. Getoor. Markov Processes and Potential Theory. Academic Press, New York, 1968. MR0264757 (41:9348)
12. E. A. Carlen, S. Kusuoka, and D. W. Stroock. Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaŕe Probab. Statist., 23(2, suppl.) (1987), 245-287. MR898496 (88i:35066)
13. Z.-Q. Chen. On reflected Dirichlet spaces. Probab. Theor. Rel. Fields, 94, 135-162, 1999. MR1191106 (93m:31014)
14. Z.-Q. Chen and T. Kumagai. Heat kernel estimates for stable-like processes on d-sets. Stoch. Proc. Applic., 108 (2003), 27-62. MR2008600 (2005d:60135) (Pubitemid 37155246)
15. Z.-Q. Chen and T. Kumagai. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Rel. Fields, 140 (2008), 277-317. MR2357678
16. Z.-Q. Chen and R. Song. Estimates on Green functions and Poisson kernels of symmetric stable processes. Math. Ann., 312 (1998), 465-501. MR1654824 (2000b:60179) (Pubitemid 128382021)
17. S. Chanillo and R. L. Wheeden. Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations. Comm. Part. Diff. Eq., 11 (1986), 1111-1134. MR847996 (87k:35097)
18. E. B. Davies. Explicit constants for Gaussian upper bounds on heat kernels. Amer. J. Math., 109 (1987), 319-333. MR882426 (88g:58174)
19. E. DeGiorgi. Sulla differenziabilit̀a e l'analiticit̀a delle estremali degli integrali multi-pli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3 (1957), 25-43. MR0093649 (20:172)
20. E. B. Fabes. Gaussian upper bounds on fundamental solutions of parabolic equations; the method of Nash. In Dirichlet forms (Varenna, 1992), volume 1563 of Lecture Notes in Math., pages 1-20. Springer, Berlin, 1993. MR1292275 (95j:35096)
21. E. B. Fabes and D. W. Stroock. A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash. Arch. Rat. Mech. Anal., 96 (1986), 327-338. MR855753 (88b:35037)
22. B. Fristedt. Sample functions of stochastic processes with stationary, independent increments. In: Advances in probability and related topics 3, pp. 241-396. Dekker, New York, 1974. MR0400406 (53:4240) ̄
23. M. Fukushima, Y. Oshima, and M. Takeda. Dirichlet Forms and Symmetric Markov Processes, volume 19 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994. MR1303354 (96f:60126)
24. C. E. Gutíerrez and R. L. Wheeden. Mean value and Harnack inequalities for degenerate parabolic equations. Colloq. Math., 60/61 (1990), 157-194. MR1096367 (92d:35025)
25. J. Hu and T. Kumagai. Nash-type inequalities and heat kernels for non-local Dirichlet forms. Kyushu J. Math 60 (2006) 245-265. MR2268236
26. R. Husseini and M. Kassmann. Jump processes, L-harmonic functions and continuity estimates. preprint; see http://www.iam.uni-bonn.de/~kassmann/ publications.html.
27. N. Ikeda, N. Nagasawa, and S. Watanabe. A construction of Markov processes by piecing out. Proc. Japan Acad., 42 (1966), 370-375. MR0202197 (34:2070)
28. V. Kolokoltsov. Symmetric stable laws and stable-like jump-diffusions. Proc. London Math Soc., 80 (2000), 725-768. MR1744782 (2001f:60016)
29. N. V. Krylov and M. V. Safonov. A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175 and 239. MR563790 (83c:35059)
30. T. Leviatan. Perturbations of Markov processes. J. Funct. Anal., 10 (1972), 309-325. MR0400409 (53:4243)
31. P.-A. Meyer. Renaissance, recollements, ḿelanges, ralentissement de processus de Markov. Ann. Inst. Fourier, 25 (1975), 464-497. MR0415784 (54:3862)
32. U. Mosco. Composite media and asymptotic Dirichlet forms. J. Funct. Anal., 123 (1994), 368-421. MR1283033 (95d:47088)
33. J. Moser. On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math., 14 (1961), 577-591. MR0159138 (28:2356)
34. J. Moser. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math., 17 (1964), 101-134. MR0159139 (28:2357)
35. J. Moser. On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math., 24 (1971), 727-740. MR0288405 (44:5603)
36. J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math., 80 (1958), 931-954. MR0100158 (20:6592)
37. L. Saloff-Coste and D. W. Stroock. Oṕerateurs uniforḿement sous-elliptiques sur les groupes de Lie. J. Funct. Anal., 98 (1991), 97-121. MR1111195 (92k:58264)
38. R.L. Schilling and T. Uemura. On the Feller property of Dirichlet forms generated by pseudo differential operators. Tohoku Math. J., (2) 59 (2007), 401-422. MR2365348
39. R. Schneider and C. Schwab. Wavelet solutions of variable order pseudodifferential equations. preprint.
40. M.L. Silverstein. Symmetric Markov Processes. Springer-Verlag, Berlin-New York, 1974. MR0386032 (52:6891)
41. R. Song and Z. Vondracek. Harnack inequality for some classes of Markov processes. Math. Z., 246 (2004), 177-202. MR2031452 (2004j:60158) (Pubitemid 38157183)
42. D. W. Stroock. Diffusion semigroups corresponding to uniformly elliptic divergence form operators. In S ́eminaire de Probabilit́es, XXII, volume 1321 of Lecture Notes in Math., pages 316-347. Springer, Berlin, 1988. MR960535 (90b:35071)
43. M. Takeda and K. Tsuchida. Differentiability of spectral functions for symmetric α-stable processes. Trans. Amer. Math. Soc., 359 (2007), 4031-4054. MR2302522 (2008a:35045)