Approximation of jump processes on fractals. (English) Zbl 1171.60019
The paper is devoted to the approximations for jump processes on \(d\)-sets, in particular, on self-similar sets.
Starting with the set of Dirichlet forms on approximating \(\varepsilon\)-sets, it is shown that there exists the Mosco–limit Dirichlet form (in the sense of K. Kuwae and T. Shioya [Commun. Anal. Geom. 11, No. 4, 599–673 (2003; Zbl 1092.53026)]). Then the strong convergence (in the sense of Kuwae-Shioya) of the resolvents and the operator semigroups naturally follows by the techniques from [Kuwae and Shioya, loc. cit.] and [U. Mosco, J. Funct. Anal. 123, No. 2, 368–421 (1994; Zbl 0808.46042)].
In the case of a self-similar set, it is proved that the approximate and limit forms satisfy the Nash’s inequality, which implies the existence of transition densities for both the approximating and the limit processes. To prove the tightness the estimate on the first exit time from a ball of certain radius is proved, which together with D. Aldous’ Theorem [Ann. Probab. 6, 335–340 (1978; Zbl 0391.60007)] implies the weak convergence of the related processes in Skorokhod space.
Starting with the set of Dirichlet forms on approximating \(\varepsilon\)-sets, it is shown that there exists the Mosco–limit Dirichlet form (in the sense of K. Kuwae and T. Shioya [Commun. Anal. Geom. 11, No. 4, 599–673 (2003; Zbl 1092.53026)]). Then the strong convergence (in the sense of Kuwae-Shioya) of the resolvents and the operator semigroups naturally follows by the techniques from [Kuwae and Shioya, loc. cit.] and [U. Mosco, J. Funct. Anal. 123, No. 2, 368–421 (1994; Zbl 0808.46042)].
In the case of a self-similar set, it is proved that the approximate and limit forms satisfy the Nash’s inequality, which implies the existence of transition densities for both the approximating and the limit processes. To prove the tightness the estimate on the first exit time from a ball of certain radius is proved, which together with D. Aldous’ Theorem [Ann. Probab. 6, 335–340 (1978; Zbl 0391.60007)] implies the weak convergence of the related processes in Skorokhod space.
Reviewer: Victoria Knopova (Kiev)
MSC:
| 60J75 | Jump processes (MSC2010) |
| 28A80 | Fractals |
| 60J35 | Transition functions, generators and resolvents |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Mosco convergence; fractal sets; self-similar sets; Dirichlet forms; jump processes on fractal setsReferences:
| [1] | D. Aldous: Stopping times and tightness , Ann. Probability 6 (1978), 335–340. · Zbl 0391.60007 · doi:10.1214/aop/1176995579 |
| [2] | M.T. Barlow: Diffusions on fractals ; in Lectures on Probability Theory and Statistics (Saint-Flour, 1995), Lecture Notes in Math. 1690 , Springer, Berlin, 1998, 1–121. · Zbl 0916.60069 · doi:10.1007/BFb0092537 |
| [3] | M.T. Barlow, R.F. Bass, Z.-Q. Chen and M. Kassmann: Non-local Dirichlet forms and symmetric jump processes , Trans. Amer. Math. Soc. 361 (2009), 1963–1999. · Zbl 1166.60045 · doi:10.1090/S0002-9947-08-04544-3 |
| [4] | R.F. Bass and T. Kumagai: Symmetric Markov chains on \(\mathbb{Z}^{d}\) with unbounded range , Trans. Amer. Math. Soc. 360 (2008), 2041–2075. · Zbl 1130.60076 · doi:10.1090/S0002-9947-07-04281-X |
| [5] | R.F. Bass and D.A. Levin: Transition probabilities for symmetric jump processes , Trans. Amer. Math. Soc. 354 (2002), 2933–2953 JSTOR: · Zbl 0993.60070 · doi:10.1090/S0002-9947-02-02998-7 |
| [6] | P. Billingsley: Convergence of Probability Measures, second edition, Wiley, New York, 1999. · Zbl 0944.60003 |
| [7] | E.A. Carlen, S. Kusuoka and D.W. Stroock: Upper bounds for symmetric Markov transition functions , Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), 245–287. · Zbl 0634.60066 |
| [8] | Z.-Q. Chen and T. Kumagai: Heat kernel estimates for stable-like processes on \(d\)-sets , Stochastic Process. Appl. 108 (2003), 27–62. · Zbl 1075.60556 · doi:10.1016/S0304-4149(03)00105-4 |
| [9] | 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. · Zbl 1131.60076 · doi:10.1007/s00440-007-0070-5 |
| [10] | P. Diaconis and L. Saloff-Coste: Nash inequalities for finite Markov chains , J. Theoret. Probab. 9 (1996), 459–510. · Zbl 0870.60064 · doi:10.1007/BF02214660 |
| [11] | S.N. Ethier and T.G. Kurtz: Markov Processes, Wiley, New York, 1986. |
| [12] | K. Falconer: Fractal Geometry, Wiley, Chichester, 1990. |
| [13] | M. Fukushima, Y. Ōshima and M. Takeda: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, Berlin, 1994. · Zbl 0838.31001 |
| [14] | M. Hinz: Heat kernel estimates for stable like processes on \(h\)-sets , preprint, Univ. Jena, 2006. |
| [15] | W. Hansen and M. Zähle: Restricting isotropic \(\alpha\)-stable Lévy processes from \(\mathbb{R}^{n}\) to fractal sets , Forum Math. 18 (2006), 171–191. · Zbl 1106.60063 · doi:10.1515/FORUM.2006.011 |
| [16] | R. Husseini and M. Kassmann: Markov chain approximations for symmetric jump processes , Potential Analysis 27 (2007), 353–380. · Zbl 1128.60071 · doi:10.1007/s11118-007-9060-6 |
| [17] | A. Jonsson: Besov spaces on closed subsets of \(\mathbf{R}^{n}\) , Trans. Amer. Math. Soc. 341 (1994), 355–370. JSTOR: · Zbl 0803.46035 · doi:10.2307/2154626 |
| [18] | A. Jonsson and H. Wallin: Function spaces on subsets of \(\mathbf{R}^{n}\) , Math. Rep. 2 (1984), xiv+221. · Zbl 0875.46003 |
| [19] | J. Kigami: Analysis on Fractals, Cambridge Univ. Press, Cambridge, 2001. · Zbl 0998.28004 |
| [20] | P. Kim: Weak convergence of censored and reflected stable processes , Stochastic Process. Appl. 116 (2006), 1792–1814. · Zbl 1105.60007 · doi:10.1016/j.spa.2006.04.006 |
| [21] | T. Kumagai: Some remarks for stable-like jump processes on fractals ; in Fractals in Graz 2001, Trends in Mathematics, Birkhäuser, Basel, 2003, 185–196. · Zbl 1029.60067 |
| [22] | T. Kumagai and K.-T. Sturm: Construction of diffusion processes on fractals, \(d\)-sets, and general metric measure spaces , J. Math. Kyoto Univ. 45 (2005), 307–327. · Zbl 1086.60052 |
| [23] | K. Kuwae and T. Shioya: Convergence of spectral structures , Comm. Anal. Geom. 11 (2003), 599–673. · Zbl 1092.53026 · doi:10.4310/CAG.2003.v11.n4.a1 |
| [24] | T. Leviatan: Perturbations of Markov processes , J. Functional Analysis 10 (1972), 309–325. · Zbl 0239.60068 · doi:10.1016/0022-1236(72)90029-8 |
| [25] | U. Mosco: Composite media and asymptotic Dirichlet forms , J. Funct. Anal. 123 (1994), 368–421. · Zbl 0808.46042 · doi:10.1006/jfan.1994.1093 |
| [26] | E.M. Stein: Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. · Zbl 0207.13501 |
| [27] | A. Stós: Symmetric \(\alpha\)-stable processes on \(d\)-sets , Bull. Polish Acad. Sci. Math. 48 (2000), 237–245. · Zbl 0984.60087 |
| [28] | D.W. Stroock and W. Zheng: Markov chain approximations to symmetric diffusions , Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), 619–649. · Zbl 0885.60065 · doi:10.1016/S0246-0203(97)80107-0 |
| [29] | H. Triebel: Fractals and Spectra, Birkhäuser, Basel, 1997. · Zbl 0898.46030 |
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