Gaussian estimates for spatially inhomogeneous random walks on \(\mathbb Z^d\). (English) Zbl 1102.60062

Let \(\Gamma\) be a symmetric finite subset of \(\mathbb{Z}^d\), let \((S_j)_{j\in \mathbb{N}}\) be a spatially inhomogeneous random walk on \(\Gamma\) and let \(p_n(x,y)\) (\(n\in \mathbb{N}; x,y\in \mathbb{Z}^d\)) be the associated transition kernel. The author proves an upper and a lower estimation of \(p_n(x,y)\) in terms of the Gaussian density \(\exp(-{|x-y|^2}/{n})\).


60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J35 Transition functions, generators and resolvents
60J45 Probabilistic potential theory
31C20 Discrete potential theory
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[1] Aronson, D. G. (1968). Non-negative solutions of linear parabolic equations. Ann. Sci. Norm. Super. Pisa 22 607–694. · Zbl 0182.13802
[2] Auscher, P. and Coulhon, T. (1999). Gaussian lower bounds for random walks from elliptic regularity. Ann. Inst. H. Poincaré Probab. Statist. 35 605–630. · Zbl 0933.60047 · doi:10.1016/S0246-0203(99)00109-0
[3] Bass, R. F. and Burdzy, K. (1994). The boundary Harnack principle for nondivergence form elliptic operators. J. London Math. Soc. 50 157–169. · Zbl 0806.35025 · doi:10.1112/jlms/50.1.157
[4] Bauman, P. E. (1984). Positive solutions of elliptic equations in nondivergence form and their adjoints. Ark. Mat. 22 536–565. · Zbl 0557.35033 · doi:10.1007/BF02384378
[5] Delmotte, T. (1999). Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 181–232. · Zbl 0922.60060 · doi:10.4171/RMI/254
[6] Escauriaza, L. (2000). Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form. Comm. Partial Differential Equations 25 821–845. · Zbl 0946.35004 · doi:10.1080/03605300008821533
[7] Fabes, E. B., Garofalo, N. and Salsa, S. (1986). A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations. Illinois J. Math. 30 536–565. · Zbl 0625.35006
[8] Fabes, E. B. and Safonov, M. V. (1997). Behavior near the boundary of positive solutions of second order parabolic equations. J. Fourier Anal. Appl. 3 871–882. · Zbl 0939.35082 · doi:10.1007/BF02656492
[9] Fabes, E. B., Safonov, M. V. and Yuan, Y. (1999). Behavior near the boundary of positive solutions of second order parabolic equations. II. Trans. Amer. Math. Soc. 351 4947–4961. · Zbl 0976.35031 · doi:10.1090/S0002-9947-99-02487-3
[10] Fabes, E. B. and Stroock, D. W. (1986). A new proof of Moser’s parabolic Harnack inequality via the old idea of Nash. Arch. Rational Mech. Anal. 96 327–338. · Zbl 0652.35052 · doi:10.1007/BF00251802
[11] Grigor’yan, A. (1991). The heat equation on noncompact Riemannian manifolds. Mat. Sb. 182 55–87. [Translation in Russian Acad. Sci. Sb. Math. 72 (1992) 47–77.] · Zbl 0743.58031
[12] Kozlov, S. M. (1985). The method of averaging and random walks in inhomogeneous environments. Russian Math. Surveys 40 73–145. · Zbl 0615.60063 · doi:10.1070/RM1985v040n02ABEH003558
[13] Kuo, H. J. and Trudinger, N. S. (1998). Evolving monotone difference operators on general space–time meshes. Duke Math. J. 91 587–607. · Zbl 0940.65089 · doi:10.1215/S0012-7094-98-09122-0
[14] Lawler, G. F. (1992). Estimates for differences and Harnack inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous, increments. Proc. London Math. Soc. 63 552–568. · Zbl 0774.39004 · doi:10.1112/plms/s3-63.3.552
[15] Safonov, M. V. and Yuan, Y. (1999). Doubling properties for second order operators. Ann. of Math. ( 2 ) 150 313–327. · Zbl 1157.35391 · doi:10.2307/121104
[16] Saloff-Coste, L. (1995). Parabolic Harnack inequality for divergence-form second-order differential operators. Potential theory and degenerate partial differential operators (Parma). Potential Anal. 4 429–467. · Zbl 0840.31006 · doi:10.1007/BF01053457
[17] Saloff-Coste, L. and Hebisch, W. (1993). Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 673–709. JSTOR: · Zbl 0776.60086 · doi:10.1214/aop/1176989263
[18] Varopoulos, N. Th. (2000). Potential theory in conical domains. II. Math. Proc. Cambridge Philos. Soc. 129 301–319. · Zbl 0980.31007 · doi:10.1017/S0305004100004503
[19] Varopoulos, N. Th., Saloff-Coste, L. and Coulhon, T. (1992). Analysis and Geometry on Groups . Cambridge Univ. Press. · Zbl 0813.22003
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