Abstract
The notion of degree and related notions concerning recurrence and transience for a class of Lévy processes on metric Abelian groups are studied. The case of random walks on a hierarchical group is examined with emphasis on the role of the ultrametric structure of the group and on analogies and differences with Euclidean random walks. Applications to separation of time scales and occupation times of multilevel branching systems are discussed.
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Bingham, N.H., Goldie, C.M. and Teugels, J.L.: Regular Variation, Cambridge Univ. Press, Cambridge, U.K., 1989.
Cox, J.T., Dawson, D.A. and Greven, A.: ‘Mutually catalytic super branching random walks: Large finite systems and renormalization analysis’, Memoirs Amer. Math. Soc. (to appear).
Cox, J.T. and Griffeath, D.: ‘Large deviations for Poisson systems of independent random walks’, Probab. Theory Related Fields 66 (1984), 543–558.
Cox, J.T. and Griffeath, D.: ‘Occupation times for critical branching Brownian motions’, Ann. Probab. 13 (1985), 1108–1132.
Cox, J.T. and Griffeath, D.: ‘Diffusive Clustering in the two-dimensional voter model’, Ann. Probab. 14 (1986), 347–370.
Darling, D.A. and Erdös, P.: ‘On the recurrence of a certain chain’, Proc. Amer. Math. Soc. 19 (1968), 336–368.
Darling, D.A. and Kac, M.: ‘On occupation times for Markoff processes’, Trans. Amer. Math. Soc. 84 (1957), 444–458.
Dawson, D.A., Gorostiza, L.G. and Wakolbinger, A.: ‘Occupation time fluctuations in branching systems’, J. Theory Probab. 14 (2001), 729–796.
Dawson, D.A., Gorostiza, L.G. and Wakolbinger, A.: ‘Hierarchical equilibria of branching populations’, Elect. J. Probab. 9 (2004), Paper 12, 316–381.
Dawson, D.A., Gorostiza, L.G. and Wakolbinger, A.: ‘Hierarchical random walks’, in Asymptotic Methods in Stochastics, Fields Institute Communications and Monograph Series, Amer. Math. Soc. (to appear).
Dawson, D.A. and Hochberg, K.J.: ‘A multilevel branching model’, Adv. in Appl. Probab. 23 (1991), 701–705.
Dawson, D.A., Hochberg, K.J. and Vinogradov, V.: ‘High-density limits of hierarchically structured branching-diffusing populations’, Stochastic Process. Appl. 62 (1996), 191–222.
Deuschel, J.-D. and Wang, K.: ‘Large deviations for the occupation time functional of a Poisson system of independent Brownian particles’, Stochastic Process. Appl. 52 (1994), 183–209.
Etheridge, A.M.: ‘Limiting behaviour of two-level measure-branching’, Adv. in Appl. Probab. 25 (1993), 773–782.
Evans, S.: ‘Local properties of Lévy processes on totally disconnected groups’, J. Theory Probab. 2 (1989), 209–259.
Evans, S. and Fleischmann, K.: ‘Cluster formation in a stepping-stone model with continuous, hierarchically structured sites’, Ann. Probab. 24 (1996), 1926–1952.
Flatto, L. and Pitt, J.: ‘Recurrence criteria for random walk on countable Abelian groups’, Illinois J. Math. 18 (1974), 1–19.
Fleischmann, K. and Greven, A.: ‘Diffusive clustering in an infinite system of hierarchically interacting diffusions’, Probab. Theory Related Fields 98 (1994), 517–566.
Gorostiza, L.G.: ‘Asymptotic fluctuations and critical dimension for a two-level branching system’, Bernoulli 2 (1996), 109–132.
Gorostiza, L.G.: ‘A note on a divergent series related to the Riemann Zeta function’, Bol. Soc. Mat. Mexicana 8 (2002), 18–23.
Gorostiza, L.G., Hochberg, K.J. and Wakolbinger, A.: ‘Persistence of a critical super-2 process’, J. Appl. Probab. 32 (1995), 534–540.
Greven, A. and Hochberg, K.J.: ‘New behavioural patterns for two-level branching systems’, in L.G. Gorostiza and B.G. Ivanoff (eds), Stochastic Models, CMS Conference Proceedings 26, AMS, 2000, pp. 205–215,.
Hughes, B.D.: Random Walks in Random Environments, Vol. 1, Oxford Science Publ., Oxford, 1995.
Hughes, B.D., Montroll, E.W. and Shlesinger, M.F.: ‘Fractal random walks’, J. Statist. Phys. 28 (1982), 111–126.
Hughes, B.D., Shlesinger, M.F. and Montroll, E.W.: ‘Random walks with self-similar clusters’, Proc. Nat. Acad. Sci. U.S.A. 78(6) (1981), 3287–3291.
Klenke, A.: ‘Different clustering regimes in systems of hierarchically interacting diffusions’, Ann. Probab. 24 (1996), 660–697.
Marchal, P.: ‘Stable processes on the boundary of a regular tree’, Ann. Probab. 29 (2001), 1591–1611.
Port, S.C. and Stone, C.J.: ‘Infinitely divisible processes and their potential theory (Part 1)’, Ann. Inst. Fourier 21 (1971), 157–275.
Rvačeva, E.L.: ‘On domains of attraction of multi-dimensional distributions’, Select. Transl. Math. Statist. Probab. 2 (1962), 183–205.
Sato, K. and Watanabe, T.: ‘Last exit times for transient Lévy processes’, Cooperative Research Project, Inst. Statist. Math. 137 (2001), 72–82.
Sato, K. and Watanabe, T.: ‘Moments of last exit times for Lévy processes’, Ann. Inst. H. Poincaré, Probab. Statist. (to appear).
Sawyer, S. and Felsenstein, J.: ‘Isolation by distance in a hierarchically clustered population’, J. Appl. Probab. 20 (1983), 1–10.
Spitzer, F.: Principles of Random Walk, Van Nostrand, Princeton, 1976.
Wu, Y.: ‘Asymptotic behaviour of two level measure branching processes’, Ann. Probab. 22 (1994), 854–874.
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Mathematics Subject Classifications (2000)
60G50, 60B15, 60F05, 60J80.
D.A. Dawson: Research supported by NSERC (Canada) and a Max Planck Award for International Cooperation.
L.G. Gorostiza: Research supported by CONACYT grant 37130-E (Mexico).
A. Wakolbinger: Research supported by DFG (SPP 1033) (Germany).
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Dawson, D.A., Gorostiza, L.G. & Wakolbinger, A. Degrees of Transience and Recurrence and Hierarchical Random Walks. Potential Anal 22, 305–350 (2005). https://doi.org/10.1007/s11118-004-1327-6
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DOI: https://doi.org/10.1007/s11118-004-1327-6