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Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 6, Pages 123–152 (Mi izv7965)  

This article is cited in 2 scientific papers (total in 3 papers)

Limit theorems for the Green function of the lattice Laplacian under large deviations of the random walk

S. A. Molchanovab, E. B. Yarovayab

a University of North Carolina Charlotte
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We carry out a resolvent analysis of the lattice Laplacian (the generator of a simple random walk on the $d$-dimensional integer lattice) under large deviations of the random walk. This enables us to obtain asymptotic representations for the transition probability of the simple random walk and the corresponding Green function. We explicitly describe the asymptotic behaviour of the transition probability as the spatial and temporal variables jointly tend to infinity. The resulting Cramér-type expansion for the transition probability is ‘universal’ in this sense. In particular, it enables us to construct a scale for measuring the transition probability as a function of the time $t$ assuming that the spatial variable is of order $t^{\alpha}$ for various values of $\alpha\ge0$. We prove limit theorems on the asymptotic behaviour of the Green function of the transition probabilities under large deviations of the random walk.

Keywords: branching random walk, difference Laplacian, large deviations, spatio-temporal scale, asymptotics of the Green function, limit theorems.

DOI: https://doi.org/10.4213/im7965

Full text: PDF file (653 kB)
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English version:
Izvestiya: Mathematics, 2012, 76:6, 1190–1217

Bibliographic databases:

UDC: 519.21
MSC: 60F10, 60J35, 35A08, 35B40, 35K15
Received: 16.02.2012

Citation: S. A. Molchanov, E. B. Yarovaya, “Limit theorems for the Green function of the lattice Laplacian under large deviations of the random walk”, Izv. RAN. Ser. Mat., 76:6 (2012), 123–152; Izv. Math., 76:6 (2012), 1190–1217

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. A. Molchanov, E. B. Yarovaya, “Large deviations for a symmetric branching random walk on a multidimensional lattice”, Proc. Steklov Inst. Math., 282 (2013), 186–201  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. Antonenko E.A., “A weakly supercritical mode in a branching random walk”, Mosc. Univ. Math. Bull., 71:2 (2016), 68–70  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    3. “Tezisy dokladov, predstavlennykh na Chetvertoi mezhdunarodnoi konferentsii po stokhasticheskim metodam”, Teoriya veroyatn. i ee primen., 65:1 (2020), 151–210  mathnet  crossref
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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