Abstract:
An important role in the theory of branching random walks is played by the problem of the spectrum of a bounded symmetric operator, the generator of a random walk on a multidimensional integer lattice, with a one-point potential. We consider operators with potentials of a more general form that take nonzero values on a finite set of points of the integer lattice. The resolvent analysis of such operators has allowed us to study branching random walks with large deviations. We prove limit theorems on the asymptotic behavior of the Green function of transition probabilities. Special attention is paid to the case when the spectrum of the evolution operator of the mean numbers of particles contains a single eigenvalue. The results obtained extend the earlier studies in this field in such directions as the concept of a reaction front and the structure of a population inside a front and near its boundary.
Citation:
S. A. Molchanov, E. B. Yarovaya, “Large deviations for a symmetric branching random walk on a multidimensional lattice”, Branching processes, random walks, and related problems, Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences, Trudy Mat. Inst. Steklova, 282, MAIK Nauka/Interperiodica, Moscow, 2013, 195–211; Proc. Steklov Inst. Math., 282 (2013), 186–201
\Bibitem{MolYar13}
\by S.~A.~Molchanov, E.~B.~Yarovaya
\paper Large deviations for a~symmetric branching random walk on a~multidimensional lattice
\inbook Branching processes, random walks, and related problems
\bookinfo Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences
\serial Trudy Mat. Inst. Steklova
\yr 2013
\vol 282
\pages 195--211
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\crossref{https://doi.org/10.1134/S0371968513030163}
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2013
\vol 282
\pages 186--201
\crossref{https://doi.org/10.1134/S0081543813060163}
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https://www.mathnet.ru/eng/tm3485
https://doi.org/10.1134/S0371968513030163
https://www.mathnet.ru/eng/tm/v282/p195
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