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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\jour Proc. Steklov Inst. Math.
\yr 2013
\vol 282
\pages 186--201
\crossref{https://doi.org/10.1134/S0081543813060163}
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This publication is cited in the following 12 articles:
Xiaoyun Chen, Dan Han, Stanislav A. Molchanov, “Phase transitions in the non-stationary lattice Anderson model”, Journal of Mathematical Physics, 65:12 (2024)
D. M. Balashova, E. B. Yarovaya, “Structure of the Population of Particles for a Branching Random Walk in a Homogeneous Environment”, Proc. Steklov Inst. Math., 316 (2022), 57–71
D. M. Balashova, “Clustering effect for multitype branching random walk”, Theory Probab. Appl., 67:3 (2022), 352–362
Iuliia Makarova, Daria Balashova, Stanislav Molchanov, Elena Yarovaya, “Branching Random Walks with Two Types of Particles on Multidimensional Lattices”, Mathematics, 10:6 (2022), 867
Balashova D., Molchanov S., Yarovaya E., “Structure of the Particle Population For a Branching Random Walk With a Critical Reproduction Law”, Methodol. Comput. Appl. Probab., 23:1 (2021), 85–102
Feng Ya., Molchanov S., Yarovaya E., “Stability and Instability of Steady States For a Branching Random Walk”, Methodol. Comput. Appl. Probab., 23:1 (2021), 207–218
Random Motions in Markov and Semi‐Markov Random Environments 1, 2021, 205
Yulia Makarova, Vladimir Kutsenko, Elena Yarovaya, Springer Proceedings in Mathematics & Statistics, 371, Recent Developments in Stochastic Methods and Applications, 2021, 255
“Abstracts of talks given at the 4th International Conference on Stochastic Methods”, Theory Probab. Appl., 65:1 (2020), 121–172
D. M. Balashova, “Branching random walks with alternating sign intensities of branching sources”, J. Math. Sci., 262:4 (2022), 442–451
Molchanov S., Vainberg B., “Population Dynamics With Moderate Tails of the Underlying Random Walk”, SIAM J. Math. Anal., 51:3 (2019), 1824–1835
A. Grigor'yan, Yu. Kondratiev, A. Piatnitski, E. Zhizhina, “Pointwise estimates for heat kernels of convolution-type operators”, Proc. London Math. Soc., 117:4 (2018), 849–880