Abstract
A time-continuous random walk on a multidimensional lattice which underlies the branching random walk with an infinite number of phase states is considered. The random walk with a countable number of states can be reduced to a system with a finite number of states by aggregating them. The asymptotic behavior of the residence time of the transformed system in each of the states depending on the lattice dimension under the assumption of a finite variance and under the condition leading to an infinite variance of jumps of the original system is studied. It is shown that the aggregation of states in terms of the described process leads to the loss of the Markov property.
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ACKNOWLEDGMENTS
The authors thank the reviewer for useful remarks.
Funding
The work is carried out at the Steklov Mathematical Institute of the Russian Academy of Sciences and is supported by the Russian Science Foundation, project no. 23-11-00375.
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Translated by E. Oborin
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Popov, G.A., Yarovaya, E.B. Phase States Aggregation of Random Walk on a Multidimensional Lattice. Moscow Univ. Math. Bull. 79, 60–70 (2024). https://doi.org/10.3103/S0027132224700074
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DOI: https://doi.org/10.3103/S0027132224700074