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.
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A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov, “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem,” Byul. Mosk. Gos. Univ., Mat. Mekh. 1(6), 1–26 (1937); French transl.: A. Kolmogoroff, I. Pretrovsky, and N. Piscounoff, “ Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,” Bull. Univ. État Moscou, Math. Méc. 1 (6), 1–25 (1937).
S. A. Molchanov and E. B. Yarovaya, “Branching processes with lattice spatial dynamics and a finite set of particle generation centers,” Dokl. Akad. Nauk 446(3), 259–262 (2012) [Dokl. Math. 86 (2), 638–641 (2012)].
S. A. Molchanov and E. B. Yarovaya, “Limit theorems for the Green function of the lattice Laplacian under large deviations of the random walk,” Izv. Ross. Akad. Nauk, Ser. Mat. 76(6), 123–152 (2012) [Izv. Math. 76, 1190–1217 (2012)].
S. A. Molchanov and E. B. Yarovaya, “Population structure inside the propagation front of a branching random walk with finitely many centers of particle generation,” Dokl. Akad. Nauk 447(3), 265–268 (2012) [Dokl. Math. 86 (3), 787–790 (2012)].
B. A. Sevastyanov, “Branching stochastic processes for particles diffusing in a bounded domain with absorbing boundaries,” Teor. Veroyatn. Primen. 3(2), 121–136 (1958) [Theory Probab. Appl. 3, 111–126 (1958)].
B. A. Sevastyanov, “Minimal points of a supercritical branching walk on the lattice N r0 and many-type Galton-Watson branching processes,” Diskret. Mat. 12(1), 3–6 (2000) [[Discrete Math. Appl. 10, 1–4 (2000)].
M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].
E. B. Yarovaya, Branching Random Walks in Nonhomogeneous Environment (Tsentr Prikl. Issled. Mekh.-Mat. Fak. MGU, Moscow, 2007) [in Russian].
E. B. Yarovaya, “Criteria of exponential growth for the numbers of particles in models of branching random walks,” Teor. Veroyatn. Primen. 55(4), 705–731 (2010) [Theory Probab. Appl. 55, 661–682 (2011)].
E. B. Yarovaya, “Symmetric branching walks with heavy tails,” in Modern Problems of Mathematics and Mechanics, Vol. 7: Mathematics and Mechanics (Mosk. Gos. Univ., Moscow, 2011), Issue 1, pp. 77–84 [in Russian].
E. B. Yarovaya, “Spectral properties of evolutionary operators in branching random walk models,” Mat. Zametki 92(1), 123–140 (2012) [Math. Notes 92, 115–131 (2012)].
S. Albeverio, L. V. Bogachev, and E. B. Yarovaya, “Asymptotics of branching symmetric random walk on the lattice with a single source,” C. R. Acad. Sci. Paris, Sér. 1: Math. 326(8), 975–980 (1998).
H. Cramér, “Sur un nouveau théorème-limite de la théorie des probabilités,” Actual. Sci. Ind. 736, 5–23 (1938).
M. Cranston, L. Koralov, S. Molchanov, and B. Vainberg, “Continuous model for homopolymers,” J. Funct. Anal. 256(8), 2656–2696 (2009).
J. Gärtner and S. A. Molchanov,, “Parabolic problems for the Anderson model. I: Intermittency and related topics,” Commun. Math. Phys. 132(3), 613–655 (1990).
J. Gärtner, W. König, and S. Molchanov, “Geometric characterization of intermittency in the parabolic Anderson model,” Ann. Probab. 35(2), 439–499 (2007)
S. Molchanov, “Lectures on random media,” in Lectures on Probability Theory: Ecole d’Eté de Probabilités de Saint-Flour XXII-1992 (Springer, Berlin, 1994), Lect. Notes Math. 1581, pp. 242–411.
K. Uchiyama, “Green’s functions for random walks on Z N,” Proc. London Math. Soc., Ser. 3, 77(1), 215–240 (1998).
V. A. Vatutin, V. A. Topchiı, and E. B. Yarovaya, “Catalytic branching random walks and queueing systems with a random number of independent servers,” Theory Probab. Math. Stat. 69, 1–15 (2004).
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Original Russian Text © S.A. Molchanov, E.B. Yarovaya, 2013, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 282, pp. 195–211.
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Molchanov, S.A., Yarovaya, E.B. Large deviations for a symmetric branching random walk on a multidimensional lattice. Proc. Steklov Inst. Math. 282, 186–201 (2013). https://doi.org/10.1134/S0081543813060163
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DOI: https://doi.org/10.1134/S0081543813060163