Geodesic
Catalytic branching random walk on three-dimensional lattice
Teoriâ slučajnyh processov, Tome 16 (2010) no. 2, pp. 23-32.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a critical catalytic continuous time branching random walk on the integer lattice under the assumption that the birth and the death of particles occur at a single source of branching located at the origin. For the introduced joint generating function of the number of particles, the differential and integral equations are obtained in case of $\mathbb{Z}^{d}$ with arbitrary $d\in\mathbb{N}$. The limit value of the population survival probability as $t\rightarrow\infty$ is found as a function of the starting point $x\in\mathbb{Z}^{3}$. We establish the asymptotic behavior of the probability that the number of particles at the origin at time $t$ is positive. The Yaglom-type conditional limit theorem for the number of particles at the origin is proved. A joint conditional limit distribution of the number of particles at the source and the number of particles outside of it with finite lifetime is studied as well.
Keywords: Critical branching random walk, survival probability, conditional limit theorems. 
@article{THSP_2010_16_2_a3,
     author = {E. V. Bulinskaya},
     title = {Catalytic branching random walk on three-dimensional lattice},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {23--32},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a3/}
}
TY  - JOUR
AU  - E. V. Bulinskaya
TI  - Catalytic branching random walk on three-dimensional lattice
JO  - Teoriâ slučajnyh processov
PY  - 2010
SP  - 23
EP  - 32
VL  - 16
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a3/
LA  - en
ID  - THSP_2010_16_2_a3
ER  - 
%0 Journal Article
%A E. V. Bulinskaya
%T Catalytic branching random walk on three-dimensional lattice
%J Teoriâ slučajnyh processov
%D 2010
%P 23-32
%V 16
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a3/
%G en
%F THSP_2010_16_2_a3
E. V. Bulinskaya. Catalytic branching random walk on three-dimensional lattice. Teoriâ slučajnyh processov, Tome 16 (2010) no. 2, pp. 23-32. http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a3/

[1] F. den Hollander, M. V. Menshikov, S. Yu Popov, “A note on Transience Versus Recurrence for a Branching Random Walk in Random Environment”, J. of Stat. Phys., 95:3-4 (2004), 587–614

[2] N. Yoshida, “Central Limit Theorem for Branching Random Walks in Random Environment”, Ann. Appl. Probab., 18:4 (2008), 1619–1635

[3] E. B. Yarovaya, “Use of the Spectral Methods to Study Branching Processes with Diffusion in a Noncompact Phase Space”, Theor. Math. Phys., 88:1 (1991), 25–30

[4] L. V. Bogachev, E. B. Yarovaya, “Moment Analysis of a Branching Random Walk on a Lattice with a Single Source”, Doklady RAN, 363:4 (1998), 439–442

[5] L. V. Bogachev, E. B. Yarovaya, “A Limit Theorem for Supercritical Branching Random Walk on $\mathbb{Z}^d$ with a Single Source”, Russ. Math. Surveys, 53:5 (1998), 229–230

[6] S. Albeverio, L. V. Bogachev, E. B. Yarovaya, “Asymptotics of Branching Symmetric Random Walk on the Lattice with a Single Source”, C. R. Acad. Sci. Paris. Sér. I., 326:9 (1998), 975–980

[7] E. B. Yarovaya, “About Limit Theorems for Branching Symmetric Random Walk on $\mathbb{Z}^{d}$”, Kolmogorov and Contemporary Mathematics (Moscow, June 16-21, 2003), 2003, 592–593

[8] E. B. Yarovaya, “A Limit Theorem for Critical Branching Random Walk on $\mathbb{Z}^{d}$ with a Single Source”, Russ. Math. Surveys, 60:1 (2005), 175–176

[9] E. B. Yarovaya, “Ctitical Branching Random Walks on Low Dimensional Lattices”, Discr. Math., 21 (2009), 117–138

[10] E. B. Yarovaya, Branching Random Walks in Inhomogeneous Medium, MSU, Moscow, 2007 (in Russian)

[11] V. A. Vatutin, V. A. Topchii, E. B. Yarovaya, “Catalytic Branching Random Walk and Queueing Systems with Random Number of Independent Servers”, Theory Probab. Math. Statist., 2004, no. 69, 1–15

[12] V. A. Topchii, V. A. Vatutin, “Individuals at the Origin in the Critical Catalytic Branching Random Walk”, Discrete Math. Theoret. Comput. Sci. (electronic), 6 (2003), 325–332

[13] V. A. Topchii, V. A. Vatutin, “A Limit Theorem for the Critical Catalytic Branching Random Walk”, Theory Probab. Appl., 49:3 (2004), 461–484