Page 173 - Special Topic Session (STS) - Volume 4

STS579 Elena Yarovaya
4. Discussion and Conclusion
Based on Theorems 2–5 we can derive some assertions for a weekly
supercritical BRW under the conditions (3) or (4). Note that the related proofs
for a weekly supercritical BRW are essentially based on how the higher terms
of the asymptotic representations (6) depend on .
Further in the section, we shortly discuss the results on the structure of the
population inside the propagating front for recurrent underlying random walk,
that is, when G0 = ∞, where by the population front we mean the set

= { ∶ (, 0, ) ≤ }.
1

The following theorem gives the description of the population inside the
front and near to its boundary for = 1 and = 2 for β := β1 = · · · = βN. It
was shown in (Yarovaya, 2017a) that there exists Ꞓ0 > 0 such that for β ∈ (βc,
βc + Ꞓ0) the operator has a unique eigenvalue λ0(β).

Theorem 6 Let > 0, βc < 0 < βc + , and x ∈ Z , d = 1 or d = 2. If for some
d
0
0
c1, c2 > 0 we have c1t ≤ |y| ≤ c2t, as t →∞ , and then

where the distribution of is independent on and obeyed the relation
∞
{ > 0}. Moreover,
∞

References
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on the convergence of multidi-mensional random walks to stable
processes. Stoch. Dyn., 15(3):1550024, 14.
2. Cranston, M., Koralov, L., Molchanov, S., and Vainberg, B. (2009).
Continuous model for homopolymers. J. Funct. Anal., 256(8):2656–2696.
3. Ermakova, E., Makhmutova, P., and Yarovaya, E. (2019). Branching random
walks and their applications for epidemic modeling. Stochastic Models,
1–18, DOI: 10.1080/15326349.2019.1572519.
4. Gikhman, I. and Skorokhod, A. (2004). The theory of stochastic processes.
II. Classics in Mathematics. Springer-Verlag, Berlin. Translated from the
Russian by S. Kotz, Reprint of the 1975 edition.
5. Khristolyubov, I. and Yarovaya, E. (2019). A limit theorem for supercritical
branching random walks with branching sources of varying intensity.
ArXiv.org e-Print archive.

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