Page 187 - Special Topic Session (STS) - Volume 4

STS579 Anastasiia Rytova et al.
(0, 2), whereas in the case of a finite variance of jumps, the relation < ∞
0

is satisfied only in dimensions ≥ 3. Then in ℤ and ℤ the can be strictly
positive, i.e. BRW population growth can be subcritical even under a
supercritical branching regime at the source, in contrast to the case of BRW
with finite variance. On high-dimensional lattices, the intensity of the
branching process at the source is not determines the criticality of the BRW
(see Table 1 in Yarovaya (2010)). For example, under condition of jumps
variance finiteness, the supercritical branching process at the source in
combination with transient random walk on ℤ , ≥ 3, can lead to either a

supercritical, or a critical or subcritical BRW. If the branching process at the
source is subcritical or critical, then on lattices ≥ 3only subcritical BRW is
possible.
The analysis of such BRW models is implemented by studying the
spectrum of operators , (see Yarovaya (2007), Yarovaya (2012),

Khristolyubov, & Yarovaya (2019)), which determines the limiting behavior of
solutions of differential equations (6), in particular, the tendency of the norm
to a constant and the fact of monotonicity. To find the asymptotics of
solutions, the Laplace transform of (, , ), (, , ), (, ), 1 — (, ) ,
integral equations and Tauberian theorems (see Ch. XIII in Feller (1971)) are
used. As a result, the asymptotics are expressed through the Green’s function.

3. Results
For convenience, we introduce the classification of possible combinations
of the lattice dimension d and the random walk jump parameter :
= = = ≥
(a) ∈ (1, 2)
(b) = 1
(c) ∈ (1/2,1) ∈ (1, 2) ∈ (3/2,2)
(d) = 1/2 = 1 = (3/2)
(e) ∈ (0, 1/2) ∈ (0, 1) ∈ (0, 3/2) ∈ (0, 2)
In Khristolyubov, & Yarovaya (2019), a supercritical symmetric continuous-
time BRW on ℤ , ≥ 1, with a < ∞ number of particle generation sources

of varying positive intensities without any restrictions on the variance of jumps
of the underlying random walk has been investigated. In Theorem 7, it was
found that if the operator have finite (counting multiplicity) number of

positive eigenvalues, and is the largest of them with the corresponding
0
normalized vector , then as → ∞ the following asymptotic relations hold
(, , )~ (, ) 0 , (, )~ () 0 , (8)
1
1
where (, ) = ()(), () = () −1 ∑ ( ). As a consequence,
=1

1
1
0

(, )~1.
Now describe the asymptotic behavior as → ∞ of the mean number of
particle at the ∈ ℤ point (, , ) , of the mean number particles of

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