Page 185 - Special Topic Session (STS) - Volume 4

STS579 Anastasiia Rytova et al.
As shown, for example, in Yarovaya (2007), for this model, the probabilities
(, , ) of a particle transition from the point to a point in time satisfy
the Kolmogorov’s backward equation
(, , ) = ∑ (, )(, , ), (0, , ) = ( − ), (2)
′

′
where δ(·) is the discrete Kronecker delta-function on ℤ .

Consider the heavy-tailed random walk, when the following asymptotic condition as
|| → ∞ on the transition intensities holds
(/||)
()~ , (3)
|| +
where | · | is the Euclidean norm on , (/||) = (−/||) is a positive

continuous function on −1 = { ∈ : || = 1}, and ∈ (0,2). This implies
that the variance of the jumps (1) becomes infinite.
We also assume that branching process is possible at special points ,
1
,…, called branching sources, where each particle can die or give a

2
random number of descendants. The reproduction law at the source , =
1
1,2, … , is defined by the continuous-time Bienaymé-Galton-Watson
branching process by the following infinitesimal generation function
∞
(, ) = ∑ ( ) , 0 ≤ ≤ 1,

=0
where ( ) ≥ 0 for ≠ 1 , ( ) < 0 and ∑ ( ) = 0 . We assume

1
() (1, ) < ∞ for every ∈ . For the future investigation an important role

will play the values
( )

′
= (1, ) = ∑ ( ) = (− ( )) (∑ (− ( )) − 1),

1

≠1 1
for = 1,2, … , , called intensity of the branching source , where the last sum

is the mean number of descendants born at the point .

The main objects of interest are the behavior of the local particle numbers
() at an arbitrary point ∈ ℤ and the total population size . We

consider their conditional expectation (, ) ≔ () under condition that

at the initial time there was only one particle in the system, located at the point
. In Yarovaya (2013b), it was shown that the following equations hold

(, , ) = ∑ (, )(, , ) + ∑ ( − )(, , ), (4)
′
′

′ =1

(, )
= ∑ (, )(, ) + ∑ ( − )(, , ), (5)
′
′

′ =1

where (0, , ) = ( − ), (0, ) = 1. . Then it was noted that equations
(2), (4), (5) can be treated as the linear differential equations in a Banach space
(, , ) (, , ) (, )
= ((,∙, ))(), = ( (,∙, )) (), = ( (,∙)) () (6)

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