Page 177 - Special Topic Session (STS) - Volume 4

STS579 Daria Balashova
As is known from [6], the transition probabilities satisfy the system of inverse
Kolmogorov equations:

where δ — discrete δ-function of Kronecker on ℤ .

We assume that branching occurs in the sources and is determined by the

infinitesimal generating functions

()
where ∑ ( ) = 0, ( ) ≥ 0 for ≠ 1 and ( ) < 0, (1) < ∞ for all

1

∈ ℕ. Denote the intensity of the source :

characterising the average number of descendants that are born in it.
Let () — is the number of particles at time at the point and

(, , ) ≔ Ε () — is the mathematical expectation of the number of
1

particles at the point at time under the condition, that at the initial instant
of time = 0 there was one particle in the system located at the point .
According to [6],

(2)

On the set of functions (), ∈ ℤ we consider the operator

and for each of the sources ∈ ℤ the operators

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