Page 169 - Special Topic Session (STS) - Volume 4

STS579 Elena Yarovaya
∆ = , and δx = δx(·) denotes the column vector on the lattice assuming

unit value at the point x and zero value at the rest of points. The perturbation
∑ ∆ of the linear operator may give rise to occurrence in the

=1

spectrum of the operator ℋ of positive eigenvalues, the number of such
1 ,…,
eigenvalues not exceeding the number of the summands in the last sum
counted with their multiplicity (Yarovaya, 2012; Yarovaya, 2017b).
The multipoint perturbations of the generator of symmetrical random walk
like (1) occur in the operator equations for the moments of particle
numbers. For example, let () be the number of particles at the time instant

at the point . Then, the condition that at the initial time instant = 0 the
system consists of a single particle situated at the point is equivalent to the
equality () = ( − ). At that, the total number of particles on the lattice
0
obeys the equality = ∑ ∈ℤ () . Denote by (, , ) = ( ())the

1
expectation of the number of particles at the time instant at the point ,
provided that () ≡ ( − )., that is, at the initial time instant the system
0
had one particle at the point . As was shown in (Yarovaya, 2012; Yarovaya,
2013), the evolution of (, , ) obeys the operator equation in the space
1
(ℤ ):

2

Evolution of the mean number of particles (, ) = ( ()) (total size

1
of the population) over the entire lattice (see, for example, (Yarovaya, 2007))
satisfies the operator equation in the corresponding space (ℤ ):
∞

Now we notice that the issue of the rate of growth or decrease of the mean
number of particles (, , ) is tightly bound to the spectral properties of
1
the operator ℋ . For example, if the operator ℋ has the maximal eigenvalue

> 0 , then (, , ) grows at infinity as , see (Khristolyubov and
1
Yarovaya, 2019).
Theorem 1 Let > 0 for = 1,2, … , and the operator has an isolated

eigenvalue λ0 > 0. Moreover, the remaining part of its spectrum be located on
() −1
the halfline {λ ∈ R : λ ≤ λ0 − }, where > 0. If = (! )for = 1,2, … ,
and ∈ ℕ, then in the sense of convergence in distribution the following
statements hold:

where ψ(y) is a non-negative non-random function and ξ is a proper random
variable.

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