Page 178 - Special Topic Session (STS) - Volume 4

STS579 Daria Balashova
where δ(·) — discrete δ-function of Kronecker on ℤ . The operator , as an

operator in the Hilbert space (ℤ ), is self-adjoint, the operators Δ act in
2

each of the function spaces (ℤ ), ∈ [1, ∞], see [10].

The behavior of the mean number of particles both at an arbitrary point
and on the entire lattice can be described in terms of an evolutionary
operator of a special type

(3)

which is a perturbation of the generator of symmetric random walk. This
operator can be treated as a linear bounded operator acting in each of the
function spaces spaces (ℤ ), ∈ [1, ∞], see [10].

According to [10] the evolution equations for the transition probabilities
(1) and the moments of particle numbers (2) can be represented as the

2

following differential equation in the space (ℤ ) and spaces (ℤ ), ∈
[1, ∞], respectively:

The Green’s function of the operator can be represented as the Laplace
transform of the transition probability (, , ):

where ∅() = ∑ ∈ () (,) for ∈ [−, ]. For further research, the value
≔ (0,0) plays an important role. If inequality

where || is the Euclidean norm of the vector , is fulfilled then the variance of
jumps is finite and < ∞ for ≥ 3 [11].
0

We are interested in the description of the behavior of particles on ℤ in terms
of the total number of particles (, ) = ∑ ∈ (, , ) on the lattice.
1

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