Page 170 - Special Topic Session (STS) - Volume 4

STS579 Elena Yarovaya
The behavior of processes whose parameters are situated near the critical
point is crucial for many applications. Previously, such problems were
considered for branching processes in non-random and random
environments, see (Limnios and Yarovaya, 2019) and the bibliography therein.
The concept of weakly supercritical BRW was introduced in (Yarovaya, 2015)
for the equal intensities ≔ = ⋯ = .
1

Definition 1 If there exists ε0 > 0 such that for β ∈ (βc, βc + ε0) the operator
ℋ has one (counting multiplicity) positive eigenvalue λ0(β) satisfying the

condition λ0(β) → 0 for β ↓ βc, then the supercritical BRW is called weakly
supercritical for β close to βc.
As was established in (Yarovaya, 2015; Yarovaya, 2017b), for β ↓ βc each
supercritical BRW is weakly supercritical. Denote now by (, , ) the
transition probability of the random walk. Clearly, the function (, , ) is
determined by the transition intensities (, ) (see, for example, (Gikhman
and Skorokhod, 2004; Yarovaya, 2007)). Then the Green function of the
operator is representable as the Laplace transform of the transition
probability (, , ):

(2)

Of special interest for the weakly supercritical BRWs are the asymptotics of the
Green function (2) and the eigenvalue λ0(β) for the evolutionary operator (1)
for β ↓ βc, that is, forβ → βc,β > βc.

3. Results
In this paper, we generalize the notion of a weakly supercritical BRW for
unequal intensities.
Definition 2 If there exists ε0 > 0 and a set (β1, β2,..., βN) of the branching
source intensities such that for β1 ∈ (βc1, βc1+ε0), β2 ∈ (βc2, βc2 +ε0),...,βN ∈ (βcN,
βcN + ε0) the operator ℋ has at least one (counting multiplicity)
1 , 2 …,
positive eigenvalue λ0(β1, β2,..., βN) satisfying the condition λ0(β1, β2, ..., βN) → 0
for βi ↓ βc,i, = 1, 2, . . . , , then the supercritical BRW is called weakly
supercritical for the branching source intensities (β1, β2,..., βN) close to (βc1, βc2,
..., βcN).
One can obtain that the theorems for the Green function (2), see (Yarovaya,
2017c), remain valid. For studying (2) the key role plays the asymptotic
behavior of transition probabilities (, , ) for underlying random walk
based on the properties of (), ∈ ℤ . As was shown in (Molchanov and

Yarovaya, 2012), the following assertion for Gλ:= Gλ(0, 0) is valid under the
condition of a finite variance of underlying random walk jumps.

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