On Uniform Stability with Growth Rates of Stochastic Skew-Evolution Semiflows in Banach Spaces

The main purpose of this paper is to study a more general concept of uniform stability in mean in which the uniform behavior in the classical sense is replaced by a weaker requirement with respect to some probability measure. This concept includes, as particular cases, the concepts of uniform exponential stability in mean and uniform polynomial stability in mean. Extending techniques employed in the deterministic case, we obtain variants of some results for the general cases of uniform stability in mean for stochastic skew-evolution semiflows in Banach spaces.


Introduction
During the last decades, considerable attention has been devoted to the problem of asymptotic behaviors of nonautonomous differential equations in Banach spaces. Many results can be carried out not only for differential equations and evolution operators but also for skew-evolution semiflows. The notion of skew-evolution semiflow was introduced in [1] and includes some particular cases of many well-known concepts in dynamical system theory, such as C 0 -semigroups, evolution operators and skew-product semiflows.
In this paper, we consider the case of stochastic skew-evolution semiflows studied in [2]. We note that the stochastic cocycles studied in [3] are particular cases of the concept below.
Several important examples of stochastic evolution semiflows give rise to stochastic evolution equations, and the reader can refer to the monographs by [4,5].
The main purpose of the present paper is to study a general concept of uniform asymptotic stability in mean, which we call "uniform h-stability in mean" where h : R + → [1, ∞) is a growth rate (i.e., h is nondecreasing and bijective).
In particular cases, we obtain the concepts of uniform exponential stability in mean and uniform polynomial stability in mean for stochastic skew-evolution semiflows in Banach spaces.
Thus, we obtain three types of characterizations for each stability in mean concept considered in our study. Connections between these concepts are given. For some other approaches to the study of uniform exponential stability in mean and uniform polynomial stability in mean, we refer to [17][18][19][20].
The paper is organized as follows. In Section 2, we review some preliminaries on stochastic skew-evolution semiflows, which will be used in the paper. A general concept of uniform asymptotic stability in mean for stochastic skew-evolution semiflows is defined. In particular, the results of the concept of uniform exponential stability in mean and uniform polynomial stability in mean. In Section 3, connections between these concepts are presented. In Section 4, we state and prove the main results of our paper. Thus, we obtain three types of characterizations for each stability in mean concept considered in our study.

Preliminaries
Let (Ω, B, P) be a probability space. Denote ∆ = {(t, s) : t ≥ s ≥ 0}. We also denote by X a real or complex Banach space and by B(X) the Banach algebra of all bounded linear operators on X.
is called a stochastic evolution cocycle associated with a stochastic evolution semiflow ϕ : In this case, the pair (Φ, ϕ) is called a stochastic skew-evolution semiflow on X 1 = Ω × X Example 1. If Φ : ∆ × Ω → B(X) is a stochastic evolution cocycle associated with the stochastic evolution semiflow ϕ : ∆ × Ω → Ω and g : R + → [1, ∞) is a growth rate, then is a stochastic evolution semiflow on Ω and is a stochastic evolution cocycle associated with the stochastic evolution semiflow ϕ h .

Example 2.
If Φ : ∆ × Ω → B(X) is a stochastic evolution cocycle associated with the stochastic evolution semiflow ϕ : ∆ × Ω → Ω and h : R + → [1, ∞) is a growth rate, then is a stochastic evolution semiflow on Ω and is a stochastic evolution cocycle associated with the stochastic evolution semiflow ψ h .
We denote by L(Ω, X, P) the Banach space of all Bochner measurable functions f : We also denote Y = R + × L(Ω, X, P) and Z = ∆ × L(Ω, X, P).
The main stability in mean concept studied in this paper is introduced by for all (t, s, x) ∈ Z.
As particular cases, we have we obtain the uniform exponential stability in mean (u.e.s.m.) concept.
Another concept used is given by for all (t, s, X) ∈ Z.
As particular cases, we have we obtain the concept of uniform exponential growth in mean (u.e.g.m.). The next theorem presents the connection between uniform h-stability in mean and uniform exponential stability in mean.

Theorem 1. The stochastic skew-evolution semiflow (Φ, ϕ) is uniformly h-stable in mean if and only if the stochastic skew-evolution semiflow
is uniformly exponentially stable in mean.

Proof.
Necessity. If (Φ, ϕ) is uniformly h-stable in mean and (t, s, ω) ∈ ∆ × Ω then there are the constants N > 1 and ν > 0 such that is uniformly exponentially stable in mean and (t, s, ω) ∈ ∆ × Ω then there are the constants N > 1 and ν > 0 such that and hence (Φ, ϕ) is uniformly h-stable in mean.
The connection between uniform polynomial stability in mean and uniform exponential stability in mean is given by is uniformly exponentially stable in mean.
The next theorem presents the connection between uniform h-stability in mean and uniform polynomial stability in mean.
is uniformly polynomially stable in mean.
Proof. It is similar to the proof of Theorem 1.
As a particular case, we obtain Corollary 2. The stochastic skew-evolution semiflow (Φ, ϕ) is uniformly exponentially stable in mean if and only if the stochastic skew-evolution semiflow (Φ 2 , ϕ 2 ), where is uniformly polynomially stable in mean.
Proof. It follows from Theorem 2 for h(t) = e t .

The Main Results
A first characterization of uniform exponential stability in mean is given by Theorem 3. If the stochastic skew-evolution (Φ, ϕ) has uniform exponential growth in mean, then (Φ, ϕ) is uniformly exponentially stable in mean if and only if there are r > 1 and c ∈ (0, 1) such that Proof. Necessity. If (Φ, ϕ) is u.e.s.m. then so are the constants N > 1 and ν > 0 with We have for all (s, x) ∈ Y where c de f = Ne −νr ∈ (0, 1) because e νr > e ln N = N Sufficiency. If (t, s) ∈ ∆, then there exists n ∈ N and r 1 ∈ [0, r) such that t = s + nr + r 1 . Then for all (s, x) ∈ Y. If we denote ν = − ln c r then ν > 0 and c n = e −νrn and Ω Φ(t, s, ω)x(ω) dP(ω) ≤ Me ωr e −ν(t−s−r 1 ) for all (t, s, x) ∈ Z.

Corollary 4.
If the stochastic skew-evolution semiflow (Φ, ϕ) has uniform h-growth in mean, then (Φ, ϕ) is uniformly h-stable in mean if and only if there are r > e and c ∈ (0, 1) such that Proof. It follows from Theorems 1 and 3.
Another characterization of polynomial uniform stability in mean is given by Theorem 4. If the stochastic skew-evolution semiflow (Φ, ϕ) has uniform polynomial growth in mean, then (Φ, ϕ) is uniformly polynomially stable in mean if and only if there exist L > 1 with for all (t, s, x) ∈ Z.
Proof. Necessity. If (Φ, ϕ) is uniformly polynomially stable in mean, then there are the constants N > 1 and ν > 0 such that for all (t, s, x) ∈ Z.
Using the inequality for all (t, s) ∈ ∆ and all x ∈ L(Ω, X, P).

Corollary 5.
If the stochastic skew-evolution semiflow (Φ, ϕ) has uniform h-growth in mean, then (Φ, ϕ) is uniformly h-stable in mean if and only if there exists L > 1 with for all (t, s, x) ∈ Z.
Proof. Necessity. If (Φ, ϕ) is uniformly h-stable in mean, then from Theorem 2 it follows that is uniformly polynomially stable in mean.
From Theorem 4, it results that there exists L > 1 with Let (t, s) ∈ ∆. Then for u Sufficiency. Let s ≥ 1 and r = e 4L . Then r ≥ e and from the hypothesis there exists From Corollary 4, we have that (Φ, ϕ) is uniformly h-stable in mean.

Corollary 6.
If the stochastic skew-evolution semiflow (Φ, ϕ) has uniform exponential growth in mean, then (Φ, ϕ) is uniformly exponentially stable in mean if and only if there exists L > 1 with for all (t, s, x) ∈ Z.
Proof. It follows from Corollary 5, taking h(t) = e t .
A majorization criterion for uniform exponential stability in mean is given in the next theorem.

Theorem 5.
If the stochastic skew-evolution semiflow (Φ, ϕ) has uniform exponential growth in mean, then (Φ, ϕ) is uniformly exponentially stable in mean if and only if there exist M > 1 and a nondecreasing application g : [1, ∞) → R + with lim t→∞ g(t) = ∞ and for all (t, s, x) ∈ Z Proof. Necessity. From Corollary 5, for the case h(t) = e t , it results that if Φ is uniformly exponentially stable in mean, then there exists L > 1 with for all (t, s) ∈ ∆, thus the condition of the theorem is satisfied for g(t) = t Sufficiency. From lim t→∞ g(t) = ∞, we have that there exists δ > 0 with M < g(δ). Then, for all (t, s) ∈ ∆, there are n ∈ N and r ∈ [0, δ) with t = s + nδ + r.

Corollary 7.
If the stochastic skew-evolution semiflow (Φ, ϕ) has uniform polynomial growth in mean, then (Φ, ϕ) is uniformly polynomially stable in mean if and only if there exists M > 1 and a nondecreasing application g : [1, ∞) → R + with lim t→∞ g(t) = ∞ and for all (t, s, x) ∈ Z.
From Theorem 5, we obtain that (Φ 2 , ϕ 2 ) is uniformly exponentially stable in mean and from Corollary 2 it results that (Φ, ϕ) is uniformly polynomially stable in mean.

Corollary 8.
If the stochastic skew-evolution semiflow (Φ, ϕ) has uniform h-growth in mean, then (Φ, ϕ) is uniformly h-stable in mean if and only if there exist M > 1 and a nondecreasing application g : for all (t, s, x) ∈ Z Proof. Necessity. It follows from Corollary 5 for g(t) = ln t.

Conclusions
In this note, we have considered three concepts of uniform stability in mean for stochastic skew-evolution semiflows. These notions are natural generalizations from the deterministic context. The established relation between these concepts represents the first main goal. Thus, the relation between uniform h-stability in mean and uniform exponential stability in mean is established in Theorem 1, the relation between uniform polynomial stability in mean and uniform exponential stability in mean is given in Corollary 1, the relation between uniform h-stability in mean and uniform polynomial stability in mean is established in Theorem 2, and finally, the connection between uniform exponential stability in mean and uniform polynomial stability in mean is done in Corollary 2. Then, based on the present results from this work, characterizations of these concepts are exposed. The second main goal of the paper is to give three types of characterizations for these concepts of uniform stability in mean. These results are presented in