In this paper, we consider the following of nonlinear strongly damped stochastic Kirchhoff equations with additive white noise:
with the Dirichlet boundary condition
and the initial value conditions
where is a positive integer, is a normal number, is a bounded region with smooth boundary in , M is a general real-valued function, is a nonlinear nonlocal source term, and is a random term. The assumptions about M and g will be given later.
Xintao Li and Lu Xu  have studied the following stochastic delay discrete wave equation
The existence of random attractors for this equation is proved by means of tail-cutting technique and energy estimation under appropriate dissipative conditions.
Ailing Ban  have considered the following of stochastic wave equations
with the Dirichlet boundary condition
and the initial value conditions
where is the real function on , is the strong damping coefficient, is the damping coefficient, and is the dissipation coefficient. In this paper, they mainly discuss the asymptotic behavior of strongly damped stochastic wave equation with critical growth index. By using the weighted norm, they prove that for any positive strong damping coefficient and dissipation coefficient, there is a compact attractor for the stochastic dynamical system determined by the solution of the equation.
Caidi Zhao, Shengfan Zhou  studied the sufficient conditions for the existence of global random attractors for a class of binary systems and their applications
They first give some sufficient conditions for the existence of global random attractor for general stochastic dynamical systems, and then use these sufficient conditions to give a simple method for finding the global random attractors of the upper bound of Kolmogorov ε-entropy. Finally, these results are applied to stochastic Sin-Gordon equation.
Guoguang Lin, Ling Chen and Wei Wang  have studied the existence of random attractors for higher-order nonlinear strongly damped Kirchhoff equations
They mainly use the Ornstein-Uhlenbech process to deal with the stochastic term of Equation (11), thus obtain the global well-posedness of the solution, and then prove the existence of the global random attractor.
As we all know, attractors have absorptivity and invariance, and have a clear description of the long-term behavior and the asymptotic stability of the solution of the equation. Because the long-term behavior of the system develops within the overall attractor, and then on this compact set, through the study of the overall behavior characteristics of the system, we can find the most common rules of the system and the basic information of future development. In real life, the evolution of many problems will be disturbed by some uncertain factors. At this time, the deterministic dynamic system can no longer describe these problems. Therefore, it is necessary to study the attractors of stochastic equations with additive noise terms.
In recent years, stochastic attractors for stochastic nonlinear equations with white noise have been favored by many scholars, and many scholars have done a lot of research on these problems and obtained good results. Xiaoming Fan, Donghong Cai and Jianjun Ye  studied stochastic attractors for dissipative KdV equations with multiplicative noise; Fuqi Yin, Shengfan Zhou, Hongyan Li and Hongjuan Hao   by introducing weighted norm and orthogonal decomposition of linear operators corresponding to the first-order evolution equation with respect to time, the existence of stochastic attractors for stochastic Sine-Gordon equation with strong damping is proved. More research on stochastic Kirchhoff equation with white noise is detailed in reference      .
The structure of this paper is as follows: in Section 2, some basic assumptions and knowledge of dynamical system required in this paper are introduced; in Section 3, the existence of random attractor family subfamilies is proved by using the isomorphism mapping method.
2. Basic Hypothesis and Elementary Knowledge
In this section, some symbols, definitions and assumptions about Kirchhoff type stress term and nonlinear nonlocal source term are given. In addition, some basic definitions of stochastic dynamical systems are also introduced.
For narrative convenience, we introduce the following symbols:
It is assumed that the Kirchhoff type stress term and the nonlinear non-local source term satisfy the following conditions, respectively:
A1) ; and , where is a constant;
A2) is Lipschitz continuous and satisfies
i) for any ;
ii) There exists a constant , such that for any , have
The following will introduce some basic knowledge about random attractor.
Let be a probabilistic space and define a family of transformations of the sum and ergodic of a family of measures preserving
Then is an orbiting metric dynamical system.
Let be a complete separable metric space and be a Borel -algebra on.
Definition 1 (Following as  ) Let be a metric dynamical system, suppose that the mapping
is -measurable mapping and satisfies the following properties:
1) The mapping satisfies
for any .
2) is continuous, for any .
Then is a continuous stochastic dynamical system on .
Definition 2 (Following as  ) It is said that random set is tempered, for , , we have
where , for any .
Definition 3 (Following as  ) Let be the set of all random sets on X, and random set is called the suction collection on , if for any and , there exists such that
Definition 4 (Following as  ) Random set is called the random attractor of continuous stochastic dynamical system on , if random set satisfies the following conditions:
1) is a random compact set;
2) is the invariant set , that is, for any , we have ;
3) attracts all sets on , that is, for any and , with the following limit:
where is Hausdorff half distance. (There ).
Definition 5 (Following as  ) Let random set be a random suction set for stochastic dynamical system , and random set satisfies the following conditions:
1) Random set is a closed set on Hilbert space ;
2) For , random set satisfies for any sequence , there is a convergence subsequence in space , when . Then stochastic dynamical system has a unique global attractor
The Ornstein-Uhlenbeck process  is given as follows:
Let , where . It can be seen that for any , the stochastic process satisfies the Ito equation
According to the nature of the O-U process, there exists a probability measure , -invariant set, and the above stochastic process
satisfies the following properties:
1) The mapping is a continuous mapping, for any given ;
2) The random variable is tempered;
3) There exist a tempered set , such that
3. Existence of Random Attractor Family
In this section, we mainly consider the existence of random attractor family of problem (1)-(3). At first, Young inequality and Holder inequality are used to prove the positive definiteness of operator ; and then the weak solution of the equation is established by Ornstein-Uhlenbeck process to deal with the random term, thus a bounded random absorption collection is obtained. Finally, the existence of random attractor family of this problem is proved by isomorphism mapping method.
The problem (1)-(3) can be rewritten to
Let , then the question (14) can be simplified to
Let , Then the question (14) may read as follows:
Lemma 1 Let , for any , if , we have
Proof: For any , we have
From hypothesis (A1), we have
Choose , then we have
Therefore, Lemma 1 is proved.
Lemma 2 Let is a solution of the problem (15), then there is a bounded random compact set , such that for any random set , existence a random variable , so that
Proof: Let is a solution of the problem (16), taking inner product of two sides of the Equation (15) is obtained by using in , we have
From Lemma 1, we have
According to Holder inequality, Young inequality and Poincare inequality, we have
from assumption (A2), we have
Combining (22)-(28) yields, we have
From Gronwal inequality , then
And because is tempered, and is continuous about t, so according to reference  , we can get a temper random variable , so that for any , we have
Replace in Equation (30) with , we can obtain that
Available from (32)
Because is tempered, and is also tempered, so we can let
then is also tempered, put is a random absorb set, and because of
then is a random absorb set of , and .
Thus, the whole proof is complete.
It is shown below that there exists a compact suction collection for stochastic dynamical system .
Lemma 3 When , for any , let is a solution of the Equation (15) with the initial value , and it can decompose , where satisfy
then , for any , there exists a temper random radius , such that
for any .
Proof: When , let is a solution of Equation (16), then according to Equation (39) and Equation (40), we know meet separately
Taking inner product Equation (41) with in , we have
From Lemma 1 and Gronwall inequality, we have
substituting by in (43), and because is tempered, then
Taking inner product (42) with in , and from Lemma 1 and Lemma 2, we have
Substituting by in (46) and from Gronwall’s Inequality and (32), we have
So there exists a temper random radius
for any . (49)
This completes the Proof of Lemma 3.
Lemma 4 The stochastic dynamical system while determined by Equation (15) has a compact attracting set .
Proof: Let be a closed sphere in space with a radius of . According to embedding relation , then is a compact set in , for any temper random set , for , according to Lemma 3, , so for any , we have
So, the whole proof is complete.
According to Lemma 1 to Lemma 4, there are the following theorems
Theorem 1 The stochastic dynamical system has a random attractor family , for any , and there exists a temper random set , such that
In this paper, starting from the positive definiteness of the operator, the weak solution of the equation established by O-U process is used to deal with the stochastic term, and a bounded stochastic absorption set is obtained, thus tempered random set is obtained. Then, the isomorphic mapping method is used to prove that the stochastic dynamical system has a attractor family .
The authors express their sincere thanks to the scholars who have provided references, and to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions.