An Averaging Principle for Mckean–Vlasov-Type Caputo Fractional Stochastic Differential Equations

In this paper, we want to establish an averaging principle for Mckean–Vlasov-type Caputo fractional stochastic diﬀerential equations with Brownian motion. Compared with the classic averaging condition for stochastic diﬀerential equation, we propose a new averaging condition and obtain the averaging convergence results for Mckean–Vlasov-type Caputo fractional stochastic diﬀerential equations.


Introduction
For complex systems, we usually want to locate an effective simplified model to approximate the original complex system or extract the main dynamical behavior of the original system. Based on these ideas, a lot of effective methods have been generated in dynamical systems, such as invariant manifolds, averaging principle, and homogenization principle. ese effective methods have now been extended to deal with stochastic systems, such as stochastic invariant manifolds see [1,2] and stochastic averaging principle, see [3][4][5][6][7][8][9].
Currently, the problem of averaging for stochastic differential equations have received a lot of attention and various types of stochastic differential equations have been studied, see [4,6,7,[10][11][12]. However, there are no relevant results of averaging principle for distribution dependent-type stochastic differential equations which we will consider in this paper.
On the contrary, the problem of averaging for stochastic fractional order differential equations have received a lot of attention in recent years, and some results [13] have been obtained under averaging condition consistence with the classic case (see [4,5,14]). Noting that the fractional order derivative is a nonlocal operator, therefore, the fractional order differential equation is more effective for describing certain phenomena in the real world (see [15][16][17]). Current research studies on stochastic fractional order differential equations mainly focused on the existence and uniqueness of the solutions, with fewer results from the dynamical system perspective.
Based on the above discussion, we shall study the averaging principle for the following Mckean-Vlasov-type Caputo fractional stochastic differential equations: where α ∈ (1/2, 1] and B t is a scalar Brownian motion. Nonlinear terms f and g are H-valued functions defined on R + × H × M c 2 (H), and M c 2 (H) denotes a proper subset of probability measure on H. If the terms f and g do not depend on the probability distribution μ(t) of the process X at time t, such equations have been studied by [13] and other authors. If α � 1, the equation becomes a classical Mckean-Vlasov-type stochastic differential equations which have been considered by many authors with different approaches (see [18][19][20]). In this paper, we just focused on α ∈ (1/2, 1), and more details can be seen in Section 2. e paper is structured as follows. We introduce some notation and assumptions in Section 2. e existence and unique solution for distribution dependent fractional stochastic differential equations will be discussed in Section 3. An averaging principle for the above equation is established in Section 4.

Preliminaries
First, we introduce some notation. Let C(H) be the space of continuous functions on H. Let B(H) be the Borel σ-algebra of subsets of H. M(H) is the space of probability measures on B(H) and carries the usual topology of weak convergence. (μ, ϕ) denotes H ϕ(x)μ(dx). Let c(x) � 1+ |x|, ∀x ∈ H, and then, define the Banach space (2) (  More details can be seen in [18]. In order to obtain the existence and uniqueness of the solution of (1), we introduce the following conditions.
In this paper, we assume there existence of a constant k such that max k 1 (t), k 2 (t) ≤ k.
First, we give an important lemma, which is a type of promotion form of Gronwall's inequality with singular kernels.

and suppose u(t) is nonnegative and locally in-
on this interval. en,

Existence and Uniqueness
Consider the integral form of equation (1): Under the assumptions of H1 and H2, we will prove the existence and uniqueness of solution for the above equation.
Theorem 1. Assume that H1 and H2 hold; then, for ∀x 0 ∈ L 2 (Ω, H), equation (10) has a unique solution We will proof the theorem by several steps. ρ)). Using the following inequality, We see that For I 2 , applying Cauchy-Schwarz's inequality and H2, it follows For I 3 , by It o's isometry formula and H2, we have Combining the above estimate results, we finally obtain where With the help of Lemma 1, it follows for ∀t ∈ [0, T], and E 2α− 1,1 (·) is a two-parameter function of the Mittag-Leffler type [21]. en, For J 1 , we have By the Cauchy-Schwartz inequality, For J 12 , we have For J 2 , using the It o isometry formula, in the similar way as J 1 , we can prove that Results of J 1 and J 2 combined together show that which implied X t ∈ C([0, T], L 2 (Ω; H)) for each fixed μ ∈ C([0, T], (M c 2 , ρ)).
as t ⟶ s. By the definition of ρ, we have which implies Hence, we verify that L( In the following, we will show that the operator Ψ has a unique fixed point in C([0, T], (M c 2 , ρ)). Take μ, ] ∈ C([0, T], (M c 2 , ρ)), and let X μ (t) and X ] (t) be the corresponding solutions of the following equations: (40) us, 6 Journal of Mathematics After simple calculation, we have that Select the appropriate T � T 0 > 0, such that en, it follows By the definition of ρ(μ, ]) and D 2 T (μ, ]), we can obtain Taking sup-norm on both sides, we obtain Combine this result with equation (44), and we finally derive Since Ψ is a contraction in C([0, T 0 ], (M c 2 (H), ρ)), it has a unique fixed point. us, equation (10)

An Averaging Principle
In this section, we study an averaging principle for the following distribution dependent fractional stochastic differential equations in H: where x 0 ∈ L 2 (Ω; H). We will show that the solution of (48) will be approximated by the following simpler or averaged process under certain conditions: Equation (49) is called the averaged equation for (48). Now, we prove that the solution of (49) converges to the solution of the original equation (48) under the following additional conditions. H3: Remark 1. Note that when we take α � 1, then this condition is consistence with the classic case, see [4]. Theorem 2. Let H1 − H4 hold. en, for ∀δ 1 > 0, there exist constants L > 0, ϵ 1 ∈ (0, ϵ 0 ] and β ∈ (0, 1) such that, for any ϵ ∈ (0, ϵ 1 ], 1/2 < α < 1, we have Let us consider By the arithmetic inequality, it follows that For Q 1 , we have