Strong Convergence of Neutral Stochastic Functional 1 Diﬀerential Equations with Two Time-Scales

The purpose of this paper is to discuss the strong convergence of neutral stochastic 6 functional diﬀerential equations (NSFDEs) with two time-scales. The existence and 7 uniqueness of invariant measure of the fast component is proved by using Wasserstein 8 distance and the stability-in-distribution argument. The strong convergence between 9 the slow component and the averaged component is also obtained by the the averaging 10 principle in the spirit of Khasminskii’s approach.

3 Generic constants will be denoted by c, we use the shorthand notation a b to mean a ≤ cb, 4 we use a T b to emphasize the constant c depends on T. 5 Let ε ∈ (0, 1), we consider a class of NSFDEs with two time-scales with the initial value Y ε 0 = η ∈ C , where b 1 : C × C → R n , b 2 : C × R n × R n → R n , 6 σ 1 : C → R n×m , σ 2 : C × R n × R n → R n×m are Gâteaux differentiable, D 1 : C → R n , 7 D 2 : R n → R n are measurable, locally bounded and continuous, (W 1 (t)) t≥0 and (W 2 (t)) t≥0 8 are two mutually independent m-dimensional Brownian motions defined on a probability 9 space (Ω, F , P), equipped with a reference family (F t ) t≥0 satisfying the usual conditions 10 (i.e., for each t ≥ 0, F t = F t+ := s>t F s , and F 0 contains all P-null sets). X ε (t) is called 11 the slow component, while Y ε (t) is called the fast component.

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The first main result in this paper is stated as below which is concerned with ergodicity 13 of the frozen Eq. (2.3) .
14 Theorem 2.1. Under (A1)-(A4), Y ζ t (η) has a unique invariant measure µ ζ , and there exists λ > 0 such that The next proposition, which plays a crucial role in discussing strong convergence for the 15 averaging principle, states that b 1 enjoys a Lipschitz property.

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Our main aim is to discuss the strong deviation between the slow component X ε (t) and the averaged component X(t), which satisfies the following NSFDE Proof of Theorem 2.1. The proof is rather technical so we divide it into six steps.
Observe that the laws of Y ζ t 2 (η) and Y ζ from (A3) and (A4) that By carrying out a similar argument to obtain (3.11), we have (3.16) In the same way as (3.12) and (3.13), we arrive at as required. 1 Step 4. Let η, η ∈ C , we prove that Consider the difference of the solution process of (2.3) starting from differential initial value. It follows that By the Itô formula and the fundamental inequality (3.6), it follows from (A3) and (A4) that Following the steps of (3.1), we obtain Also, by the Itô formula and the B-D-G inequality, one gives (3.18) In the same way as (3.12) and (3.13), we derive for any t ≥ 0, Sept 5. We shall show the existence and uniqueness of invariant measure of Y ζ t . 1 Let P(C ) be the set of all probability measures on C . d 2 denotes the L 2 -Wasserstein distance on P(C ) induced by the bounded distance ρ(ξ, η) : where C (µ 1 , µ 2 ) is the set of all coupling probability measures with marginals µ 1 and µ 2 .
which goes to zero as t 1 (hence t 2 ) tends to ∞. Therefore, {P ζ,η t } t≥0 is a Cauchy sequence w.r.t. the distance d 2 . By the completeness of P(C ) w.r.t. the distance d 2 , there is µ ζ η ∈ P(C ) such that Moreover, for fixed ζ ∈ C and arbitrary η, η ∈ C , observing that and using (3.17), we obtain for any η, η ∈ C and frozen ζ ∈ C The existence and uniqueness of invariant measure of Y ζ t follows by (3.19) and (3.21). 1 Step 6. We are now going to prove (2.4).

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Let T > 0 be fixed and set δ := τ N ∈ (0, 1) for a sufficiently large positive integer N . For any t ∈ [0, T ], consider the following auxiliary two-time-scale systems of NSFDEs where t δ := t/δ δ, the nearest breakpoint preceding 6 t, with t/δ being the integer part of t/δ.

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Lemma 4.2. Assume that (A1) and (A2) hold and suppose further ε/δ ∈ (0, 1). Then, there exists β > 0 such that Proof. Note that In view of Hölder's inequality, B-D-G's inequality, it follows from (2.1) and (A4) that By using Lemma 4.1 and (A1), one gives Therefore, to finish the argument of Lemma 4.2, it suffices to show that there exists β > 0 such that In the sequel, we shall claim (4.8) by an induction argument. For any t ∈ [0, τ ), due to By means of Itô's formula and B-D-G's inequality, together with Y ε (t δ ) = Y ε (t δ ), we obtain from (A2) that Consequently, we conclude that (4.9) This, combining Lemma 4.1 with Gronwall's inequality, gives that for some c > 0. Next, for any t ∈ [τ, 2τ ), thanks to (4.10), it is immediate to note that By using Itô's formula and B-D-G's inequality again, for any t ∈ [τ, 2τ ), we deduce from (4.10) that and, Thus, the Gronwall inequality reads where we have used ε/δ ∈ (0, 1) in the second step . Finally, (4.8) follows by repeating the 1 previous procedure.

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The following consequence explores a uniform estimate w.r.t. the parameter ε for the 3 segment process associated with the auxiliary fast motion.

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Lemma 4.3. Assume that (A1) and (A3) hold. Then, there exists C T > 0, independent of ε, such that (4.11) sup where we used the fact that W (t) := 1 √ ε W 2 (εt) is a Brownian motion. For fixed ε > 0 and Observe that (4.12) can be rewritten as Then, following an argument to deduce (3.9), for any s > 0 we can deduce that In particular, taking s = t/ε we arrive at This, together with (4.3), yields that for some C T > 0. Observe from (4.8) and Höder's inequality that Next, taking δ = ε(− ln ε) 1 2 in the estimate above and letting y = (− ln ε) Then, the desired assertion follows since the leading term e y 2 (e −y 2 y) p−2 2 e βy → 0 as y ↑ ∞ 1 whenever p > 4. In what follows, let t ∈ [0, T ] be arbitrary and assume p > 4. By using the inequality (4.7), we have that for any > 0 which goes to zero by taking p > 4 and letting y ↑ ∞. 1 Next, we intend to claim (4.15). Set Applying Hölder's inequality, B-D-G's inequality, Lipschitz property of b 1 due to Corollary 2.2, and Lemma 4.1, we derive that which, together with Gronwall's inequality, leads to (4.16) where we have utilized the fact that Γ p (t, δ, ε) is nondecreasing with respect to t. By a comparison (4.15) with (4.16), we need only to prove (4.17) Γ p (t, δ, ε) ε δ ν for some ν ∈ (0, 1).