Averaging principle for two-time-scale stochastic differential equations with correlated noise

1 Introduction

Almost all dynamical systems in sciences, such as materials science, fluid dynamics, climate dynamics, celestial mechanics, and radiophysics, have a time hierarchy where the components evolve at quite different rates. In other words, some components evolve rapidly, while others vary slowly. Moreover, all important and interesting systems are essentially nonlinear and disturbed by inner and outer noises. Thus, to explore the interactions between multiscale, nonlinearity and uncertainty, many authors propose a lot of different methods in which the theory of averaging principle stands out. The averaging principle provides an effective tool to explore the asymptotic behavior of the multiscale dynamical systems.

The primary target of the averaging principle is to find the averaged equation, when the scale parameter ε tends to 0, for the slow component x of original multiscale system, and then to demonstrate the approximation between x and the solution of this averaged equation.

The averaging principle was initiated by Bogoliubov and Mitropolsky [1] for ordinary differential equations and then this theory is developed to deal with partial differential equations (PDEs). It is well known that realistic models incorporate uncertainty as an indispensable ingredient. The first result in this direction for stochastic differential equations (SDEs) was obtained by Khasminskii [2], which demonstrated the correctness of averaging principle in a weak sense. Now, the averaging principle for stochastic systems is attracting more and more interest and has a wide range of applications, see, e.g., [3,4, 5,6,7, 8,9] and references therein for the SDEs and [10,11, 12,13,14, 15,16,17, 18,19,20, 21,22] and references therein for stochastic partial differential equations (SPDEs).

However, as far as we know, all these works assume that the noises driving fast and slow components are independent. No result has been obtained for stochastic dynamical systems with two timescales as the noises are correlated. In order to fill the gap in this field of research, we consider a stochastic slow-fast dynamical system where the noise driving the fast components is correlated with the noise driving the slow components. More exactly, we consider the following SDEs driven by correlated noises on Z :

where the small parameter ε > 0 is the ratio between the fast component y and the slow component x . B 1 and B 2 are independent standard Brownian motions. We take Z = R d , thus x ∈ R l , y ∈ R d − l , or x ∈ T l , y ∈ T d − l , then we have Z = T d . T d denotes the d -dimensional unit torus.

This article is organized as follows. In Section 2, some notations, hypotheses, and useful results we need in later sections are introduced. We present the main result of this article in Section 3, and the proof of this article is given in Section 4. An example is given to illustrate our result in Section 5.

2 Preliminaries

For the convenience of the following description, we first introduce some notations. For two d × d matrices A = ( a i j ) , B = ( b i j ) denote the inner product by A : B = t r ( A T B ) = ∑ i , j a i j b i j . Note that S : T = S T : T = 1 2 ( S + S T ) : T , if matrix T is symmetric. For vectors a , b , c ∈ R d , define the outer product, which is a matrix, of a and b by ( a ⊗ b ) c = ( b ⋅ c ) a .

The drift and diffusion coefficients should satisfy some appropriate conditions to derive the existence and uniqueness of the solution to the SDEs (1.1) and (1.2), as is stated in the following result (Theorem 6.1 in [23]):

The backward Kolmogorov equation for equations (1.1) and (1.2) is

where

with

The significance of the backward Kolmogorov equation above is displayed in the proof of the main result of this work in Section 4. Roughly speaking, we first write the solution to the backward Kolmogorov equation above in multiple-scale expansion. Then we identify the PDE satisfied by the main term of the expansion. Through the correspondence between the SDE and its backward Kolmogorov equation, we can finally derive a simplified equation to approximate the dynamics of x .

The adjoint operator of ℒ 0 is denoted by ℒ 0 ∗ . We impose the following assumptions:

A1: B ( x , y ) is a strictly and uniformly positive-definite matrix;

A2: ℒ 0 satisfies the Fredholm alternative. To be exact, let ℒ 0 : H → H be an operator in H . Then

(i) Either

have unique solutions for every γ , Γ ∈ H ;

or

(ii) the corresponding homogeneous equations

satisfy

where N ( ℒ 0 ) and N ( ℒ 0 ∗ ) denote the null spaces of ℒ 0 and ℒ 0 ∗ , respectively. In the latter case, then equations (2.8) and (2.9) have a solution iff

and

A3 (Centering condition): ∫ Y f 1 ( x , y ) ρ ∞ ( y ; x ) d y = 0 , ∀ x ∈ X .

With the help of the assumption (A1), the following result (Theorem 6.16 in [23]) holds:

3 Main result

As preparation for the main result of this article, we define the cell problem as follows:

This is a PDE in y , with x as a parameter. The solution Φ ( x , y ) will be crucial to the derivation in the next section. By assumptions (A2) and (A3), the existence and uniqueness of the solution to the cell problem (3.1) is guaranteed.

Then define ρ by

and σ ( x ) by

where

Now we state our main result, the derivation of which is in the next section.

Note that ρ and σ in equation (3.6) for X both depend on f 0 in the x equation.

4 Derivation

In this section, we give the derivation of Theorem 3.1. By multiscale expansion, we seek the solution of (2.1) with the form

Substitute this expansion into (2.1) and equate the coefficients of equal powers in ε . The first three of the hierarchy of equations are

By (2.12), we can deduce, from equation (4.2), v 0 ( x , y , t ) is independent of y , i.e., v 0 = v 0 ( x , t ) . As for equation (4.3), the solvability condition is satisfied by assumption (A3). From the expression of (2.4), we have

Equation (4.3) becomes

The general solution of (4.5) has the form

since ℒ 0 can be viewed as a differential operator in y with x being a parameter.

We set the function Φ 1 to 0 because it plays no role in what follows. Thus, solution v 1 is represented as a linear operator acting on v 0 . This form for v 1 is a useful representation since we aim to find a closed equation for v 0 . Substituting for v 1 in (4.5) indicates that Φ is the solution to the cell problem (3.1). The assumptions (A2) and (A3) ensure the existence of the solution to the cell problem and the normalization condition, the second equation in (3.1), makes it unique. The right-hand side equation (4.4) becomes

Hence for each fixed x , solvability of (4.4) leads to

Using the symbols we introduce, the first term on the right-hand side of the equation above is

As for the term I 2 , note that

The last equality is from the definition of the inner product between matrices.

By direct calculation, we have

and

The last equality holds when β α 1 T is symmetric. The first term of the right-hand side of the last equality is from the property A : ( B ⋅ c ) = ( A : B ) ⋅ c . The second term of the right-hand side of the last equality is from the property A : ( B C ) = ( A B ) : C while A , C are both symmetric matrices.

Hence, I 2 = I 3 + I 4 where

and

Thus,

By equations (4.7), (4.8), and (4.15), we obtain the following equation:

This is exactly the backward Kolmogorov equation corresponding to the reduced dynamics given in (3.5). This completes the proof.

5 Example

To illustrate our result, consider the following system in T 2 :

Hence, we have the corresponding functions f 1 ( x , y ) = ( 1 − y 2 ) x , f 2 ( x , y ) = x y , α 1 ( x , y ) = x , α 2 ( x , y ) = y , g 1 ( x , y ) = α y , g 2 ( x , y ) = 0 , β ( x , y ) = 2 α .

In this case, ℒ 0 = − α y ∂ ∂ y + α ∂ 2 ∂ y 2 and the cell problem becomes

This equation has a unique centered solution

It is well known that the second and fourth moments take values 1 and 3, respectively, under the standard normal distribution, by which we obtain

Thus, we derive the simplified equation as follows to approximate the slow component x ( t ) :

where W ( t ) is a standard Brownian motion.

6 Conclusion

In this article, we study the averaging principle for systems of SDEs with two widely separated time scales, which is driven by correlated noises. By means of perturbation expansion of the solution to the backward Kolmogorov equation and consequent elimination of variables, we obtain the backward equation corresponding to the reduced simplified system. The solution to reduced SDE (3.6) approximates the slow variable x ( t ) of the initial systems (1.1) and (1.2) in the weak sense.