Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise

In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system driven by colored noise with a nonlinear diffusion term. We demonstrate that the colored noise is much easier to deal with than the white noise for studying the pathwise dynamics of stochastic systems. In addition, we show the attractors of the random FitzHugh-Nagumo system driven by a linear multiplicative colored noise converge to that of the corresponding stochastic system driven by a linear multiplicative white noise.


1.
Introduction. This paper is concerned with the asymptotic behavior of the random non-autonomous FitzHugh-Nagumo system driven by a colored noise in a bounded domain O: ∂u ∂t − ∆u + αv = f (t, x, u) + g(t, x) + R(t, x, u)ζ δ (θ t ω), ∂v ∂t + σv − βu = h(t, x) + λζ δ (θ t ω)v, (1) where α, β, σ and λ are positive constants, g and h are in L 2 loc (R, L 2 (O)), f and R are nonlinear functions which satisfy certain dissipative conditions. Given δ > 0, the process ζ δ in (1) is an Ornstein-Uhlenbeck (O-U) process (also known as a colored noise). To describe this process, we introduce a probability space (Ω, F, P), where Ω = {ω ∈ C(R, R) : ω(0) = 0} with the open compact topology, F is its Borel σ-algebra, and P is the Wiener measure. The classical transformation {θ t } t∈R on Ω is given by θ t ω(·) = ω(t + ·) − ω(t) for ω ∈ Ω. Let W be a two-sided real-valued Wiener process on (Ω, F, P). For each δ > 0, define a random variable ζ δ : Ω → R by ζ δ (ω) = 1 δ 0 −∞ e s δ dW . Then the process O δ (t, ω) = ζ δ (θ t ω) is called an O-U process, which is a stationary Gaussian process with E(ζ δ ) = 0 and is the unique stationary solution of the stochastic equation The O-U process O δ is also called a colored noise because its power spectrum is not flat compared to the white noise, see for instance [4,25,42,43,55]. Furthermore, the O-U process is the only existing Markovian Gaussian colored noise, see, e.g., [17] and [43]. In general, the Wiener process W can be chosen as a stochastic process to represent the position of the Brownian particle, but the velocity of the particle cannot be obtained from the Wiener process because of the nowhere differentiability of the sample paths of W . In such a case, the O-U process was originally constructed in [42,55] to approximately describe the stochastic behavior of the velocity and hence it can be further used to determine the position of the particle. On the other hand, as demonstrated in [43], in many complex systems, stochastic fluctuations are actually correlated and hence should be modeled by colored noise rather than white noise.
Indeed, one of the most crucial issues in studying stochastic dynamics arises from the modeling of the random forcing. To represent such random forcing, we need to consider two time scales: the time scale τ d of the deterministic system and the time scale τ r of the random forcing. The stochastic forcing is modeled in different ways based on the ratio of τ r /τ d . If τ r /τ d 1, the dynamical system is very slow with respect to the temporal variability of its random drivers, and hence the random forcing could be modeled as white noise. If τ r /τ d 1, then the dynamics of the system is sensitive to the autocorrelation of the random forcing, and hence the random forcing should be modeled by colored noise. Based on these considerations, the colored noise has been used in many publications to study the dynamics of physical and biological systems, see, e.g., [4,25,34,35,38,42,43,55] and the references therein.
In this paper, we will study the dynamics of system (1) driven by colored noise. For a wide class of nonlinear functions R, we will prove the random system (1) is pathwise well-posed in L 2 (O) = L 2 (O) × L 2 (O) and hence generates a continuous non-autonomous cocycle. Moreover, this cocycle possesses a unique tempered random attractor in L 2 (O) (see Theorem 2.8). This is in sharp contrast with the corresponding stochastic system driven by a white noise: ∂u ∂t − ∆u + αv = f (t, x, u) + g(t, x) + R(t, x, u) • dW dt , ∂v ∂t + σv − βu = h(t, x) + λv • dW dt , where the symbol • indicates the system is understood in the sense of Stratonovich's integration. As far as the authors are aware, currently, we can only define a random dynamical system for (2) when the diffusion term R(·, ·, u) is a linear function in u ∈ R. In other words, for a general nonlinear function R, we are unable to define a random dynamical system for (2) and hence cannot investigate the dynamics of the stochastic equations by the random dynamical systems approach. This is the reason why there is no result available in the literature on the existence of random attractors for (2) with a nonlinear function R. Our results indicate that the colored noise is much easier to handle than the white noise for studying pathwise dynamics of stochastic equations. In the present paper, when R(t, x, u) ≡ u, we will also investigate the limiting behavior of system (1) as δ → 0. In this special case, we will show the solutions of (1) converges to that of (2) in L 2 (O) as δ → 0 (see Corollary 3), and the random attractors of (1) converge to that of (2) in terms of the Hausdorff semi-distance in L 2 (O) as δ → 0 (see Theorem 4.10). This demonstrates that, when R(t, x, u) ≡ u, the random system (1) is is closely related to the stochastic system (2) for sufficiently small δ > 0.
We mention that colored noise has already been extensively used in the literature to study the solutions of random equations, see e.g., [1,25,34,35,38,43] and the references therein. However, there is no result available regarding the existence of random attractors for equations like (1) driven by colored noise. We also remark that the Wong-Zakai approximations can be used to study the solutions and dynamics of stochastic equations, for which the reader is referred to [33,36,45,46,47,48,56,57,39] for more details.
This paper is organized as follows. In the next section, we prove the existence and uniqueness of tempered random attractors for system (1) with a nonlinear diffusion term R. We then prove the existence of such attractors for the stochastic system (2) when R(t, x, u) ≡ u. In the last section, we prove the convergence of solutions and random attractors of system (1) with R(t, x, u) ≡ u, as δ → 0.
Hereafter, we will denote the inner product and norm of L 2 (O) by (·, ·) and · , respectively. We also write . Similarly, we write F(t, x) = (g(t, x), h(t, x)) for t ∈ R and x ∈ O, and F(t) 2 = g(t) 2 + h(t) 2 . The letters c and c i are used for positive constants whose values may change from line to line. 2. Attractors of random FitzHugh-Nagumo systems. In this section, we study the dynamics of the random FitzHugh-Nagumo system driven by a colored noise. We first define a continuous non-autonomous cocycle for the system, and then prove the existence of pullback random attractors in L 2 (O) for a wide class of nonlinear functions f and R.
2.1. Definition of continuous cocycles. Let O be a bounded domain in R n and τ, δ ∈ R with 0 < δ ≤ 1. Consider the following non-autonomous random FitzHugh-Nagumo system defined in O with homogeneous Dirichlet boundary condition and initial condition: where g, h ∈ L 2 loc (R, L 2 (O)), ζ δ (θ t ω) is a colored noise and f, R : R × O × R → R are continuous functions such that for all t, s ∈ R and x ∈ O, (G) f (t, x, s) is differentiable in x, s and there exist positive constants p > 2, α 1 , α 2 and α 3 such that x, s) is differentiable in x, s and there exist positive constants 2 ≤ q < p, α 4 and α 5 such that ). Recall that there exists a {θ t } t∈R -invariant subset of full measure (see, e.g., [5]), which is still denoted by Ω, such that for ω ∈ Ω, Throughout this paper, for every ω ∈ Ω and δ ∈ (0, 1], we write Then ζ δ (θ t ω) is the so-called O-U process (also known as colored noise) with E(ζ δ ) = 0. In addition, this process has the following properties.
For later sections, we now prove the following uniform convergence of colored noise on a finite interval.
Since the following integral is convergent, i.e.
As an immediate consequence of Lemma 2.2 and the continuity of ω, we have the following estimates.
Note that (3) is a deterministic equation parametrized by ω ∈ Ω, by the Galerkin method as in [49], we can prove that for every τ ∈ R, ω ∈ Ω and Furthermore, the solution is continuous with respect to initial conditions in L 2 (O) and is (F, B(L 2 (O)))-measurable in ω ∈ Ω. Then, we can define a continuous cocycle Φ : Let D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} be a family of bounded nonempty subsets of L 2 (O) with the property: for every γ > 0, τ ∈ R and ω ∈ Ω, Such a family is called tempered in L 2 (O). Let D be the collection of all tempered families of bounded nonempty subsets of L 2 (O), i.e.
Recall Poincare's inequality on O: there exists a positive constant λ 1 such that Let κ = min{λ 1 , σ} and κ 0 be a fixed number in (0, κ). Hereafter, we assume and for every positive number c, 2.2. Existence of pullback random attractors. In this subsection, we prove the existence of tempered pullback random attractors for system (3). We first prove the existence of tempered absorbing sets and then show the pullback asymptotic compactness of solutions.
Proof. Taking the inner product of (3) We infer from (G1), (H1) and Young's inequality that and By (50)-(54) and κ = min{λ 1 , σ} we get where Applying Gronwall's Lemma to (2.2) over (r, ϑ) with ϑ ≥ r and for every ω ∈ Ω, we get Now, replacing r by τ − t and ω by θ −τ ω in (56), we get Next, we estimate every term on the right-hand side of (57). First, by Lemma 2.1, we find which implies that there exists s 0 < 0 such that for all s ≤ s 0 , Also we know from (47) The second term on the right-hand side of (57) is well-defined due to (6), and for the first term, since w τ −t ∈ D(τ − t, θ −t ω) and D ∈ D, we have which along with (60) completes the proof.
Also, we can obtain the following estimates from Lemma 2.3 for later purpose.
where w τ −t ∈ D(τ − t, θ −t ω) and M 2 is a positive constant independent of τ and D.

ANHUI GU AND BIXIANG WANG
where w τ −t ∈ D(τ − t, θ −t ω) and M 3 is a positive constant depending on τ and ω.
Next, we will show the asymptotic compactness of solutions of system (3). To that end, we write v = v 1 + v 2 where v 1 and v 2 solve the following systems and dv 2 dt By (7) and (77) we find that for Next, we derive uniform estimates of v 2 in H 1 0 (O) for which we further assume where M 4 is a positive constant independent of τ, ω and D.
We now prove the existence of D-pullback attractors of Φ. Proof. Since Φ has a closed measurable D-pullback absorbing set K ∈ D by Corollary 2 and is D-pullback asymptotically compact in L 2 (O) by Lemma 2.7, then the existence and uniqueness of D-pullback attractor A of Φ follows immediately.
3. Attractors of stochastic FitzHugh-Nagumo system. In this section, we investigate the dynamics of the stochastic FitzHugh-Nagumo system driven by a linear multiplicative noise. More precisely, we will prove the existence and uniqueness of tempered random attractors for the system. The results of this section will be used for studying the limiting behavior of solutions of the random system (3) when δ → 0. Indeed, in the next section, we will prove, under certain conditions, the limiting dynamics of system (3) is governed by that of a stochastic system as δ → 0. Given τ ∈ R, consider the stochastic FitzHugh-Nagumo system where f , g and h are the same as in the previous section.
By Lemma 3.1 we obtain the following estimates. (47) hold. Then for every τ ∈ R, ω ∈ Ω and D ∈ D, there exists T = T (τ, ω, D) > 0 such that for all t ≥ T ,

Lemma 3.2. Suppose (G) and
with L 1 being the same constant as in Lemma 3.1 which is independent of τ and ω.
Proof. The proof is similar to that of Lemma 2.6 and hence omitted here.
Proof. The proof is similar to that of Lemma 2.7, and hence omitted here.
We now in the position to show the existence of D-pullback attractors of Φ 0 .
4. Convergence of random attractors. In this section, we study the limiting behavior of solutions of the random system (3) when δ → 0. Under certain conditions, we will show the solutions and attractors of system (3) converge to that of the corresponding stochastic system when δ → 0. Consider the following random system From now on, we write the solution of (114) as w δ = (u δ , v δ ) to indicate its dependence on δ. Note that system (114) is a special case of (3) and can be obtained by formally replacing W (t) by t 0 ζ δ (θ r ω)dr in (89). We will establish the relations between the solutions of systems (89) and (114) and show that the limiting behavior of system (114) is governed by the stochastic system (89) as δ → 0.
By (114) and (115), we obtain First, we derive the uniform estimates on the solutions of system (116) on finite time intervals.
Next, we derive uniform estimates of the solutions when t → ∞.
Proof. For t ∈ R + , τ ∈ R and ω ∈ Ω, integrating (119) from τ − t to τ , we get where c 2 is a positive constant independent of τ, ω and δ. Due to (7), we find On the other hand, since D ∈ D, again by (7), we see that there exists T = T (τ, ω, D, δ) > 0 such that for all t ≥ T , which together with (4) and (122) completes the proof.
As an immediate consequence of Lemma 4.2, we obtain the following estimates.
with L 5 being the same positive constant as in Lemma 4.2 which is independent of τ , ω and δ.
Proof. Due to (115), we have , which along with Lemma 4.2 implies the desired estimates.
To establish the uniform compactness of random attractors, we need the following estimates.
Proof. The proof is similar to Lemma 2.6. Of course, Lemma 4.2 instead of Lemma 2.3 should be used this time, and the details are omitted.
Recall that for each δ > 0, A δ is the unique D-pullback attractor of Φ δ in L 2 (O). To obtain the uniform compactness of these attractors with respect to δ, we need further estimates on Φ δ,2 as given below.