Moderate Deviations for Stochastic Fractional Heat Equation Driven by Fractional Noise

Since the work of Freidlin and Wentzell [1], the large deviation principle (LDP) has been extensively developed for small noise systems and other types of models (such as interacting particle systems) (see [2–7]). Cardon-Weber [2] proved a LDP for a Burgers-type SPDE driven by white noise. Marquez-Carreras and Sarra [3] proved a LDP for a stochastic heat equation with spatially correlated noise, and Mellali and Mellouk [4] extended Marquez-Carreras and Sarra’s [3] to a fractional operator. Jiang et al. [5] proved a LDP for a fourth-order stochastic heat equation driven by fractional noise. Budhiraja et al. [6] studied large deviation properties of systems of weakly interacting particles. Budhiraja et al. [7] proved a large deviation for Brownian particle systems with killing. Similar to the large deviation, the moderate deviation problems also come from the theory of statistical inference. Using the moderate deviation principle (MDP), we can get the rate of convergence and an important method to construct asymptotic confidence intervals, for example, Liming [8], Guillin and Liptser [9], Cattani and Ciancio [10], and other references therein. There are also many works about MDP about stochastic (partial) differential equations; some surveys and literatures could be found in Budhiraja et al. [11], Wang and Zhang [12], Li et al. [13], Yang and Jiang [14], and the references therein. On the other hand, fractional equations have attracted many physicists and mathematicians due to various applications in risk management, image analysis, and statistical mechanics (see Droniou and Imbert [15], Bakhoum and Toma [16], Levy and Pinchas [17], Mardani et al. [18], Niculescu et al. [19], Paun [20], and Pinchas [21] for a survey of applications). Stochastic partial differential equations involving a fractional Laplacian operator have been studied by many authors; see Mueller [22], Wu [23], Liu et al. [24], Wu [25], and the references therein. Motived above, we investigated the moderate deviations about the stochastic fractional heat equation with fractional noise as follows:


Introduction
Since the work of Freidlin and Wentzell [1], the large deviation principle (LDP) has been extensively developed for small noise systems and other types of models (such as interacting particle systems) (see [2][3][4][5][6][7]). Cardon-Weber [2] proved a LDP for a Burgers-type SPDE driven by white noise.Marquez-Carreras and Sarra [3] proved a LDP for a stochastic heat equation with spatially correlated noise, and Mellali and Mellouk [4] extended Marquez-Carreras and Sarra's [3] to a fractional operator.Jiang et al. [5] proved a LDP for a fourth-order stochastic heat equation driven by fractional noise.Budhiraja et al. [6] studied large deviation properties of systems of weakly interacting particles.Budhiraja et al. [7] proved a large deviation for Brownian particle systems with killing.
Similar to the large deviation, the moderate deviation problems also come from the theory of statistical inference.Using the moderate deviation principle (MDP), we can get the rate of convergence and an important method to construct asymptotic confidence intervals, for example, Liming [8], Guillin and Liptser [9], Cattani and Ciancio [10], and other references therein.There are also many works about MDP about stochastic (partial) differential equations; some surveys and literatures could be found in Budhiraja et al. [11], Wang and Zhang [12], Li et al. [13], Yang and Jiang [14], and the references therein.On the other hand, fractional equations have attracted many physicists and mathematicians due to various applications in risk management, image analysis, and statistical mechanics (see Droniou and Imbert [15], Bakhoum and Toma [16], Levy and Pinchas [17], Mardani et al. [18], Niculescu et al. [19], Paun [20], and Pinchas [21] for a survey of applications).Stochastic partial differential equations involving a fractional Laplacian operator have been studied by many authors; see Mueller [22], Wu [23], Liu et al. [24], Wu [25], and the references therein.
Motived above, we investigated the moderate deviations about the stochastic fractional heat equation with fractional noise as follows: where t ∈ 0, T , x ∈ D = 0, 1 , D δ,α is the fractional Laplacian operator which is defined in Appendix, and B H dt, dx denotes a fractional noise which is fractional in time and white in space with Hurst parameter H ∈ 1/2, 1 ; that is, B H is a mean zero Gaussian random field on 0, T × D with covariance.
Cov B H t, x B H s, y = Assume that the coefficients satisfy the following.
Assumption 1. Function f is Lipschitz; that is, there exist an m > 0 satisfying Under the conditions of Assumption 1, (1) possesses a unique solution in the sense of Walsh [26] as follows: As the parameter ε → 0, the solution v ε t, x of (1) will tend to v 0 t, x which is the solution to the following equation: This paper mainly devotes to investigate the deviations of v ε from the deterministic solution v 0 , as ε → 0, that is, the asymptotic behavior of the trajectories.
where a ε is the same deviation scale that strongly influences the asymptotic behavior of V ε .If a ε = 1/ ε, we are in the domain of large deviation estimate, which can be proved similarly to Jiang et al. [5].
The case a ε ≡ 1 provides the central limit theorem.As ε↓0, we will prove that v ε − v 0 / ε converges to a random field in this paper.
To fill the gap between scale a ε = 1 and scale a ε = 1/ ε, we mainly devote to the moderate deviation when the scale satisfies the following: This paper is organized as follows.In Section 2, the definition of the fractional noise B H ds, dz is given.In Section 3, the main result is given and proved.In Appendix, some results about the Green kernel are given.

Fractional Noise
Let H ∈ 1/2, 1 , and B H 0, t × A t,A ∈ 0,T ×B ℝ is a centered Gaussian family of random variables with the covariance satisfying , where |A| denotes the Lebesgue measure of the set A ∈ B ℝ and B ℝ denotes the class of Borel sets in ℝ.
We denote φ as the set of step functions on 0, T × ℝ.Let H be the Hilbert space defined as the closure of φ with respect to the scalar product.
According to Nualart and Ouknine [27], the mapping 1 0,t ×A → B H t, A can be extended to an isometry between H and the Gaussian space H 1 B H associated with B H and denoted by Define the linear operator K * H φ↦L 2 0, T by

11
where K H is defined by Moreover, K H satisfies the following: Then, since one can get where ϕ and ψ in φ are any step functions.So the operator K * H gives an isometry between the Hilbert space ℋ and L 2 0, T × 0, 1 .Hence, W t, A , t ∈ 0, T , A ∈ ℬ 0, 1 defined by is a space-time white noise, and B H has the following form: Therefore, the mild formulation of (4) has the following form: That is, the last term of ( 4) is equal to The following embedding proposition is given by Nualart and Ouknine [27].
More precisely, for any Borel measurable subset B of where B o and B denote the interior and the closure of B, respectively.
We furthermore suppose that the coefficients satisfy the following.
Assumption 2. f is differentiable, and the derivative f ′ of f is Lipschitz.That is to say, there exist positive constant m and m ′ which satisfy the following: Together with the Lipschitz of f , we conclude that Now, we give the following central limit theorem.
Theorem 2. Let f and its derivative f ′ satisfy Assumptions 1 and 2.Then, for p ≥ 1, v e − v 0 / ε converges in L p to a random field U on C γ 0, T × D with 0 < γ < 1, determined by Let the function U e be the solution to the following partial differential equation: Under Assumptions 1 and 2, by Theorem 1, one can get U/a ε which satisfies large deviation principles on C γ 0, T × D with the speed e 2 ε and the good rate function satisfies the following: Now, the second result is given as follows: Under the Assumptions1and2, then, the random field 1/ εa ε v ε − v 0 satisfies a large deviation principle on the space C γ 0, T × D with speed a 2 ε and the good rate function I φ defined by (36), where 0 < γ < 1.

The Proof of the Main Results
. We first prove Theorem 1.
Our proof is based on the following proposition (see Doss and Priouret [28]).
(3) For any R, δ, a > 0, there exist ρ > 0 and ϵ 0 > 0 satisfying I 1 ≤ a and ϵ ≤ ϵ 0 for all h ∈ E 1 , and Then, X ϵ 2 , ϵ > 0 satisfies a LDP with the rate functions To prove Theorem 1, one only needs to prove (i) Under some topology, Z e : e e E ≤ a → C 0, T , L p D is continuous for any a > 0.
Proof.One only needs to prove that for fixed a > 0, e, h ∈ ∥e∥ E ≤ a , Note that Using (A.5) in Appendix with p = q = 2, then ρ = 1, one can get 6 Complexity Now, we deal with II t, x , together with Recalling K * H is defined by (11), one can get Using Gronwall's inequality, we can get The proof of the theorem is completed.
We now prove the Freidin-Wentizell inequality as follows: For e ∈ E and ϵ > 0, we define and Using Girsanov's theorem, the process W is a Brownian sheet under P. Suppose v ε t, x is a solution of (1) under p.Then, Now, one can prove (36).Note that, under p, then, So under p, by Gronwall's Lemma, one can get 7 Complexity Now, one can change (34) the proof to the following theorem.
Theorem 5. Suppose e ∈ E and e ∈ E ≤ a.For each R > 0, η > 0, and e ∈ E, there exists a constant δ > 0 satisfying In the following, we give a key Lemma to prove Theorem 1, which is similar to Candon-Weber [2], and the proof is omitted.
Proof of Theorem 2.
To this end, we need to prove that Assumptions 3 and 4 are satisfied for

79
and Using Taylor's formula, there exists a ξ ε t, x such that Note that f ′ is Lipschitz continuous and ξ ε t, x ∈ 0, 1 ; one can get 83 Using Hölder's inequality, for p < 2 and 1/α < q < α, one can get

84
where 1/p + 1/q = 1.Together with (5) of Lemma A.1 and Proposition 1, there exists a constant C p, m, m′, T only depending on p, m, m′, T satisfying Since f ′ ≤ M, for p > 2, together with Hölder's inequality and (5) of Lemma A.1, we can get
The operator D δ,α is a closed, densely defined operator on L 2 ℝ , and it is the infinitesimal generator of a semigroup which is in general not symmetric and not a contraction.This operator is a generalization of various well-known operators, such as the Laplacian operator (when α = 2), the inverse of the generalized Riesz-Feller potential (when α > 2), and the lim sup ε→0 e −2 ε log P III T,μ ,θ Riemann-Liouville differential operator (when δ = 2 + α 2 or δ = α − α ).It is self-adjoint only when δ = 0, and in this case, it coincides with the fractional power of the Laplacian.We refer the readers to Debbi and Dozzi [31] for more details about this operator.According to Komatsu [32], D δ,α can be represented for 1 < α < 2, by and for 0 < α < 1, by where κ δ − are κ δ + two nonnegative constants satisfying κ δ − + κ δ + > 0, φ is a smooth function for which the integral exists, and φ′ is its derivative.This representation identifies it as the infinitesimal generator for a nonsymmetric α-stable Lévy process.Suppose G δ α t, x is the fundamental solution to the following equation: where δ 0 • is the Dirac distribution.Using Fourier transform, one can get G δ α t, x which is given by Let us list some known facts on G δ α t, x which will be used later on (see, e.g., Debbi and Dozzi [31]).
Proof.For any x, y ∈ ℝ and t ∈ 0, T , 13 Complexity By the mean-value theorem, for θ ∈ 0, 1 , one can get that and Similarly, we can check Hence, the inequality (A.9) holds.As for the inequality (A.10), for any x ∈ ℝ and t, s ∈ 0, T , By mean-value theorem, it holds that Note that Hence, one can attain So The proof of the lemma is completed.
Lemma A.3.Suppose p ∈ 1, ∞ , q ∈ 1, p , and ρ ∈ 1, ∞ satisfying Suppose G δ α = G δ α t, x − z is the Green kernel, Q = ∂/∂z G δ α or Q = G δ α with t, x, z ∈ 0, T × ℝ × ℝ.We define I by with u ∈ L 1 0, T ; L q .Then, I L 1 0, T ; L q → L ∞ 0, T ; L q is a bounded linear operator which satisfies the following: The proof of ( 2) is omitted since it is similar to case (1).The proof of this lemma is completed.

Theorem 4 .
When the level set e E ≤ a is endowed with the topology of uniform convergence on 0, T × D , Z e : e e E ≤ a → C 0, T , L p D 55