Quantitative central limit theorems for the parabolic Anderson model driven by colored noises

In this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use It\^o's calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-order Gaussian Poincar\'e inequality to overcome this lack of martingale structure and our work improves the results by Nualart-Zheng (2020 \emph{Electron. J. Probab.}) and Nualart-Song-Zheng (2021 \emph{ALEA, Lat. Am. J. Probab. Math. Stat.}).


Introduction
In the recent years, the study of spatial averages of the solution to certain stochastic partial differential equations (SPDEs) has received growing attention. The paper [16], being the first of its kind, investigated the nonlinear stochastic heat equation on R + × R driven by a space-time white noiseẆ : Consider the stochastic heat equation on R + × R d driven by a Gaussian noiseẆ : where the nonlinearity is encoded into a deterministic Lipschitz continuous function σ : R → R. In Duhamel's formulation (mild formulation), this equation is equivalent to u(t, x) = 1 + t 0 R p t−s (x − y)σ(u(s, y))W (ds, dy), (1.2) where the stochastic integral against W (ds, dy) is an extension of the Itô stochastic integral, and p t (x) = (2πt) −1/2 e −|x| 2 /(2t) for (t, x) ∈ R + × R denotes the heat kernel. One of the main results in [16] provides the following estimate. Suppose also σ(1) = 0, which excludes the trivial case u(t, x) ≡ 1. Let and let d TV (X, Y ) denote the total variation distance between two real random variables X and Y (see (2.11)). Then, it holds that for any t > 0, there is some constant C t , that does not depend on R, such that the following quantitative Central Limit Theorem (CLT) holds where Z ∼ N (0, 1) is a standard normal random variable and σ R (t) = Var(F R (t)) > 0 for each t, R ∈ (0, ∞). The key ideas to obtain (1.3) can be roughly summarized as follows: (i) By the mild formulation (1.2) and applying stochastic Fubini's theorem, one can write where δ denotes the Skorohod integral (the adjoint of the Malliavin derivative operator; see Section 2.1) and V t,R is the random kernel given by (ii) Via standard computations, one can obtain σ R (t) ∼ constant × R 1/2 as R ↑ ∞.
(iii) The Malliavin-Stein bound (c.f. [16,Proposition 2.2]), being the most crucial ingredient, indicates that where DF R (t) denotes the Malliavin derivative of F R (t), which is a random function and belongs to the space L 2 (R + × R) under the setting of [16]. Then the obtention of (1.3) follows from a careful analysis of the inner product DF R (t), V t,R L 2 (R + ×R) .
We remark here that for a general nonlinearity σ, the computations mentioned in points (ii) and (iii) are made possible through applications of the Clark-Ocone formula and Burkholder-Davis-Gundy inequality, which are valid only in the white-in-time setting. The noiseẆ that is white in time, naturally gives arise to a martingale structure so that Itô calculus techniques come into the picture and enable the careful analysis of the variance term in (1.4).
The above general strategy has also been exploited in several other papers, see [3,4,5,17,20,21] for results on stochastic heat equations and see [2,7,28] for results on stochastic wave equations, to name a few. The common feature of these papers is that they consider the case where the driving Gaussian noise is white in time so that the aforementioned strategy of [16] is working very well. To the best of our knowledge, the colored-in-time setting has only been considered in [26,27] for heat equations and in [1] for wave equations.
In the present paper we are interested in the following parabolic Anderson model (that means σ(u) = u) on R + × R d driven by a Gaussian noiseẆ , which is colored in time and space, with flat initial condition: ∂u ∂t = 1 2 ∆u + u ⋄Ẇ , u(0, •) = 1, (1.5) where ⋄ denotes the Wick product (c.f. [9,Section 6.6]). Because we allow the noise to be colored in time, we need to take σ(u) = u, otherwise it is not clear how to show the existence and uniqueness of a solution.
Let us now introduce some notation to better facilitate the discussion as well as to state our main results. Fix a positive integer d. Heuristically,Ẇ = {Ẇ (t, x), (t, x) ∈ R + × R d } will be a centered Gaussian family of random variables with covariance structure given by where γ 0 , γ 1 are (generalized) functions that satisfy one of the following two conditions: is a nonnegative-definite locally integrable function and γ 1 ≥ 0 is the Fourier transform of some nonnegative tempered measure µ on R d (called the spectral measure), satisfying Dalang's condition (see [6]), (1.6) Hypothesis 2. (d = 1) There are H 0 ∈ [ 1 2 , 1) and H 1 ∈ (0, 1 2 ) with H 0 + H 1 > 3 4 , such that where δ is the Dirac delta function at 0 and γ 1 is the Fourier transform of µ(dξ) = c H 1 |ξ| 1−2H 1 dξ with c H 1 = π −1 R (1 − cos x)|x| 2H 1 −2 dx; see (2.8) for the choice of c H 1 . We will call the setting under the Hypothesis 1 the regular case, since the spatial correlation function γ 1 is a function as opposed to the setting under Hypothesis 2. Oppositely, we call the setting under Hypothesis 2 the rough case, because the spatial correlation corresponds to fractional Brownian motion with Hurst index H 1 ∈ (0, 1/2) (thus rougher than the standard Brownian motion or white noise).
In order to define rigorously the noise, we need some definitions. Let C ∞ c (R + ) and C ∞ c (R d ) denote the set of real smooth functions with compact support on R + and R d , respectively. Then, we define Hilbert spaces H 0 and H 1 to be the completion of C ∞ c (R + ) and C ∞ c (R d ) with respect to the inner products which can be also written using the Fourier transform as follows, where φ(t, ξ) and ψ(t, ξ) stand for the Fourier transform in the space variable. We also introduce the following hypotheses that will be used to state our main results.
We remark here that the restriction for β in Hypothesis 3b ensures Dalang's condition (1.6).
With these preliminaries, we consider a centered Gaussian family of random variables for all φ, ψ ∈ H. The family W is called an isonormal Gaussian process over H. Heuristically, the noiseẆ (t, x 1 , . . . , x d ) = ∂ d+1 W (t,x) ∂t∂x 1 ···∂x d is the (formal) derivative of W in time and space and the mild formulation of equation (1.5) is given by where the stochastic integral against W (ds, dy) is a Skorohod integral (see [25,Section 1.3.2]). It has been proved that under either Hypothesis 1 or 2, the parabolic Anderson model (1.5) admits a unique mild solution; see [11,14,15,30]. Due to the temporal correlation in time of the driving noise, we do not have the playground to apply martingale techniques for obtaining central limit theorem for the spatial statistics Fortunately, because of the explicit chaos expansion (see (2.5) and (2.6)), one can express F R (t)/σ R (t), with σ 2 R (t) = Var F R (t) , as a series of multiple stochastic integrals. This series falls into the framework of applying the so-called chaotic CLT. The chaotic CLT roughly means that once we have some control of the tail in the series, it would be enough to show the convergence of each chaos, which can be further proved by using the fourth moment theorems; see [23,Section 6.3] for more details. In fact, in the papers [26,27], the authors investigated the Gaussian fluctuations of F R (t) along this idea and proved the following results. Theorem 1.1. Assume Hypothesis 1 or 2, and let u be the solution to the parabolic Anderson model (1.5). Recall the definition of F R (t) from (1.10) and let σ R (t) = Var(F R (t)). Then the following results hold.
(1) Assume Hypotheses 1 and 3a. Then, for any fixed t ∈ (0, ∞), as R ↑ ∞, (2) Assume Hypotheses 1 and 3b. Then, for any fixed t ∈ (0, ∞), as R ↑ ∞, where g(z) = e z − z − 1 is strictly positive except for z = 0. Then the above limit vanishes if and only if almost surely I z = 0 for almost every z ∈ R. Taking into account the explicit expression of I z = I 1,2 t,t (z) and equation (1.8) in [26,Proposition 1.3], we can conclude that the limit in (1.11) is strictly positive and thus σ R (t) ∼ R 1/2 . Indeed, by (1.8) and L 2continuity, E[u(t, x)u(t, 0)] = E exp(I x ) > 1 for x near 0 so that with positive probability I x = 0 for x near zero.
Note that the above CLT results are of qualitative nature and there are also functional version of these results where the limiting objects are centered Gaussian process with explicit covariance structures. Both CLT in (1) and (3) are chaotic, meaning that each 1 chaos contributes to the Gaussian limit, while CLT in (2) is not chaotic. More precisely, in (2) the first chaotic component, which is Gaussian, dominates the asymptotic behavior as R ↑ ∞; see the above references for more details. Here we point out that the application of the chaotic CLT does not yield the rate of convergence, that is, the error bound like (1.3) is not accessible through this method. Our paper is devoted to deriving quantitative versions of the above CLT results as stated in the following theorem. Then we have for any fixed t, R ∈ (0, ∞), (1.12) where the constant C t > 0 is independent of R and Z ∼ N (0, 1) denotes a standard normal random variable.
In a recent paper [1], the authors face the problem of establishing a quantitative CLT for the hyperbolic Anderson model driven by a colored noise. A basic ingredient in this paper is the so-called second-order Gaussian Poincaré inequality (see Proposition 2.3). With this inequality in mind, it is not difficult to see that, in order to obtain the desired rate of convergence, we need to equip ourselves with fine L p -bounds of the Malliavin derivatives valued at (almost) every space-time points. This will be our approach in the regular case, and the majority of the effort will be allocated to show these L p -bounds, with which we will apply Proposition 2.3 to get the quantitative CLT.
However, in the rough case the spatial correlation γ 1 is a generalized function and if one understands the inner product •, • H using the Fourier transform (1.8), Proposition 2.3 does not fit. This is a highly nontrivial difficulty the we overcome by taking advantage of an equivalent formulation of the inner product •, • H based on fractional calculus (see Section 2.2). Starting from such an equivalent expression, we derive in the rough case another version of the second-order Gaussian Poincaré inequality (see Proposition 2.4) that is better adapted to our purpose. We also refer the readers to Remark 2.5 for a detailed discussion.
Let us complete this section with a few more remarks on (quantitative) CLT in other settings.
1) The authors of [26] study the situation where the noise W is colored in space-time R + × R with the spectral density ϕ satisfying a modified Dalang's condition and the concavity condition: for all x, y ∈ R. Using the chaotic CLT, they are able to establish the CLT results of qualitative nature. It is not clear to us how to derive the moment estimates for Malliavin derivatives in this setting. A more intrinsic problem is that unlike in our rough case, we are not aware of any equivalent real-type expression for the inner product of the underlying Hilbert space H and thus we do not see how to put the potential moment estimates in use.
2) In the recent paper [29], the spatial ergodicity for certain nonlinear stochastic wave equations with spatial dimension not bigger than 3 is established under some mild assumptions. The condition on the driving noise W can be roughly summarized as follows: W is white in time, the spatial correlation satisfies Dalang's condition and the spectral measure has no atom at zero. So far the CLT is open when the spatial dimension is three, while there have been several works [2,7,28] in dimensions 1 or 2.
When the spatial dimension is three, the wave kernel is a measure (not a function any more) and then the Malliavin derivative Du(t, x) of the solution shall be a random measure supported on some sphere, which can not be valued pointwisely.
The rest of the paper is organized as follows: Section 2 contains some preliminaries that will be used in this paper. We study the regular case in Section 3 and leave the rough case to Section 4.

Preliminaries and technical lemmas
In this section, we provide some preliminaries and useful lemmas.
In the first step, we introduce some notation that will be used frequently in this paper. Let n be a positive integer, we make use of notation x n = (x 1 , . . . , x n ) ∈ R nd and t n = (t 1 , . . . , t n ) ∈ R n + . Given x n ∈ R nd , we write x k:n ∈ R n−k+1 short for (x k , . . . , x n ) with k = 1, . . . , n, and t k:n ∈ R n−k+1 + is defined in the same way. Let 0 ≤ s < t < ∞. We put T s,t n = {s n ∈ R n + : s < s 1 < · · · < s n < t} and T t n = T 0,t n . We use c, c 1 , c 2 and c 3 for some positive constants which may vary from line to line. Finally, we write A 1 A 2 if there exists a constant c such that A 1 ≤ cA 2 .

Wiener chaos and parabolic Anderson model
Let H be a Hilbert space of (generalized) functions on R + × R, and let W = {W (h), h ∈ H} be an isonormal Gaussian process over H. Denote by F = σ(W ) the smallest σ-algebra generated by W . Then, any F-measurable and square integrable random variable F can be unique expanded into a series of multiple Itô-Wiener integrals (c.f. [25, Theorem 1.1.2]), where for n = 1, 2, . . . , I n (f n ) is the multiple Itô-Wiener integral of f n , which is a symmetric function on (R + × R d ) n , meaning for all permutation σ on {1, . . . , n} .
In the space-time white noise case, I n (f n ) can be understood as the n-folder iterate Itô-Walsh's integral (c.f. [32]), For any n, we denote by H n the n-th Wiener chaos of W , that is the collection of random variables of the form F = I n (f n ), with f n ∈ H ⊙n . In any fixed Wiener chaos H n , the following inequality of hypercontractivity (c.f. [23, Corollary 2.8.14]) holds for p ≥ 2 and for all n = 1, 2, . . . and F ∈ H n . Assume Hypothesis 1. Set X = H when γ 0 = δ. Then, X = L 2 (R + ; H 1 ), and we have the following lemmas that are very helpful in Section 3.2.
Lemma 2.1. For any nonnegative function f ∈ X ⊗n supported in ([0, t] × R d ) n , we have . Let m, n ≥ 1 be integers and let f and g in X ⊙n and X ⊙m respectively. Then, where I X n denotes the the multiple Itô-Wiener integral with respect to an isonormal Gaussian process over X and f ⊗ r g denotes the r-th contraction between f and g, namely, an element in X n+m−2r defined by In particular, if in addition f, g have disjoint temporal support 2 , then and I X n (f ), I X m (g) are independent. In the rest of this subsection, we provide the definition for the solution to the parabolic Anderson model (1.5) 2)). Then, u is said to be Skorohod integrable with respect to W , if the following series is convergent in L 2 (Ω), is Skorohod integrable and the following equation holds almost surely, It has been proved (c.f. [10, Section 4.1]) that u is a solution to (1.5), if and only if it has the following Wiener chaos expansion where the integral kernels f t,x,n are given by with σ the permutation of {1, . . . , n} such that 0 < s σ(1) < · · · < s σ(n) < t and p t (x) being the heat kernel in R d . The chaos expansion (2.5) and the expression (2.6) will be used frequently in this paper.

Fractional Sobolev spaces and an embedding theorem
In this subsection, we give a basic introduction to fractional Sobolev spaces. They are closely related to H 1 under Hypothesis 2. We also provide an embedding theorem for H 0 . These will be used in Section 4. For a more detailed account on applications of this topic to SPDEs, we refer the readers to papers [11,12,13] and the references therein. Following the notation in [8], given parameters s ∈ (0, 1) and p ≥ 1, the fractional In particular, if p = 2, W s,2 turns out to be a Hilbert space. By using the Fourier transformation, one can write (c.f. Here the constant C(s) is slightly different from that in [8], because we use another version of the Fourier transformation. Therefore, assuming Hypothesis 2, we see immediately that In this paper, we will use both representations of the norm in H 1 via the Fourier transformation and the Gagliardo formulation.
In the next part of this subsection, we introduce an embedding property for the Hilbert space H 0 . Assume Hypothesis 2 with H 0 ∈ ( 1 2 , 1), then there exist a continuous embedding for all f, g ∈ H 0 . Combining this fact and Cauchy-Schwarz's inequality on the Hilbert space By iteration, we can write for all φ, ψ ∈ H ⊗n and n = 1, 2, 3, . . . .

Second-order Gaussian Poincaré inequalities
In this subsection, we provide two versions of the second-order Gaussian Poincaré inequality (see [24,Theorem 1.1] for the first version). As stated in Section 1, they will be used in estimating the total variance distance in regular and rough cases respectively. Denote by D the Malliavin differential operator (c.f. [25, Section 1.2]). Let D 2,4 stand for the set of twice Malliavin differentiable random variables F with We denote by D 2,4 * the set of random variables F ∈ D 2,4 such that we can find versions of the derivatives DF and D 2 F , which are measurable functions on Let F and G be random variables. The total variance distance between F and G is defined by where µ, ν are the probability laws of F and G respectively. The next proposition cited from [1, Proposition 1.8] will be used in the regular case. Proposition 2.3. Assume Hypothesis 1. Let F ∈ D 2,4 * be a random variable with mean zero and standard deviation σ ∈ (0, ∞). Then where Z ∼ N (0, 1) and Inspired by [31, Theorem 2.1], we also have the following proposition, that will be used in the rough case. Proposition 2.4. Assume Hypothesis 2. Let F ∈ D 2,4 * be a random variable with mean zero and standard deviation σ ∈ (0, ∞). Then where Z ∼ N (0, 1) and Proof. We begin with the Malliavin-Stein bound 3 where the two terms on the right-hand side can be dealt with in the same manner. In what follows, we only estimate the second term. Put where we have used the expression (2.7) for the inner product in H 1 . With this notation we can write Therefore, by using Hölder inequality and the contraction property of P t on L 4 (Ω), we get We have the same bound for the first term (2.13) except for the multiplicative constant 2, due to 2 ∞ 0 dte −t = 2. Hence, the proof of Proposition 2.4 is completed.
Remark 2.5. (i) Compared to the regular case, the expression of A is much more complicated in the rough case, where we need not only to control D r,z u(t, x) p , but we also have to estimate the more notorious differences . This is the current paper's highlight in regard of the technicality.
(ii) When γ 0 is the Dirac delta function at zero, the expression of A reduces to This case corresponds to the white-in-time setting where the driving noise W behaves like Brownian motion in time, so as an alternative to using Proposition 2.4, one may adapt the general strategy based on the Clark-Ocone formula (see [16]) to establish the quantitative CLT for F R (t); however, the roughness in space will anyway force one to use the Gagliardo formulation (see (2.7)) of the inner product on H when estimating the variance of DF R (t), V t,R H . This will lead to almost the same level of difficulty as in bounding the expression A, while our computations will be done for a broader range of temporal correlation structures that include the Dirac delta function (white-in-time case).

Technical lemmas
In this subsection, we provide some useful results related to the heat kernel and gamma functions. All of them will be used in Section 4. Let us first introduce a few more notation. Set for all t ∈ R + and x, x ′ , x ′′ ∈ R. The next lemma provides further estimates for ∆ t and R t . This lemma, as well as operator Λ (see (2.27) below), will be used in Proposition 4.1 combined with the simplified formulas in Lemmas 4.2 and 4.3.
Lemma 2.6. Let ∆ t , R t and N t be given as in (2.14) -(2.16). Then, the following results hold: and x ′ is the operator acting on M(R), the real measurable functions on R, given by with θ denoting the shift function, that is, Remark 2.7. For any t > 0 and z ∈ R, operator Φ β t,z can be expressed as with Ig = g. It is easy to check that the following commutativity property holds Proof of Lemma 2.6. Equality (2.17) follows from direct computations, which we omit here.
In what follows, we first derive the estimate (2.18) for ∆ t . If |x ′ | > √ t, it follows immediately that (2.20) On the other hand, suppose now that |x ′ | ≤ √ t. Notice firstly that By the mean value theorem, there exists a point z between 0 and x ′ such that, In view of the fact that for all α > 0, we know that |x+z| √ t e − (x+z) 2 4t is uniformly bounded, from which it follows that and thus for some universal constant c > 0. As a consequence, if |x ′ | ≤ √ t, for any β ∈ [0, 1], we have Next, we prove the inequality (2.19) by considering the following four cases.
then the same argument from Case 2 leads to dt for a > 0. By mean value theorem, we can write where z ′ is some number between 0 and y. Using (2.21) and It follows that We also introduce the operator Λ : M(R) × M(R) → M(R 2 ) as follows. This operator will be used in Section 4. Let 0 < r < s and let z ′ , y ′ ∈ R. Then, for any (g 1 , g 2 , x, y) ∈ M(R) × M(R) × R 2 .
We complete this subsection by the following results about the gamma functions.
As a result of Lemma 2.9 (i), we can deduce the following corollary.

Regular cases under Hypothesis 1
In this section, we prove the first two error bounds in (1.12). As already mentioned in the introduction, the majority of this section will devoted to proving the following L p estimates of Malliavin derivatives.
where f t,x,m is the chaos coefficient defined as in (2.6) and the constant C(t) depends on (t, p, m, γ 0 , γ 1 ) and is increasing in t.
The proof of Theorem 3.1 is deferred to Section 3.2. In Section 3.1, we prove the first two error bounds in (1.12) by using Theorem 3.1.

Proof of quantitative CLTs in regular cases
Assume Hypothesis 1. With we have the following facts from [27].

Proof of (1.12) under Hypotheses 1 and 3a
Using Minkowski's inequality, we have Then it follows from (3.1) that with f t,x,2 (r, z, s, y) = 1 2 In the same way, we have where the implicit constants in (3.2) and (3.3) do not depend on (R, r, z, s, y) and are increasing in t.

Apply Proposition 2.3 and plugging (3.2) and (3.3) into the expression of A, we get
Taking the expression of f t,x,2 into account, we need to consider four terms depending on r > θ or not, and depending on s > θ ′ or not. Since the computations are similar, it suffices to provide the estimate for case r > θ and s > θ ′ . In other words, we need to show that In fact, the above estimate follows from integrating with respect to dx 1 , dx 2 , dx 4 , dy ′ , dy, dw ′ , dw, dz, dz ′ , dx 3 one by one and using the local integrability of γ 0 . The desired bound follows immediately.

Proof of (1.12) under Hypotheses 1 and 3b
Similarly as in Section 3.1.1, we need to show A * R 4d−3β . Making the change of variables and using the scaling properties of the Riesz and heat kernels 5 yields Making the following change of variables where Z 1 , . . . , Z 6 are i.i.d. standard Gaussian vectors on R d . Notice that Therefore, we deduce that Hence applying Proposition 2.3 yields the desired conclusion.

Proof of Theorem 3.1
Recall the Wiener chaos expansion (2.5) and (2.6) for u(t, x). Then, for any positive integer m, the m-th Malliavin derivative valued at (s 1 , y 1 , . . . , s m , y m ) is given by whenever the series converges in L 2 . By definition, it is easy to check that f t,x,n (s m , y m , •) ∈ H ⊙(n−m) and by symmetry again, we can assume t > s m > s m−1 > · · · > s 1 > 0.
Let h t,x,n (s m , y m ; •) be the symmetrization of f t,x,n (s m , y m ; •). Then, for any p ∈ [2, ∞), we deduce from Minkowski's inequality and (2.3) that t,x,n (s m , y m ; •).

It follows that
Additionally, due to Lemma 2.1, we deduce that Using the independence among the random variables inside the expectation, see Lemma 2.2, and the notation (i 0 , s m+1 , y m+1 ) = (0, t, x), we can write Thanks to the isometry property between the space H ⊙n (see (2.2)), equipped with the modified norm √ n! • H ⊗n , and the n-th Wiener chaos H n , we can write We first estimate f s,y,k X ⊗k and begin with with r k+1 = s and p t (ξ) = e −t|ξ| 2 /2 . In the current regular case, we can deduce from the maximal principle (c.f. [27,Lemma 4.1]) that Then, preforming change of variables w j = r j+1 − r j , and using Lemma 3.3 in [15], we have where T k (s) := {w k ∈ R k + : w 1 + · · · + w k ≤ s} and s,y,k (r, z; •) X ⊗(k−1) . It is trivial that for k = 1, For k = 2, we can write f (2) s,y,2 (r, z; •) Using the fact (c.f. [5,Formula (1.4)]) that The Fourier transform of f is given by This implies that and thus f (2) s,y,2 (r, z; 14) for any N > 0. Indeed, In order to estimate f s,y,k (r, z; r k−1 , z k−1 ) as follows, where by convention, r 0 = r and z 0 = z. In the next step, we compute the Fourier transform of f (k) s,y,k (r, z; r k−1 , •). Using the representation (3.15), we integrate the following expression f (k) s,y,k (r, z; r k−1 , z k−1 )e i ξ ξ ξ k−1 ,z k−1 subsequently in ξ k−1 , ξ k−2 , . . . , ξ 1 and get (3.16) Notice that Thus, by the the maximal principle ([27, Lemma 4.1]) again, we have s,y,k (r, z; r k−1 , ξ ξ ξ k−1 ) where, by the change of variables v i = s−r k−i s−r for all i = 1, . . . , k, We estimate the term I as follows. First we make the decomposition where D N and C N appearing below are introduced as in (3.11) and J c = {1, . . . , k − 1} \ J. Suppose J c = {ℓ 1 , . . . , ℓ j } for some j = 0, . . . , k−1 with ℓ 1 < · · · < ℓ j . Then, performing the integral with respect to v ℓ 1 , yields Applying this procedure for the integrals with respect to the variables v ℓ 2 ,. . . ,v ℓ j successively, we obtain It follows that Notice that, by using Lemma 2.9 (ii) and Corollary 2.10, we have Thus, by iteration, we conclude that Inequality (3.1) is a consequence of inequalities (3.7) and (3.19).
On the other hand, suppose that C N > 0 for all N > 0. From inequality (3.18), we deduce that Notice that by definition lim N ↑∞ C N = 0. Therefore, we can find N large enough such that This completes the proof of Theorem 3.1.

Rough case under Hypothesis 2
In this section, we will deal with the rough case. That is, we consider the parabolic Anderson model (1.5) under Hypothesis 2 and, as already mentioned in the introduction, extra effort will be poured into for the spatial roughness. Taking advantage of the Gagliardo representation (2.7) of the inner product on H 1 , we apply a modified version of the secondorder Gaussian Poincaré inequality (see Proposition 2.4). In order to estimate the quantity A in Proposition 2.4, we need the next proposition about the upper bounds of the Malliavin derivatives and their increments.
Proposition 4.1. Assume Hypothesis 2 and let u be the solution to (1.5). Given t ∈ (0, ∞), for almost any 0 < r < s < t, x, y, y ′ , z, z ′ ∈ R and for every p ≥ 2, the following inequalities hold: and where we fix Φ = Φ H 0 − 1 4 , defined as in Lemma 2.6, Λ is defined as in (2.27), and ) for all t > 0 with some constants c 1 and c 2 depending on H 0 and H 1 .
The proof of Proposition 4.1 is based on the following lemmas. We firstly show how these lemmas imply Proposition 4.1. The proofs of Lemmas 4.2 and 4.3, which heavily rely on the Wiener chaos expansion, are postponed to Section 4.2.
Lemma 4.2. Assume Hypothesis 2 and let u be the solution to (1.5). Then, for almost every (s, x, y, y ′ ) ∈ (0, t) × R 3 and for any p ≥ 2, the following inequalities hold: and where ∆ t and N t are defined as in (2.14) and (2.16), respectively, and C 1 (t) is the same as in Proposition 4.1.
where ∆ t , R t and N t are defined as in (2.14) -(2.16), respectively, and C 1 is the same as in Proposition 4.1.
In the next step, we prove inequality (4.2) that contains more terms. This is because the bound in (4.5) is a sum of 4 terms, which we denote by R 1 , R 2 , R 3 and R 4 .
Firstly, we estimate R 1 and R 2 by using inequality (4.6) and Lemma 2.6 as follows, and taking into account Remark 2.7 and the inequality p t (x) ≤ 2p 4t (x), we can write Following a similar argument, we also get and Using the above estimates, inequality (4.2) follows immediately.
In what follows, we first give the remaining proof of (1.12) in Section 4.1. Later, in Section 4.2 provides proofs of several auxiliary results.

Proof of (1.12) in the rough case
According to Proposition 2.4, we need to estimate the quantity A defined as in (2.12) with F = F R given as in (1.10). Our goal is to show A R for large R. Indeed, we already know from Theorem 1.1 that σ 2 R (t) ∼ R, then the desired bound (1.12) (in the rough case) follows.
In what follows, we only provide a detailed proof assuming γ 0 (s) = |s| 2H 0 −2 for H 0 ∈ (1/2, 1), while the other case (γ 0 = δ 0 ) can be dealt with in the same way. We first write where, using a changing of variables in space, Next, we will estimate A 0 . Suppose 0 < r < s < t, and 0 < θ < s ′ < t. We deduce from Proposition 4.1 that Now we first integrate out x 3 , x 4 and x 2 one by one (see Remark 2.8): and then integrate out w, y and z to get Applying Cauchy-Schwarz inequality, we can further deduce that Due to the fact that 2H 1 + 2H 0 − 5 2 > −1 and We can also deduce the next inequality by definition of Λ and Remark 2.7, which, together with (2.17), implies For the case γ 0 = δ 0 , the expression for B 0 reduces to and for the same reason as above, 0<r<s<t 0<θ<s<t The remaining three cases      0 < s < r < t and 0 < θ < s ′ < t 0 < r < s < t and 0 < s ′ < θ < t 0 < s < r < t and 0 < s ′ < θ < t can be estimated in an almost same way and finally we can get the same upper bounds. That is, we obtain the desired bound A R and hence conclude the proof of (1.12) under Hypothesis 2.

Proof of some auxiliary results
In this subsection, we introduce some auxiliary results and provide the proof of Lemma 4.2 and 4.3 in Section 4.2.2.

Estimates for fixed Wiener chaoses
Fix 0 < s < t < ∞ and x, y ∈ R. For any n = 1, 2, . . . , let g 1 s,t,x,n and g 2 s,y,t,x,n be functions on [s, t] n × R n given by g 1 s,t,x,n (s n , y n ) = n!f t,x,n (s n , y n )1 [s,t] n (s n ), (4.9) and g 2 s,y,t,x,n (s n , y n ) = p s σ(1) −s (y σ(1) − y)g 1 s,t,x,n (s n , y n ), (4.10) where f t,x,n is defined as in (2.6) and σ, because of the indicator function 1 [s,t] n , is now a permutation on {1, . . . , n} such that s < s σ(1) < · · · < s σ(n) < t. In Section 4.2.2 below, we will see that g 1 and g 2 are closely related to the chaos coefficients of the Mallivain derivatives of u. In fact, the next lemmas, which give some estimates for g 1 n and g 2 n , are essential to the proofs of Lemmas 4.2 and 4.3.
Lemma 4.4. Let 0 < s < t < ∞ and let x ∈ R. Fix a positive integer n, and let g 1 n be given as in (4.9). Then, the following equalities hold.