On the intermittency front of stochastic heat equation driven by colored noises

We study the propagation of high peaks (intermittency front) of the solution to a stochastic heat equation driven by multiplicative centered Gaussian noise in $\mathbb{R}^d$. The noise is assumed to have a general homogeneous covariance in both time and space, and the solution is interpreted in the senses of the Wick product. We give some estimates for the upper and lower bounds of the propagation speed, based on a moment formula of the solution. When the space covariance is given by a Riesz kernel, we give more precise bounds for the propagation speed.


Introduction
We consider the stochastic heat equation in R d driven by a general multiplicative centered Gaussian noise (parabolic Anderson model) with a continuous and nonnegative initial condition u 0 of compact support. The covariance of the noiseẆ can be informally written as E Ẇ t,xẆs,y = γ(s − t)Λ(x − y) , and the product appearing in (1.1) is interpreted in the Wick sense.
In this paper we are interested in the position of the high peaks that are farthest away from the origin. The propagation of the farthest high peaks was first considered by Conus and Khoshnevisan in [4] for a one dimensional heat equation driven by space-time white noise, where it is shown that there are intermittency fronts that move linearly with time as αt. Namely, for any fixed p ∈ [2, ∞), if α is sufficiently small, then the quantity sup |x|>αt E(|u(t, x)| p ) grows exponentially fast as t tends to ∞; whereas the preceding quantity vanishes exponentially fast if α is sufficiently large. To be more precise, the authors of [4] define for every α > 0, S (α) := lim sup and think of α L as an intermittency lower front if S (α) < 0 for all α > α L , and of α U as an intermittency upper front if S (α) > 0 whenever α < α U . In [4] it is shown that for each real number p ≥ 2, 0 < α U ≤ α L < ∞, and when p = 2, some bounds for α L and α U are given. In a later work by Chen and Dalang [1], it is proved that when p = 2, there exists a critical number α * = λ 2 2 such that S (α) < 0 when α > α * while S (α) > 0 when α < α * (this property was first conjectured in [4]). See also [9] for a discussion of these facts. This paper is inspired by the aforementioned works. We are interested in the multidimensional stochastic heat equation driven by a colored noise, both in space and time, when the solution is interpreted in the Wick sense. Our analysis will be based on the pth moment formula and Wiener chaos expansion of the solution, obtained in [7], as well as some small ball estimates. Due to the presence of the time covariance, the propagation speed of the farthest high peaks may not be linear. Thus, in contrast to (1.2), the inequality |x| > αt there needs to be replaced by |x| > αtθ t for some suitable function θ t (see Theorems 3.1 and 3.4 below for precise choice of θ t ). When Λ is the Riesz kernel, a better estimate of the intermittency lower front is obtained in Proposition 3.9. We would like to mention the work [2], where another nonlinear propagation speed (growth indices of exponential type) is studied.
This paper is organized as follows: In Section 2, we set up some preliminaries for the structure of our Gaussian noises in equation (1.1) and present some elements of Malliavin calculus. We also prove the non-negativity of the solution to equation (1.1). Section 3 contains the main results of this paper, where we obtain some upper and lower bounds for the growth index. In the special case when the space covariance is a Riesz kernel, we give a more detailed computation for the upper bound of the growth index, and we see that the orders of λ and p in the estimate of the growth index are sharp.

Preliminaries
We first introduce some basic notions. The Fourier transform is defined with the normalization so that the inverse Fourier transform is given by F −1 u(ξ) = (2π) −d Fu(−ξ). We denote by D((0, ∞) × R d ) the space of infinitely differentiable functions with compact support on (0, ∞) × R d .
On a complete probability space (Ω, F, P ) we consider a Gaussian noise W encoded by a centered Gaussian family {W (ϕ), ϕ ∈ D((0, ∞) × R d )}, whose covariance structure is given by where γ : R → R + and Λ : R d → R + are non-negative definite functions. We also assume that the Fourier transform FΛ = µ is a tempered measure, that is, there is an integer m ≥ 1 such that Our results also cover the case where γ (or Λ if d = 1) is the Dirac delta function, which corresponds to the time (or space) white noise.
Let H be the completion of D((0, ∞) × R d ) endowed with the inner product where Fϕ refers to the Fourier transform with respect to the space variable only. The mapping ϕ → W (ϕ) defined in D((0, ∞) × R d ) extends to a linear isometry between H and the Gaussian space spanned by W . We will denote this isometry by We shall make a standard assumption on the spectral measure µ, which will prevail until the end of the paper.

Hypothesis 2.1
The measure µ satisfies the following integrability condition: Now we state some basic facts about Malliavin calculus. For a detailed account on this theory, we refer to [11]. We will denote by D the Malliavin derivative. That is, if F is a smooth and cylindrical random variable of the form (namely, f and all its partial derivatives have polynomial growth), then DF is the H-valued random variable defined by The operator D is closable from L p (Ω) into L p (Ω; H) for any p ≥ 1 and we define the Sobolev space D 1,p as the closure of the space of smooth and cylindrical random variables under the norm We denote by δ the adjoint of the derivative operator given by the duality formula for any F ∈ D 1,2 and any element u ∈ L 2 (Ω; H) in the domain of δ. The operator δ is also called the Skorohod integral because in the case of the Brownian motion, it coincides with an extension of the Itô integral introduced by Skorohod. We will make use of the notation If F ∈ D 1,2 and φ is an element of H, then F φ is Skorohod integrable and, by definition, the Wick product equals the Skorohod integral of F h, that is, In view of this definition, the mild solution to equation (1.1) will be formulated below in terms of the Skorohod integral.
Next we give a short account of Wiener chaos expansion. For any integer n ≥ 0 we denote by H n the nth Wiener chaos of W . We recall that H 0 is simply R and for n ≥ 1, H n is the closed linear subspace of L 2 (Ω) generated by the random variables {H n (W (h)), h ∈ H, h H = 1}, where H n is the nth Hermite polynomial. Then we will have the orthogonal decomposition For each n ≥ 0, we will denote by J n the orthogonal projection on the nth Wiener chaos. Consider the one-parameter semigroup {T t , t ≥ 0} of contraction operators on L 2 (Ω) defined by which is called the Ornstein-Uhlenbeck semigroup. The following property of T t is taken from [12].
,p for any t > 0 and we also have We are ready to give the definition of mild solution to equation (1.1). We denote by Then we have the following definition.
The following theorem about the existence and uniqueness of the solution to equation (1.1) is taken from [7]. The next lemma states the non-negativity of the solution and will be used in the next section.
Lemma 2.5 Assume that µ satisfies Hypothesis 2.1 and γ is locally integrable. If the initial condition u 0 is nonnegative, then for each (t, Proof We will follow the procedure in Section 3.2, [7]. For any δ > 0, we define the function provides an approximation of the Dirac delta function δ 0 (t, x) as ε and δ tend to zero. Define for a standard d-dimensional Brownian motion B independent of W . Then it is obvious from the definition of u ε,δ (t, x) that u ε,δ (t, x) > 0 a.s. for each (t, x), and from Theorem 3.6 in [7] and its proof, we see that for each F ∈ D 1,2 and (t, then letting s go to 0 we obtain by Proposition 2.2 which shows that u(t, x) ≥ 0 a.s. The next result concerning the moment formula for the solution is taken from [7], see also [3], where γ is the Dirac delta function δ.
Theorem 2.6 Suppose γ is locally integrable and µ satisfies Hypothesis 2.1. Let u(t, x) be the solution to equation (1.1). Then for any integer p ≥ 2 where {B j , j = 1, . . . , p} is a family of d-dimensional independent standard Brownian motions independent of W .

Main results
We need first to introduce some notation. If µ is a measure satisfying Hypothesis 2.1, for any real number N > 0, we define On the other hand, if γ is locally integrable, we set Theorem 3.1 Let u(t, x) be the solution to equation (1.1) driven by a noise W with covariance structure (2.1). Assume that u 0 is non-negative and supported in the ball Assume that µ satisfies Hypothesis 2.1 and γ is locally integrable.
Then, for any integer p ≥ 2, we havē Proof Using the moment formula (2.12), together with Cauchy-Schwartz inequality, we can write for any integer p ≥ 2 Note that in the above expression, the first expectation in the last inequality is exactly the pth moment of the solution to equation (1.1) (denoted by v(t, x)) with noise W having a covariance functional with parameters γ and 2Λ respectively, and with initial condition 1. From [7] we see that v(t, x) admits the chaos expansion where for n ≥ 1, the kernel f n is given by (1) ) .
Therefore, have the L p norm of the v(t, x) is bounded as follows Then using Fourier transform as in [7] we have where T n (t) denotes the simplex {0 ≤ s 1 ≤ s 2 ≤ · · · ≤ s n ≤ t}, µ(dξ) = n i=1 µ(dξ i ) and ds is defined similarly. Then with the change of variable s i+1 − s i = w i and by Lemma 3.3 below applied to N = N t , we obtain where S t,n = {(w 1 , . . . , w n ) ∈ [0, ∞) n : w 1 + · · · + w n ≤ t}. Thus where the last inequality comes from the definition of N t . Thus we obtain from Lemma 3.5 below we need ̺ > √ κ + 1. Letting κ → 0 we conclude thatν(p) ≤ 1. Section 6 in [7] gives the moment upper bounds for some specific choices of γ and Λ, assuming the initial condition is a bounded function. Actually the proof of Theorem 3.1 above also gives a general upper bound for the pth moment, stated in the following corollary.

3). Then we have the moment upper bound
Eu

5)
for some constant C independent of p and t.
The next lemma is used in the proof of Theorem 3.1. For a proof, see Lemma 3.3 in [7].
The next result is a lower bound for the lower intermittency front, when Λ is bounded below by the Riesz kernel.
Proof Note that using the change of variable s → u+v 2 and r → v−u 2 we have . Then using the moment formula for the solution u(t, x) as before, where B 0 s is a standard Brownian motion, ε is a positive number satisfying ε < R 2 , which will be chosen later. In order to estimate of the above probability, notice that where G t is the filtration generated by {B 0 s : 0 ≤ s ≤ t}. Then we can choose ε ≤ M 2 (the specific choice of ε will be given below) and invoke Lemma 3.5 to get, for t large enough, where C is a universal constant. The last inequality follows from the small ball probability estimate (see Theorem 1 in [13]) Then we obtain We apply Lemma 3.6 with to maximize the right hand side of the above inequality, by choosing In this way we obtain We remark condition (3.7) implies that ε chosen above tends to 0 as t → ∞, thus the small ball estimate used above works for t large enough, in such a way that ε < M 2 . If we want lim sup Letting κ → 0 and invoking Lemma 2.5 we conclude that The theorem is proved. The next two lemmas are used in the proof of previous theorems. where ω d is the volume of the unit ball in R d .
Remark 3.7 If condition (3.7) does not hold, that is, the limit is finite Γ ∞ (which happens, for instance, if γ is a Dirac function), then, we need the following additional condition on M : (3.12) Remark 3.8 When γ is the Dirac delta function, the noiseẆ is white in time and correlated in space and Theorems 3.1 and 3.4 still hold with Γ t = 1 2 . The appearance of the functions θ t and η t in Theorems 3.1 and 3.4 come from the time covariance of the noise. In the case when Γ t → ∞ as t → ∞ (for instance, when γ(t) = |t| −α with 0 < α < 1, Γ t = t 1−α 1−α ), the restriction on M (3.12) is not needed.
In the case where Λ(x) = |x| −β with 0 < β < 2 ∧ d, we obtain the following more precise result concerning the upper bound of the intermittency front. In this case, the function θ t defined in Theorem 3.1 can be replaced by Γ 1 2−β t . Notice that this function coincides with the function η t in Theorem 3.4 except for the factor δ 2 . If we let δ tend to 1, then the lower bound in (3.8) tends to zero. Proposition 3.9 Assume that u 0 is non-negative and supported in the ball B M . Let γ be a locally integrable, positive and positive definite function, and Λ(x) = |x| −β , assume 0 < β < 2 ∧ d. Set Proof We will follow the notations and the same calculations used in the proof of Theorem 3.1.
We have Since Λ(x) = |x| −β , its Fourier transform is given by (see, e.g., Chapter 5 in [14]) Using polar coordinates we can compute the integral .

Then with the change of variable
.
To alleviate the notation we denote .
Using the asymptotic behavior of Mittag-Leffler function (see e.g., page 208 in [6]) and considering the fact that we are only interested in when t → ∞, we obtain Thus, as in the proof of Theorem 3.1, we obtain .
Recall that we have set ϑ t = Γ Finally, if we plug in the value of B and Λ β , the proposition is proved.
Remark 3.10 Proposition 3.9 still holds if we take the fractional kernel Λ(x) = d i=1 |x i | 2H i −2 with 1 2 < H i < 1 for all i and β := 2d − 2 d i=1 H i with 0 < β < 2. The order of p − 1 and λ in the upper bound ofῡ(p) will be exactly the same, although the coefficient may be different.
Theorem 3.4 does not cover the case when the noise is white in space. However, if we approximate the Dirac delta function by p ε (x), we have the following result.
Proposition 3.11 Assume d = 1. Let u(t, x) be the solution to equation (1.1) with nonnegative initial condition u 0 being uniformly bounded away from 0 in the ball B M and supported in B rM , where r ≥ 1. Assume that Λ(x) is the Dirac delta function. Set ϑ t = Γ t , fix δ ∈ (0, 1) and set η t = Γ tδ 2 . Letῡ(p) and ν(p) be defined in (3.13) and (3.8), respectively. Then we havē (3.14) If we further assume (3.7) holds, then ν(p) ≥ 2 √ 2 e 2 π 3 2 (1 − δ) Proof The proof of upper bound follows along the same lines as the proof for proposition 3.9 except now µ(dξ) = dξ. For the lower bound, we consider the approximation of the Dirac delta function by the heat kernel p ε , and define Expanding the exponential and using Fourier analysis, one can show that Eu(t, x) p ≥ I t,p,ε (see [7,8]) for any ε > 0. Then the proof follows along the same lines as the proof for Theorem 3.4 except here, we restrict the expectation on the set We omit the details of the proof.