Spatial asymptotic of the stochastic heat equation with compactly supported initial data

We investigate the growth of the tallest peaks of random field solutions to the parabolic Anderson models over concentric balls as the radii approach infinity. The noise is white in time and correlated in space. The spatial correlation function is either bounded or non-negative satisfying Dalang's condition. The initial data are Borel measures with compact supports, in particular, include Dirac masses. The results obtained are related to those of Conus, Joseph and Khoshnevisan 2013 and X. Chen 2016, where constant initial data are considered.


Introduction
We consider the stochastic heat equation in R ℓ ∂u ∂t = 1 2 ∆u + uẆ , u(0, ·) = u 0 (·) (1.1) where t ≥ 0, x ∈ R ℓ (ℓ ≥ 1) and u 0 is a Borel measure. Herein, W is a centered Gaussian field, which is white in time and it has a correlated spatial covariance. More precisely, we assume that the noise W is described by a centered Gaussian family where µ is non-negative measurable function and F denotes the Fourier transform in the spatial variables. To avoid trivial situations, we assume that µ is not identical to zero. The inverse Fourier transform of µ is in general a distribution defined formally by the expression (1.3) If γ is a locally integrable function, then it is non-negative definite and (1.2) can be written in Cartesian coordinates φ(s, x)ψ(s, y)γ(x − y)dxdyds . (1.4) The following two distinct hypotheses on the spatial covariance of W are considered throughout the paper.
Hereafter, we denote by | · | the Euclidean norm in R ℓ and by x · y the usual inner product between two vectors x, y in R ℓ . Condition (H.2b) is known as Dalang's condition and is sufficient for existence and uniqueness of a random field solution. If γ exists as a function, condition (H.2c) induces the scaling relation γ(cx) = c −α γ(x) for all c > 0. Equation (1.1) with noise satisfying condition (H.2) was introduced by Dalang in [Dal99]. In [HLN15], for a large class of initial data, we show that equation (1.1) has a unique random field solution under the hypothesis (H.2). Under hypothesis (H.1), we note that γ may be negative, but proceeding as in [Hua16], a simple Picard iteration argument gives the existence and uniqueness of the solution. In addition, in both cases, the solution has finite moments of all positive orders. We give a few examples of covariance structures which are usually considered in literatures.
Suppose for the moment thatẆ is a space-time white noise and u 0 is a function satisfying c ≤ u 0 (x) ≤ C, for some positive numbers c, C. (1.6) It is first noted in [CJK13] that there exist positive constants c 1 , c 2 such that almost surely (1.7) Later Xia Chen shows in [Che16] that indeed the precise almost sure limit can be computed, namely, (1.9) 2 Thanks to the scaling property of the space-time white noise, Xia Chen has managed to derive (1.9) from the following long term asymptotic result lim t→∞ 1 t log Eu(t, x) m = E m (1.10) where the constant E m grows as 1 24 m 3 when m → ∞. Under condition (1.6), analogous results for other kinds of noises are also obtained in [Che16]. More precisely, for noises satisfying (H. On the other hand, it is known that equation (1.1) has a unique random field solution under either (H.1) or (H.2) provided that u 0 satisfies (1.13) Hence, condition (1.6) excludes other initial data of interests such as compactly supported measures. It is our purpose in the current paper to investigate the almost sure spatial asymptotic of the solutions corresponding to these initial data. Upon reviewing the method in obtaining (1.8) described previously, one first seeks for an analogous result to (1.10) for general initial data. In fact, it is noted in [HLN15] that for every u 0 satisfying (1.13), one has (1.14) where E m is a constant whose asymptotic as m → ∞ is known. It is suggestive from (1.14) that with a general initial datum, one should normalized u(t, x) in (1.8) (and (1.9)) by the factor p t * u 0 (x). Therefore, we anticipate the following almost sure spatial asymptotic result.
In the particular case of space-time white noise, we conjecture that In the case of space-time white noise, note that if u 0 satisfies the condition (1.6), (1.17) is no different than (1.8). On the other hand, if u 0 is a Dirac delta mass at x 0 , (1.17) precisely describes the spatial asymptotic of log u(t, x): at large spatial sites, log u(t, x) is concentrated near a logarithmic perturbation of the parabola − 1 2t (x − x 0 ) 2 . More precisely, (1.17) with this specific initial datum reduces to (1.18) While a complete answer for Conjecture 1.2 (including (1.18)) is still undetermined, the current paper offers partial results, focusing on initial data with compact supports, especially Dirac masses. To unify the notation, we denotē For noises satisfying (H.2), or for initial data with compact supports, the picture is less complete.
Theorem 1.4. Assume that u 0 is a non-negative measure with compact support and either (H.1) or (H.2) holds. Then we have (1.20) For initial data satisfying (1.6), the lower bound of (1.16) is proved in [Che16] using a localization argument initiated from [CJK13]. In our situation, a technical difficulty arises in applying this localization procedure, which leads to the missing lower bound in Theorem 1.4. A detailed explanation is given at the beginning of Subsection 6.2. As an attempt to obtain the exact spatial asymptotics, we propose an alternative result which is described below. We need to introduce a few more notation. For each ǫ > 0, we denote which is a bounded non-negative definite function. Let W ǫ be a centered Gaussian field defined by In the above, p ǫ = (2πǫ) −ℓ/2 e −|x| 2 /(2ǫ) and p ǫ * φ is the convolution of p ǫ with φ in the spatial variables. The covariance structure of W ǫ is given by In other words, W ǫ is white in time and correlated in space with spatial covariance function γ ǫ , which satisfies (H.1). Under condition (H.2c), γ ǫ satisfies the scaling relation (1.24) Let u ǫ be the solution to equation (1.1) withẆ replaced byẆ ǫ . It is expected that as ǫ ↓ 0, u ǫ (t, x) converges to u(t, x) in L 2 (Ω) for each (t, x), see [CH16] for a proof when the initial data is a bounded function. The following result describes spatial asymptotic of the family of random fields {u ǫ } ǫ∈(0,1) .
Theorem 1.5. Assume that u 0 is a non-negative measure with compact support and either (H.1) or (H.2) holds. Then (1.26) Neither one of (1.16) and (1.26) is stronger than the other. While the result of Theorem 1.5 relates to the solution of (1.1) indirectly, it is certainly interesting. In Hairer's theory of regularity structures (cf. [Hai14]), one first regularizes the noise to obtain a sequence of approximated solutions. The solution of the corresponding stochastic partial differential equation is then constructed as the limiting object of this sequence. From this point of view, (1.26) provides a unified characteristic of the sequence of approximating solutions {u ǫ } ǫ∈(0,1) , which approaches the solution u as ǫ ↓ 0. The proof of (1.26) does not rely on localization, rather, on the Gaussian nature of the noise. This leads to a possibility of extending (1.26) to temporal colored noises, which will be a topic for future research.
The remainder of the article is structured as follows: In Section 2 we briefly summarize the theory of stochastic integrations and well-posedness results for (1.1). In Section 3 we introduce some variational quantities which are related to the spatial asymptotics. In Section 4 we derive some Feynman-Kac formulas of the solution and its moments, these formulas play a crucial role in our consideration. In Section 5 we investigate the high moment asymptotics and Hölder regularity of the solutions of (1.1) with respect to various parameters. The results in Section 5 are used to obtain upper bounds in (1.15) and (1.16). This is presented in Section 6, where we also give a proof of the lower bounds in Theorems 1.3, 1.4 and 1.5.

Preliminaries
We introduce some notation and concepts which are used throughout the article. The space of Schwartz functions is denoted by S(R ℓ ). The Fourier transform of a function g ∈ S(R ℓ ) is defined with the normalization so that the inverse Fourier transform is given by F −1 g(ξ) = (2π) −ℓ F g(−ξ). The Plancherel identity with this normalization reads Let us now describe stochastic integrations with respect to W . We can interpret W as a Brownian motion with values in an infinite dimensional Hilbert space. In this context, the stochastic integration theory with respect to W can be handled by classical theories (see for example, [DQS11]). We briefly recall the main features of this theory.
We denote by H 0 the Hilbert space defined as the closure of S(R ℓ ) under the inner product (2.1) which can also be written as If γ satisfies (H.1), then H 0 contains distributions such as Dirac delta masses. The Gaussian family W can be extended to an isonormal Gaussian process {W (φ), φ ∈ L 2 (R + , H 0 )} parametrized by the Hilbert space H := L 2 (R + , H 0 ). For any t ≥ 0, let F t be the σ-algebra generated by W up to time t. Let Λ be the space of H 0 -valued predictable processes g such that E g 2 H < ∞. Then, one can construct (cf. [HLN15]) the stochastic integral Stochastic integration over finite time interval can be defined easily Finally, the Burkholder's inequality in this context reads which holds for all p ≥ 2 and g ∈ Λ. A useful application of (2.4) is the following result Proof. We consider only the hypothesis (H.2), the other case is obtained similarly. In view of Burkholder inequality (2.4) and Minkowski inequality, it suffices to show Hs,y . (2.5) In fact, using (2.2) and Minkowski inequality, the left-hand side in the above is at most Note in addition that by Cauchy-Schwarz inequality, . From here, (2.5) is transparent and the proof is complete.
We now state the definition of the solution to equation (1.1) using the stochastic integral introduced previously.
is an element of Λ. We say that u is a mild solution of (1.1) if for all t ∈ [0, T ] and x ∈ R ℓ we have The following existence and uniqueness result has been proved in [HLN15] under hypothesis (H.2). Under hypothesis (H.1), one can proceed as in [Hua16], using a simple Picard iteration argument to obtain the existence and uniqueness of the solution.
When u 0 = δ(· − z), we denote the corresponding unique solution by Z(z; t, x). In particular Z(z; ·, ·) is predictable and satisfies for all t ≥ 0 and x ∈ R ℓ . Next, we record a Gronwall-type lemma which will be useful later.
Lemma 2.4. Suppose α ∈ [0, 2) and f is a locally bounded function on [0, ∞) such that where A, B are positive constants and g is non-decreasing function. Then there exists a constant C α such that It is easy to see for some suitable constant C depending only on α. We then choose ρ = (2AC) 2 . This leads to D ρ ≤ 2Bg T , which implies the result.
Let us conclude this section by introducing a few key notation which we will use throughout the article. Let B = (B(t), t ≥ 0) denote a standard Brownian motion in R ℓ starting at the origin. For each t > 0, we denote (2.8) The process B 0,t = (B 0,t (s), 0 ≤ s ≤ t) is independent from B(t) and is a Brownian bridge which starts and ends at the origin. An important connection between B and B 0,t is the following identity. For every λ ∈ (0, 1) and every bounded measurable function F on This is in fact an application of Girsanov's theorem, see [HLN15, Eq. (2.8)] for more details. Let B 1 , B 2 , . . . be independent copies of B and B 1 0,t , B 2 0,t , . . . be the corresponding Brownian bridges. An important quantity which appears frequently in our consideration is (2.10) From the proof of Proposition 4.2 in [HLN15], it is easy to see that under one of the hypotheses (H.1) and (H.2), Θ t (m) < ∞ for any t > 0. Finally, A E means A ≤ CE for some positive constant C, independent from all the terms appearing in E.

Variations
We introduce two variational quantities and give their basic properties and relations. The high moment asymptotic is governed by a variational quantity which is known as the Hartree energy (cf. [CP]). If there exists a locally integrable function γ whose Fourier transform is µ, then the Hartree energy can be expressed as 2) The subscript H stands for "Hartree". We can also write this variation in Fourier mode. Indeed, the presentation (1.3) leads to Under (H.1), from (3.1), we upper bound γ(x − y) by γ(0), it follows that E H (γ) ≤ γ(0), which is finite. The fact that this variation (either in the form (3.1) or (3.3)) is finite under the condition (H.2) is not immediate. In some special cases, this is verified in [CHSX15] and [CHNT16].
On the other hand, using the elementary inequality and Cauchy-Schwarz inequality, we also get Then, for every R > 0 we have We now choose R sufficiently large so that 4(2π) −ℓ |ξ|>R µ(ξ) for all g in A, which finishes the proof.
In establishing the lower bound of spatial asymptotic, another variation arises, which is given by or alternatively in frequency mode Before giving the proof, let us see how (3.1) and (3.4) are connected to a certain interpolation inequality. Under scaling condition (H.2c), it is a routine procedure in analysis to connect the finiteness of E H (γ) with a certain interpolation inequality. For instance, when γ = δ and ℓ = 1, the fact that is equivalent to the following Gagliardo-Nirenberg inequality for all g in W 1,2 (R). For readers convenience, we provide a brief explanation below.
In addition the constant κ can be chosen to be (3.7) (ii) If (3.6) holds for some finite constant κ > 0, then E H (γ) is finite and the best constant in (3.6) is κ(γ).
Writing these integrals back to g and using (H.2c) yields for all θ > 0. Optimizing the left-hand side (with respect to θ) leads to Removing the normalization g L 2 = 1 and some algebraic manipulation yields the result. (ii) Let κ 0 be the best constant in (3.6). Then for every g ∈ G, This shows E H (γ) is finite and at most 2−α α ( α 2 κ 0 ) 2 2−α , which also means κ(γ) ≤ κ 0 . On the other hand, (i) already implies κ 0 ≤ κ(γ), hence completes the proof.
Proof of Proposition 3.2. Reasoning as in Proposition 3.3, we see that M(γ) is finite if and only if (3.6) holds for some constant κ > 0. In addition, the best constant κ(γ) in (3.6) satisfies the relation Together with (3.7), this yields the result.
The following result preludes the connection between E H , M with exponential functional of Brownian motions.
where G D is the class of functions g in W 1,2 (R n ) such that D |g(x)| 2 dx = 1 and τ D is the exit time τ D := inf{t ≥ 0 : B t / ∈ D}.
Proof. The process {B 0,t (s) = B(s) − s t B(t)} s∈[0,t] is a Brownian bridge. We fix θ ∈ (0, 1) and consider first the limit Let M be such that |x| ≤ M for all x ∈ D. Using Girsanov theorem (see [HLN15,Eq. (2.38)]), we can write The result of [CHSX15, Proposition 3.1] asserts that This leads to lim inf Observing that we can send θ ↑ 1 in (3.9) to obtain the lower bound for (3.8). The upper bound for (3.8) is proved analogously, we omit the details.
We conclude this section with an observation: (H.2c) induces the following scaling relation on E H (γ)

Feynman-Kac formulas and functionals of Brownian Bridges
We derive Feynman-Kac formulas for the moments Eu m (t, x) for integers m ≥ 2. These formulas play important roles in proving upper and lower bounds of (1.15) and (1.26).
To discuss our contributions in the current section, let us assume for the moment thaṫ W is a space-time white noise and ℓ = 1. The most well-known Feynman-Kac formula for second moment is where B 1 , B 2 are two independent Brownian motions starting at 0. If u 0 is merely a measure, some efforts are needed to make sense of u 0 (B(t) + x), which appears on the right-hand side above. An attempt is carried out in [CHN16] using Meyer-Watanabe's theory of Wiener distributions.
The Feynman-Kac formulas presented here (see (4.13) below) have appeared in [HLN15]. However, there seems to have a minor gap in that article. Namely, Eq. (4.52) there has not been proven if u 0 is a measure. In the current article, we take the chance to fill this gap. Our approach is in the same spirit as [HLN15] and is different from [CHN16]. In particular, we do not make use of Wiener distributions.
Since W ǫ has bounded covariance, it is easy to see that the stochastic heat equation has a unique random field solution u ǫ . In addition, for each t > 0 and x ∈ R ℓ , u ǫ (t, x) admits a chaos expansion (see, for instance [HN09]) where f 0 [u 0 ](t, x) = p t * u 0 (x) and for each n ≥ 1 f n [u 0 ](t, x; s 1 , x 1 , . . . , s n , x n ) Here, σ denotes the permutation of {1, 2 . . . , n} such that 0 < s σ(1) < · · · < s σ(n) < t and I ǫ,n is the n-th multiple Itô-Wiener integral with respect to the Gaussian field W ǫ .

13
Proof. Let v(t, x) be the integral on the right-hand side of (4.4). From (2.7), integrating z with respect to u 0 (dz) and applying the stochastic Fubini theorem (cf. [DPZ92,Theorem 4.33]), we have Hence, v is a solution of (1.1) with initial datum u 0 . By unicity, Theorem 2.3, we see that u = v and (4.4) follows. Next, we show (4.5) assuming (H.1). Fix t > 0 and x ∈ R ℓ . For every u 0 ∈ C ∞ c (R ℓ ), the following Feynman-Kac formula holds Using the decomposition (2.8) and the fact that B 0,t and B(t) are independent, we see that Together with (4.4) we obtain for all u 0 ∈ C ∞ c (R ℓ ). Next we show that z → Y (z; t, x) is continuous. Fix p > 2. From the elementary relation |e x − e y | ≤ (e x + e y )|x − y| and the Cauchy-Schwarz inequality, it follows Since γ is bounded, it is easy to see that for some constant C p,t . We now resort to Minkowski inequality, our exponential bound for V t,x (z) and the relation between L p and L 2 moments for Gaussian random variables in order to obtain In addition, under (H.1), γ is Hölder continuous with order κ > 0 at 0, it follows that Thus, the process z → Y (z; t, x) has a continuous version. On the other hand, z → Z(z; t, x) is also continuous (see Proposition 5.5 below). It follows that Z(z; t, x) = Y (z −x; t, x)p t (z − x), which is exactly (4.5).
Proposition 4.2. Assuming (H.1), we have and Proof. We observe that conditioned on B, is a normal random variable with mean zero. In addition, for every x, x ′ , z, z ′ ∈ R ℓ , applying (1.23), we have For every (x 1 , . . . , x m ) ∈ (R ℓ ) m , using (4.5) and (4.9), we have Note that in the exponent above, the diagonal terms (with j = k) are removed because there are cancellations with the normalization factor − t 2 γ(0) in (4.5), which occur after taking expectation with respect to W . Finally, apply [HLN15, Lemma 4.1], we obtain (4.8) from (4.10).
To extend the previous result to nosies satisfying (H.2), we need the following result.
We are now ready to derive Feynman-Kac formulas for positive moments. (4.14) Proof. We prove the result under the hypothesis (H.2). The proof under hypothesis (H.1) is easier and omitted.
We conclude this section with the following observation.
Remark 4.5. Under (H.1), it is evident from (4.5) that Z(z; t, x) is non-negative for every z, t, x. Under (H.2), thanks to Proposition 4.3, Z(z; t, x) is the limit of non-negative random variables, hence Z(z; t, x) is also non-negative for every z, t, x. Furthermore, in view of (4.4), if u 0 is non-negative then u(t, x) is non-negative for every t, x.

Moment asymptotic and regularity
Moment asymptotic. We begin with a study on high moments. Under hypothesis (H.1), the high moment asymptotic is governed by the value of γ at the origin.
The following result is needed to obtain moment asymptotic under (H.2). Proof. For each λ ∈ (0, 1), we note that Using (2.9), we see that the expectation above is at most In addition, reasoning as in [HLN15,Lemma 4.1], we see that It follows that lim sup Applying [CP, Theorem 1.1], we get Thus we have shown Finally, we send λ → 1 − to finish the proof.

From Lemma 5.2 and the fact that
Hence, it suffices to show for every fixed q > 1, Together with (2.9), we arrive at note that the right hand side of the above inequality is the m-th moment of the solution to the equation (1.1) driven by the noise with spatial covariance 2q m (γ − γ ǫ ), i.e., Eu( tm 2 , x) m , the initial condition is u 0 (x) ≡ 2 ℓ . Using the hyper-contractivity as in [HLN15,Lê16], we have where in the last line we have used the estimate (3.7) in [HHNT15] and µ(η)dη is abbreviation for k j=1 µ(η j )dη j . Since α < 2, we can find a β > 0 such that β < 1 − α 2 . Then using the elementary inequality Hence, we have shown , from which (5.5) follows. The proof for (5.3) is complete.
Hölder continuity. We investigate the regularity of the process Z(x;t,y) pt(y−x) in the variables x and y. These properties will be used in the proof of upper bound. For each integer m ≥ 2 and t > 0, we recall that Θ t (m) is defined in (2.10).
Note that from Proposition 4.4, we have Lemma 5.4. For every r > 0 and y 1 , under (H.2); and under (H.1). In the above, the constant C does not depend on y 1 , y 2 nor r.
Proof. We denote f (·) = |p r (· − y 1 ) − p r (· − y 2 )|. Assuming first (H.2), we observe the following simple estimate Noting that sup the result easily follows. Under (H.1), we used the following inequality together with (5.7) to obtain the result.
6.1. The upper bound. This subsection is devoted to the proof of upper bounds in Theorems 1.3 and 1.4 by combining the moment asymptotic bounds and the regularity estimates obtained in Section 5. We also recall that Θ t (m) is defined in (2.10). Propositions 5.1, 5.3 together with (5.6) imply where E is defined in (1.19). The following result gives an upper bound for spatial asymptotic of Z(x; ·, ·). Proof. We begin by noting that according Remark 4.5, Z(x; t, y) is non negative a.s. for each x, y, t. Let t be fixed and put where we have omitted the dependence on t. For every n > 1 and every λ > 0, we consider the probability P n := P sup x∈K,|y|≤e n log K(x, y) > λn a .
Let b be a fixed number such that a < b < 1. We can find the points ) and M n e ℓn b . In addition, by partitioning the ball B(0, e n ) into unit balls, we see that P n is at most Applying Chebychev inequality, we see that .
Using Proposition 5.5 and (5.6), we see that For each β > 0, we choose m = ⌊βn 1−a ⌋. In addition, for every fixed ǫ > 0, (6.2) yields 1−a n for all n sufficiently large. It follows that 1−a n .
To ensure the convergence of S 2 , we choose λ such that It follows that the series on the right hand side of (6.5) is finite. By Borel-Cantelli lemma, we have almost surely lim sup n→∞ n −a sup x∈K,|y|≤e n log K(x, y) ≤ λ .
Assume now that (H.2) holds. We put t n = t 1−a n a so that ǫ n = ǫ t tn . The Brownian motion scaling and the relation (1.24) yield It follows that Let D be the set of compactly supported continuous functions on R ℓ with unit L 2 (R ℓ )-norm. For every f ∈ D, applying Cauchy-Schwarz inequality, we see that the right-hand side in the equation above is at least We now let D ↑ R ℓ to get lim inf D↑R ℓ lim inf n→∞ 1 mn a log E B Z m (n) ≥ t 1−a sup g∈G λ R ℓf ǫ (x)g 2 (x)dx − 1 2 R ℓ |∇g(x)| 2 dx .
We now link the variation on the right-hand side with M(γ) by observing that Indeed, for each fixed g ∈ G, applying Fubini's theorem, Hahn-Banach theorem and (6.28), we have This leads us the identity (6.29). We can send ǫ ↓ 0 and apply Proposition 3.2 to obtain (6.27) under hypothesis (H.2).
We now provide the proof of (6.25). where we recall η c n is defined in (6.22). Proof. Assuming first that (H.1) holds. We recall that ǫ n = 0 in this case so that γ ǫn = γ. Let B be the σ-field generated by the Brownian motions {B j } 1≤j≤m . First we will show that for any 0 < ρ < 1 2 , we can find d > 0 sufficiently large so that on the event {min 1≤j≤m τ j ≥ t}, for every z, z ′ ∈ B(0, e n ) with |z − z ′ | ≥ d.
In particular, for every |z − z ′ | ≥ d we have which verifies (6.31).