A macroscopic multifractal analysis of parabolic stochastic PDEs

TL;DR

This paper rigorously demonstrates that solutions to certain stochastic PDEs with multiplicative noise exhibit complex, multi-scale multifractal structures in both space and time, confirming long-standing conjectures about their peak behavior.

Contribution

It provides a rigorous proof that the spatio-temporal peaks of these stochastic PDEs form infinitely many multifractals across various scales, extending previous qualitative insights.

Findings

Solutions form macroscopic multifractals with large peaks

Existence of infinitely many multifractals on multiple scales

Similar structures found in constant-coefficient cases

Abstract

It is generally argued that the solution to a stochastic PDE with multiplicative noise---such as $\dot{u}=\frac12 u"+u\xi$, where $\xi$ denotes space-time white noise---routinely produces exceptionally-large peaks that are "macroscopically multifractal." See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (1989; 1992). The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as "stretch factors." A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.

Tables

Table 1. Table 1: A table of universal constants

Constant	Source
$A$	Lemma 3.5
$A_0$	Lemma 3.5
$b_0$	Proposition 3.1
$b_1$	Theorem 3.10
$B$	Lemma 3.6
$B_0$	Lemma 3.6
$c$	Lemma 4.3
$C$	Proposition 4.7
$C_0$	Lemma 3.8
$C_1$	Proof of Proposition 4.7
$C_2$	Proof of Proposition 4.7
$\tilde{C}$	Proof of Proposition 4.6
$D$	Proof of Lemma 3.13
$K$	Proposition 3.1
$K_0$	Proposition 3.1
$K_1$	Proposition 3.11
$L$	Proposition 3.1
$L_0$	Proposition 3.1
$L_1$	Proof of Lemma 3.13
$L_2$	Proof of Lemma 3.13
$L_3$	Proposition 3.14
$L_4$	Proposition 3.14
${\text{Lip}}_{\sigma }$	Proof of Proposition 3.5
$m_0$	Theorem 3.9
$m_1$	Proposition 3.11
$M$	Lemma 3.2
$M_0$	Lemma 3.2
$M_1$	Eq. (3.21)
$N$	Lemma 3.2
$N_0$	Lemma 3.2
$N_1$	Lemma 3.13
$N_2$	Lemma 3.13
$N_3$	Proposition 3.14
$N_4$	Proposition 3.14

Equations

$$\text{Cov}[\xi (t,x),\xi (s,y)]={\delta }_0(t-s)\cdot {\delta }_0(x-y)\text{for all s,t⩾0 and x,y∈R},$$

$$\dot{u}(t,x)=\frac{1}{2}{u}^{\prime \prime }(t,x)+\sigma (u(t,x))\xi (t,x),$$

$$\underline{\gamma }(k):=\underset{t\to \infty }{\mathrm{liminf}}{t}^{-1}\log \underset{x\in \mathbb{R}}{\inf }\mathrm{E}\left(|u(t,x){|}^k\right)\text{and}\overline{\gamma }(k):=\underset{t\to \infty }{\mathrm{limsup}}{t}^{-1}\log \underset{x\in \mathbb{R}}{\sup }\mathrm{E}\left(|u(t,x){|}^k\right),$$

$$\underset{|z|>0}{\inf }|\sigma (z)/z|>0.$$

$$M_0^k{\mathrm{e}}^{M{k}^{3}t}⩽\mathrm{E}\left(|u(t,x){|}^k\right)⩽N_0^k{\mathrm{e}}^{N{k}^{3}t},$$

$$M{k}^3⩽\underline{\gamma }(k)⩽\overline{\gamma }(k)⩽N{k}^3\text{for all k∈[2,∞).}$$

$$\mathcal{P}(\beta ):=\left\{(x,t)\in \mathbb{R}\times (\mathrm{e},\infty ):u(t,x)>{\mathrm{e}}^{\beta t}\right\},$$

$$S_{\vartheta }(x,t):=\left(x,{\mathrm{e}}^{t/\vartheta }\right)\text{for all (x,t)∈R×(0,∞)}.$$

$$S_{\vartheta }\left([1,2]\times [1,2]\right)=[1,2]\times \left[{\mathrm{e}}^{1/\vartheta },{\mathrm{e}}^{2/\vartheta }\right].$$

$$2-A{\beta }^{3/2}\vartheta ⩽{\mathrm{Dim}}_{{}_{\mathrm{H}}}\left[S_{\vartheta }\left(\mathcal{P}(\beta )\right)\right]⩽2-a{\beta }^{3/2}\vartheta \text{a.s.,}$$

$$\text{Epi}[f]:=\left\{(x,y)\in (0,\infty {)}^2:y⩾f(x)\right\}.$$

$$\underset{x\to \infty }{\mathrm{liminf}}\frac{f(x)}{x}>1,$$

$${\mathrm{Dim}}_{{}_{\mathrm{H}}}\left(\text{Epi}[f]\right)=1+\inf \left\{0<\alpha ⩽1:{\int }_1^{\infty }{\left(\frac{f^{-1}(x)}{x}\right)}^{\alpha }\frac{\mathrm{d}x}{x}<\infty \right\},$$

$${\mathcal{E}}_p:=\left\{(x,y)\in (0,\infty {)}^2:y⩾x({\log }_{+}x{)}^p\right\}\text{and}\widetilde{{\mathcal{E}}_q}:=\left\{(x,y)\in (0,\infty {)}^2:y⩾x^q\right\}.$$

$${\mathrm{Dim}}_{{}_{\mathrm{H}}}({\mathcal{E}}_p)=1+p^{-1}\text{and}{\mathrm{Dim}}_{{}_{\mathrm{H}}}(\widetilde{{\mathcal{E}}_q})=\{\begin{array}{ll}2& \text{if 0<q⩽1},\\ 1& \text{if q>1}.\end{array}$$

$${\mathrm{Dim}}_{{}_{\mathrm{H}}}\left(S_{\vartheta }(\mathcal{P}(\beta ))\cap {\tilde{\mathcal{E}}}_q\right)⩽{\mathrm{Dim}}_{{}_{\mathrm{H}}}({\tilde{\mathcal{E}}}_q)=1\text{a.s.}$$

$$S_{\vartheta }\left(\mathcal{P}(\beta ))\cap \left({\mathbb{R}}^2\setminus {\tilde{\mathcal{E}}}_q\right)\right).$$

$$\left({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\right)\cap S_{\vartheta }\left(\mathcal{P}(\beta ))\cap \left({\mathbb{R}}^2\setminus {\tilde{\mathcal{E}}}_q\right)\right)=:{\mathbb{R}}_{+}^2\setminus {\mathcal{R}}_1.$$

$${\mathcal{S}}_n:=\left[{\mathrm{e}}^n,{\mathrm{e}}^{n+1}\right]\times \left[{\mathrm{e}}^n,{\mathrm{e}}^{n+1}\right][n\gg 1].$$

$$\mathcal{B}:=\bigcup _{n=1}^{\infty }\left(\left[{\mathrm{e}}^n,{\mathrm{e}}^{n+1}\right]\times \left[{\mathrm{e}}^n,{\mathrm{e}}^{n+1}\right]\right)⊊{\mathbb{R}}_{+}^2\setminus ({\mathcal{R}}_2\cup {\mathcal{R}}_1).$$

$$\max \left\{1,2-A{\beta }^{3/2}\vartheta \right\}⩽{\mathrm{Dim}}_{{}_{\mathrm{H}}}\left[S_{\vartheta }\left(\mathcal{P}(\beta )\right)\cap \mathcal{B}\right]⩽\max \left\{1,2-a{\beta }^{3/2}\vartheta \right\}\text{a.s.,}$$

$${\mathcal{V}}_k:={\left(-{\mathrm{e}}^k,{\mathrm{e}}^k\right]}^2,{\mathcal{S}}_0:={\mathcal{V}}_0,{\mathcal{S}}_{k+1}:={\mathcal{V}}_{k+1}\setminus {\mathcal{V}}_k.$$

$$Q^{(n)}:=(i{\mathrm{e}}^n,(i+1){\mathrm{e}}^n]\times (j{\mathrm{e}}^n,(j+1){\mathrm{e}}^n],$$

$$\mathcal{D}:=\bigcup _{n=-\infty }^{\infty }{\mathcal{D}}_n.$$

$${\mathcal{D}}_{⩾1}:=\bigcup _{n=0}^{\infty }{\mathcal{D}}_n.$$

$${\nu }_{\rho }^k(A):=\min \sum _{i=1}^m{\mathrm{e}}^{\rho ({\ell }_i-k-1)},$$

$${\mathrm{Dim}}_{{}_{\mathrm{H}}}(A):=\inf \left\{\rho >0:\sum _{k=1}^{\infty }{\nu }_{\rho }^k(A)<\infty \right\}.$$

$$\underset{x\to \infty }{\mathrm{liminf}}\frac{f(x)}{x}>\mathrm{e}.$$

$$f_n:=f^{-1}({\mathrm{e}}^{n+1})\text{for all integers n⩾0}.$$

$$\text{Epi}[f]\cap {\mathcal{S}}_{n+1}\subset (0,f_n]\times \left({\mathrm{e}}^n,{\mathrm{e}}^{n+1}\right]\text{for all integers n⩾0}.$$

Full text

Keywords: Intermittency, space-time multifractality, macroscopic/large-scale Hausdorff dimension, stochastic partial differential equations.

AMS 2000 subject classification: Primary 60H15; Secondary. 35R60, 60K37.

1 Introduction

1.1 The main result

Let $\xi$ denote space-time white noise, normalized so that

[TABLE]

and consider, throughout the paper, the stochastic heat equation

[TABLE]

defined on $(t\,,x)\in(0,\infty)\times\mathbb{R}$ with initial datum $u(0)=u_{0}\in L^{\infty}(\mathbb{R})$. We always will assume that $u_{0}$ and $\sigma$ are nonrandom real-valued functions on $\mathbb{R}$, that $\sigma$ is Lipschitz continuous and satisfies $\sigma(0)=0$, and that $\inf_{x\in\mathbb{R}}u_{0}(x)>0$. Among many other things, these conditions ensure that (1.1) has a unique continuous and strictly-positive solution $u(t\,,x)$ that has finite moments of all order, uniformly in $x$ and locally uniformly in $t$ [12, 11, 13, 21, 25, 27].

The main objective of this paper is to study the intermittency properties of the solution to the stochastic PDE (1.1). In order to recall the meaning of this phrase, let us first define

[TABLE]

for all $k\in[2\,,\infty)$. The functions $\underline{\gamma}$ and $\overline{\gamma}$ are known respectively as the lower and the upper moment Lyapunov exponents of the solution $u$ to (1.1). According to Jensen’s inequality, both $\underline{\gamma}$ and $\overline{\gamma}$ are nondecreasing functions on $[2\,,\infty)$. The solution $u$ to (1.1) is said to be intermittent if $\underline{\gamma}$ and $\overline{\gamma}$ are both strictly increasing on $[2\,,\infty)$; see [15] and [5, 6, 16, 23, 24, 28] for earlier variations.

It is known that the solution to (1.1) is intermittent [15] under the additional constraint that $\sigma$ satisfies the following condition:

[TABLE]

For this reason, we might refer to Condition (1.2) as an “intermittency condition.” The intermittency condition (1.2) also has quantitative consequences. For example, it implies—see [5, 7, 8, 15, 18]—that there exist finite and positive constants $M_{0}<N_{0}$ and $M<N$ such that

[TABLE]

uniformly for all real numbers $x\in\mathbb{R}$, $t>0$, and $k\geqslant 2$. It follows from these bounds that

[TABLE]

Also, (1.3) suggests that the tall spatio-temporal peaks of the stochastic process $u$ might grow exponentially with time. For a heuristic argument see the Introductions of Bertini and Cancrini [5] and Camona and Molchanov [6], together with Chapter 7 of Khoshnevisan [21]. With this connection to spatio-temporal peaks in mind, let us consider the random space-time set,

[TABLE]

of peaks of height profile $t\mapsto\exp(\beta t)$ for every $\beta>0$. When $t\gg 1$ and $(x\,,t)\in\mathscr{P}(\beta)$, we have $u(t\,,x)>{\rm e}^{\beta t}\gg 1$. The present work, in a sense, implies that a “typical” such pair $(x\,,t)$ in fact satisfies $u(t\,,x)\approx{\rm e}^{\beta t}$. Thus, we see that if $\mathscr{P}(\beta)\neq\varnothing$ a.s. for infinitely-many distinct $\beta>0$, then there are infinitely-many natural length scales in which one can measure the tall peaks of the solution to (1.1). This will verify, quantitatively, a property that is believed to hold for a large class of “complex systems”; see Gibbon and Titi [17] for an argument.

In fact, the situation is more complicated still. For every $\vartheta>0$ let us define a function $S_{\vartheta}:\mathbb{R}\times(0\,,\infty)\to\mathbb{R}\times(1\,,\infty)$ as follows:

[TABLE]

It is easy to see that, for small values of $\vartheta$, the application $S_{\vartheta}$ amounts to a nonlinear stretching of $\mathbb{R}\times(0\,,\infty)$ in the $t$-direction. For instance,

[TABLE]

In this way we can see that, when $\vartheta$ is small, $S_{\vartheta}$ maps the upright square box $[1\,,2]\times[1\,,2]$ to an elongated, stretched, box in the $(x\,,t)$ plane.

Let ${\rm Dim}_{{}_{\rm H}}$ denote the Barlow–Taylor [3, 4] macroscopic Hausdorff dimension on $\mathbb{R}^{2}$. [For a detailed definition of ${\rm Dim}_{{}_{\rm H}}$ see §2 below.] The following is the main result of this paper.

Theorem 1.1**.**

If $\sigma$ satisfies the intermittency condition (1.2), then there exist finite constants $A>a>0$ and $b,\varepsilon>0$ such that

[TABLE]

valid for every $\beta>b$ and $\vartheta\in(0\,,\varepsilon\beta^{-3/2})$.

The following corollary of Theorem 1.1 clarifies the intent of that theorem.

Corollary 1.2**.**

If (1.2) holds then there exist nonrandom numbers $0<\beta_{1}<\beta_{2}<\cdots$ and $0<\vartheta_{1}<\vartheta_{2}<\cdots$ such that:

• •

$1<{\rm Dim}_{{}_{\rm H}}[S_{\vartheta_{i}}(\mathscr{P}(\beta_{j}))]<2$* a.s. for all $i,j\geqslant 1$; and*

• •

If there exist $i,j,k,l\geqslant 1$ such that ${\rm Dim}_{{}_{\rm H}}[S_{\vartheta_{i}}(\mathscr{P}(\beta_{j}))]={\rm Dim}_{{}_{\rm H}}[S_{\vartheta_{k}}(\mathscr{P}(\beta_{l}))]$ with positive probability, then $i=k$ and $j=l$.

Corollary 1.2 shows, in particular, that there exist infinitely-many different stretch factors $\vartheta_{1},\vartheta_{2},\ldots$ and infinitely-many different length scales $\beta_{1},\beta_{2},\ldots$ such that for every $i\geqslant 1$, the $\vartheta_{i}$-stretching of peaks of height $t\mapsto\exp\{\beta_{j}t\}$ [$j=1,2,\ldots$] all have distinct and nontrivial macroscopic Hausdorff dimension. This means that every $\vartheta_{i}$-stretching of the tall peaks of $u$ is macroscopically multifractal. Moreover, the said Hausdorff dimensions themselves are distinct as we vary the stretch factors. One can interpret this finding as follows: Under the intermittency condition (1.2), the tall peaks of the solution to (1.1) form different multifractals on infinitely-many different stretch scales. We believe that the quantitative statement of Theorem 1.1 and its proof are novel. However, the idea that the peaks of $u$ should form very complex macroscopic space-time multifractals has been argued much earlier in the literature. We learned that idea from an insightful paper by Gibbon and Doering [16] on the role of intermittency in turbulence. And a paper by Zimmerman et al [29] discusses this sort of complex multifractal behavior in the context of the closely-related stochastic Allen–Cahn equation with multiplicative forcing.

In a recent paper [22] we have established that, at each fixed time $t>0$, the solution to (1.1) under (1.2) and $\sigma\equiv 1$ are both multifractal. This is a somewhat counterintuitive statement because the solution to (1.1) is intermittent under (1.2)—this is close to the KPZ universality class [2, 10, 19, 20]—and is non intermittent when $\sigma\equiv 1$—this is close to the Edwards-Wilkinson universality class [2, 10, 14]. We shall see in §4 below [see Theorem 4.1] that, as can be determined by Hausdorff dimension considerations alone, the spatio-temporal peaks of the constant-$\sigma$ case are significantly smaller than the spatio-temporal peaks of the case (1.2), though both models have infinitely-many different natural length scales and stretch factors.

1.2 An outline of the proof of Theorem 1.1

The proof of Theorem 1.1 hinges on a blend of probabilistic, analytic, and geometric ideas, many of which we believe are novel. It also relies on various probability estimates of our earlier paper [22], which we will recall in due time.

In the remainder of this introduction we outline the intuition behind the proof of Theorem 1.1, though the proper proof itself contains a number of additional technical hurdles that will need to be circumvented.

First, we present the following elegant geometric result, whose proof appears in the next section of the paper.

Proposition 1.3**.**

Suppose $f:(0\,,\infty)\to(0\,,\infty)$ is a strictly increasing convex function, and recall that the epigraph of $f$ in $(0\,,\infty)^{2}$ is the planar set,

[TABLE]

If

[TABLE]

then

[TABLE]

where $\inf\varnothing:=1$.

Proposition 1.3, and its proof, together will imply the following.

Corollary 1.4**.**

Let $\log_{+}y:=\log(y\vee{\rm e})$ for all $y\geqslant 0$, and for all $p>1$ and $q>0$ define

[TABLE]

Then,

[TABLE]

Choose and fix an arbitrary $q>1$, as close to one as one would like. According to Corollary 1.4, ${\rm Dim}_{{}_{\rm H}}(\widetilde{\mathcal{E}}_{q})=1$; this and elementary properties of macroscopic Hausdorff dimension readily imply that

[TABLE]

One may notice next that Theorem 1.1 asserts that ${\rm Dim}_{{}_{\rm H}}[S_{\vartheta}(\mathscr{P}(\beta))]\geqslant 1$ a.s. over the range of $\vartheta$ and $\beta$ mentioned in the statement of the theorem. Thus, it follows that all of the interesting fractal behavior of $S_{\vartheta}(\mathscr{P}(\beta))$ occurs off the infinite set $\widetilde{\mathcal{E}}_{q}$. In other words, in order to understand the large-scale fractal structure of $S_{\vartheta}(\mathscr{P}(\beta))$, it is necessary and sufficient to understand the large-scale fractal structure of the random set

[TABLE]

A simple symmetry calculation reduces this problem to one about the analysis of the random set

[TABLE]

Figure 1 includes a depiction of the restriction of $\mathcal{R}_{1}$ to a large box $\cup_{i=1}^{n}\mathcal{S}_{i}$, where

[TABLE]

We might think of $\mathcal{S}_{n}$ as the “$n$th shell.”

The preceding discussion tells us that—as far as the macroscopic structure of $S_{\vartheta}(\mathscr{P}(\beta))$ is concerned—nothing interesting happens in $\mathcal{R}_{1}$.

Let $\mathcal{R}_{2}$ denote the symmetric reflection of $\mathcal{R}_{1}$ about the diagonal of $\mathbb{R}^{2}_{+}$; see Figure 1. A second symmetry calculation shows that nothing interesting happens in $\mathcal{R}_{2}$. Thus, it follows that all of the interesting large-scale fractal structure of $S_{\vartheta}(\mathscr{P}(\beta))$ is contained in the part of $\mathbb{R}^{2}_{+}$ that is sandwiched between $\mathcal{R}_{2}$ and $\mathcal{R}_{1}$; that is, to $\mathbb{R}^{2}_{+}\setminus(\mathcal{R}_{2}\cup\mathcal{R}_{1})$.

Next, a covering argument can be devised to reduce the domain of interest from the relatively complicated infinite set $\mathbb{R}^{2}_{+}\setminus(\mathcal{R}_{2}\cup\mathcal{R}_{1})$ to the much simpler infinite set

[TABLE]

The restriction of the unbounded set $\mathcal{B}$ to the $n$th shell appears in Figure 1 as a black upright square. In this way, we see that the proof of Theorem 1.1 is reduced to proving that

[TABLE]

for all $\beta$ sufficiently large and $\vartheta$ sufficiently small. This reduction is significant because every point $(x\,,t)$ in the restriction of $\mathcal{B}$ to the $n$th shell has the property that ${\rm e}^{-1}\leqslant x/y\leqslant{\rm e}$. That is, the spatial behavior and the temporal behavior of $u$ in $\mathcal{B}$ are, in some sense, comparable. This property turns out to, in some sense, help “homogenize” our problem on large scales.

In order to prove (1.6), we now enlarge our view of $\mathcal{B}$—see Figure 2—and do a multiscale analysis in $\mathcal{B}$.

For every large integer $n\gg 1$, let us subdivide, in the $t$ direction, the box $\mathcal{B}\cap[{\rm e}^{n},{\rm e}^{n+1}]$ using equally-spaced lines $L_{1},L_{2},\ldots$ that are one unit apart [in the $t$, or vertical, direction]. It turns out that there exist two numbers $\rho_{L}=\rho_{L}(\vartheta\,,\beta)\in(0\,,1)$ and $\rho_{U}=\rho_{U}(\vartheta\,,\beta)\in(0\,,1)$ with $\rho_{L}\leqslant\rho_{U}$ such that the following happens almost surely for all $n$ large:

For every $\rho\in(\rho_{U}\,,1)$, the equipartition of every $L_{j}$ into subintervals of length $\exp(n\varrho)$ has the property that all of the said subintervals contain at least one point where the peak of $u$ is of height $\geqslant\exp\{\beta t\};$ whereas 2. 2.

For every $\rho\in(0\,,\rho_{L})$, none of the mentioned subintervals correspond to a peak of height $\geqslant\exp\{\beta t\}$.

In other words, $S_{\vartheta}(\mathscr{P}(\beta))\cap\mathcal{B}$ behaves, on large scales, as a “random self-similar fractal.” It is easy to compute the macroscopic Hausdorff dimension of a self-similar fractal; a variation on that calculation then yields (1.6).

1.3 A brief outline of the paper

Let us conclude the Introduction by describing briefly the structure of the paper.

In §2 we recall some basic facts about the Barlow–Taylor theory of macroscopic fractals, macroscopic Hausdorff dimension, etc. [3, 4]. Proposition 1.3 and Corollary 1.4 are also proved in §2.

Section 3 is dedicated to the proof of Theorem 1.1 and its Corollary 1.2. The results of this section include large-deviations probability bounds, localization estimates, and bounds on a so-called spatial correlation length of the solution to (1.1). It is generally believed that the solution to (1.1) spatially decorrelates at length scale $\Theta(t^{3/2})$ when $t\gg 1$.111Recall that $f(t)=\Theta(g(t))$ means that there exists $a>1$ such that $a^{-1}g(t)\leqslant f(t)\leqslant ag(t)$ for all sufficiently large $t$. We have not found a carefully-stated form of this as a conjecture in print, but the fact is for example hinted at implicitly in Corwin [10], and is also believed to be true by many physicists. Here, we prove that the said correlation length is not more than $\Omega(t^{2})$;222Recall that $f(t)=\Omega(g(t))$ means that there exists a positive constant $a$ such that $f(t)\geqslant ag(t)$ for all sufficiently large $t$. for a careful statement see Theorem 3.9. This result is the best-known bound to date on the correlation length of $x\mapsto u(t\,,x)$ when $t\gg 1$.

For purposes of comparison, we derive in §4 an analogue of Theorem 1.1 that holds for the solution to (1.1) in the case that $\sigma$ is constant. The main theorem of that section is Theorem 4.1 which implies that, when $\sigma$ is a constant, the exceptionally-tall spatio-temporal peaks of the solution to (1.1) are much smaller than when for example $\sigma(u)=u$. But the complex multifractal structure of the peaks continues to pervade.

At the end of the paper we have taken care to collect a list of many of the constants that appear within proofs, particularly those of Theorem 1.1. It turns out that one has to be very careful in some cases in order to make sure that various parameter dependencies do not arise. In some cases, this is a truly non-trivial task; therefore, we have taken care to outline the important universal constants, together with where they first arise, in Table 1 in an appendix that follows the bibliography. In this way one can use Table 1 in order to keep track of the various parameter dependencies of interest.

2 Macroscopic dimension

Let us begin by recalling the Barlow–Taylor theory of macroscopic Hausdorff dimension [3, 4].

For every integer $k\geqslant 0$, let

[TABLE]

Also, for every $n\in\mathbb{Z}$ let $\mathscr{D}^{(n)}$ denote the collection of all ${\rm e}$-adic squares of the form

[TABLE]

where $i,j\in\mathbb{Z}$ range over all integers. If a square $Q^{(n)}$ has the form (2.2), then we say that $(i{\rm e}^{n},j{\rm e}^{n})$ is the southwest corner of $Q^{(n)}$, and ${\rm e}^{n}$ is the sidelength of $Q^{(n)}$. By $\mathscr{D}$ we mean the collection of all ${\rm e}$-adic squares of $\mathbb{R}^{2}$; that is,

[TABLE]

A special role is played by

[TABLE]

This is the collection of all ${\rm e}$-adic squares of sidelength not smaller than $1$.

For every integer $k\geqslant 0$, all $\rho\in(0\,,\infty)$, and each $A\subseteq\mathbb{R}^{d}$ define

[TABLE]

where the minimum is over all possible coverings of $A\cap\mathcal{S}_{k}$ by ${\rm e}$-adic squares $Q_{1},\ldots,Q_{m}\subseteq\mathcal{S}_{k}$ of sidelength ${\rm e}^{\ell_{i}}\geqslant 1$. Note in particular that these squares are all elements of $\mathscr{D}_{\geqslant 1}$.

M. T. Barlow and S. J. Taylor [3, 4] defined the macroscopic Hausdorff dimension of a set $A\subseteq\mathbb{R}^{2}$ as

[TABLE]

The papers by Barlow and Taylor [3, 4] contain further information about the macroscopic Hausdorff dimension ${\rm Dim}_{{}_{\rm H}}$. Among other things, the following result of Barlow and Taylor is noteworthy.

Proposition 2.1** (Barlow and Taylor [4]).**

Let $A\subseteq\mathbb{R}^{2}$ be a set.

Suppose we redefined $\nu^{k}_{\rho}(A)$ as in (2.3), but where the minimum is over all possible coverings of $A\cap\mathcal{S}_{k}$ by squares $Q_{1},\ldots,Q_{m}\subseteq\mathcal{S}_{k}$ of the form $[x_{1}\,,x_{1}+r)\times[x_{2}\,,x_{2}+r)$, where $r\geqslant 1$. Then this change does not alter the numerical value of ${\rm Dim}_{{}_{\rm H}}(A)$. 2. 2.

Choose and fix a real number $a>1$, and suppose we redefined $\{\mathcal{V}_{k}\}_{k\geqslant 0}$—and hence also $\{\mathcal{S}_{k}\}_{k\geqslant 0}$—in (2.1) as follows: $\mathcal{V}_{k}:=\left(a^{k}\,,a^{k+1}\right]^{2}$ for every integer $k\geqslant 0$. Then, this change does not affect the numerical value of ${\rm Dim}_{{}_{\rm H}}(A)$.

Having dispensed with an introduction to macroscopic Hausdorff dimension, we establish Proposition 1.3 next.

Proof of Proposition 1.3.

We first prove the proposition under the more restrictive condition,

[TABLE]

We then outline how to replace the preceding by the weaker condition (1.4).

As is commonly done in the local theory of dimension, one proceeds by first obtaining an upper bound and then a lower bound for ${\rm Dim}_{{}_{\rm H}}(\text{\rm Epi}[f])$. We first consider the upper bound for ${\rm Dim}_{{}_{\rm H}}(\text{\rm Epi}[f])$.

Define

[TABLE]

Because $f$ is strictly increasing, one can draw a picture—see Figure 3—in order to see that

[TABLE]

Now, the condition (2.4) ensures that $f_{n}<{\rm e}^{n}$ for all sufficiently large integers $n\gg 1$. Therefore,

[TABLE]

where the union is taken over all nonnegative integers $j\leqslant\lfloor({\rm e}^{n+1}-{\rm e}^{n})/f_{n}\rfloor+1$, and

[TABLE]

for all integers $n,j\geqslant 0$. Each $R_{j,n}$ is an upright square of side $f_{n}$. Therefore,

[TABLE]

for all $\rho>0$ and $n\gg 1$. It follows from this inequality that

[TABLE]

Cauchy’s test shows that $\sum_{n=0}^{\infty}(f_{n}/{\rm e}^{n})^{\rho-1}$ converges iff $\int_{1}^{\infty}[f^{-1}(x)/x]^{\rho-1}x^{-1}{\rm d}x$ converges. Therefore, the preceding display proves that ${\rm Dim}_{{}_{\rm H}}(\text{\rm Epi}[f])$ is bounded from above by the expression on the right-hand side of (1.5). We would like to record the fact that this part of the proof does not require $f$ to be convex.

We now derive a matching lower bound for ${\rm Dim}_{{}_{\rm H}}(\text{\rm Epi}[f])$. Define, for every integer $n\geqslant 0$,

[TABLE]

It might help to consider Figure 3 at this point, keeping in mind that the rectangle $A_{n}$ is the disjoint union $A_{n}^{u}\cup A_{n}^{l}$.

Since $f$ is convex, $A_{n}^{u}\subseteq\text{\rm Epi}[f]\cap A_{n}$ for all $n\geqslant 0$. In addition, we have

[TABLE]

thanks to mid-axial symmetry. Consequently,

[TABLE]

Therefore, it suffices to prove that

[TABLE]

This and Cauchy’s integral test together prove that ${\rm Dim}_{{}_{\rm H}}(\text{\rm Epi}[f])$ is bounded from below by the right-hand side of (1.5), and hence complete the proof.

Let us define a Borel measure $\mu$ on $\mathbb{R}^{2}$ as follows: For every Borel set $E\subset\mathbb{R}^{2}$,

[TABLE]

where $|\cdots|$ denotes the 2-dimensional Lebesgue measure. For every upright box of the form $(x_{1}\,,x_{1}+r]\times(x_{2}\,,x_{2}+r]$—where $r\geqslant 1$—and for every $\rho\in(0\,,1]$,

[TABLE]

We now use the density theorem of Barlow and Taylor [4, Theorem 4.1], and the fact that $\mu(A_{n})=f_{n}({\rm e}^{n+1}-{\rm e}^{n})$, in order to obtain the following:

[TABLE]

This inequality immediately implies (2.5) and completes the proof of the lower bound under the more restrictive condition (2.4).

To complete the argument, we outline how one changes the preceding to accommodate the more general condition (1.4). Choose and fix a real number $a>1$ such that

[TABLE]

and redefine $\{\mathcal{V}_{k}\}_{k\geqslant 0}$—hence also $\{\mathcal{S}_{k}\}_{k\geqslant 0}$—in (2.1). Now repeat the preceding argument but everywhere replace ${\rm e}^{n}$ and ${\rm e}^{n+1}$ by $a^{n}$ and $a^{n+1}$ respectively. Proposition 2.1 ensures that these changes do not affect the end result of the method. ∎

Having just completed the proof of Proposition 1.3, we can now establish Corollary 1.4. This proof will conclude the material of this section.

Proof of Corollary 1.4.

The function $f(x):=x(\log_{+}x)^{p}$ satisfies the conditions of Proposition 1.3, and one can deduce the asserted formula for ${\rm Dim}_{{}_{\rm H}}(\mathcal{E}_{p})$ from Proposition 1.3.

If $q>1$, then $f(x):=x^{q}$ satisfies (1.4), and Proposition 1.3 immediately shows that ${\rm Dim}_{{}_{\rm H}}(\widetilde{\mathcal{E}_{q}})=1$.

Finally, if $q\in[0\,,1]$, then it suffices to prove that ${\rm Dim}_{{}_{\rm H}}(\widetilde{\mathcal{E}_{q}})\geqslant 2$. Because $\widetilde{\mathcal{E}_{q}}\cap\mathcal{S}_{n}$ contains

[TABLE]

for all sufficiently large integers $n\gg 1$, it suffices to prove that

[TABLE]

Define

[TABLE]

and note that

[TABLE]

thanks to mid-axial symmetry. As was observed first by Barlow and Taylor [4],

[TABLE]

for all sets $A$ and $B$. Therefore, it remains to prove that

[TABLE]

Since $\cup_{n=0}^{\infty}[\mathcal{F}_{n}\cup\mathcal{F}_{n}^{\prime}]=\cup_{n=0}^{\infty}({\rm e}^{n},{\rm e}^{n+1}]^{2}$, we can define a measure $\mu$ on $\cup_{n=0}^{\infty}[\mathcal{F}_{n}\cup\mathcal{F}_{n}^{\prime}]$ as follows:

[TABLE]

for all $n\geqslant 0$ and Borel sets $A\subset\mathbb{R}^{2}_{+}$, where $|\,\cdots|$ denotes the planar Lebesgue measure. Since $\mu((a\,,a+r]\times(b\,,b+r])\leqslant r^{2}$ and $\mu(\mathcal{S}_{n})\geqslant\text{const}\cdot{\rm e}^{2n}$ uniformly for all $n\geqslant 1$, an appeal to a density theorem of Barlow and Taylor [4, Theorem 4.1] yields (2.6). ∎

3 Proof of Theorem 1.1 and Corollary 1.2

3.1 A large deviations estimate

The following is the main result of this section.

Proposition 3.1**.**

If (1.2) holds, then there exist positive and finite constants $b_{0}$, $K_{0}<L_{0}$, and $K<L$ such that

[TABLE]

uniformly for all $t>1$, $\beta>b_{0}$, and $x\in\mathbb{R}$.

The proof hinges on the following moment inequality that was mentioned earlier in the Introduction.

Lemma 3.2** (Joseph et al [18]).**

There exist positive and finite constants $M_{0}<N_{0}$ and $M<N$ such that (1.3) holds uniformly for all real numbers $k\geqslant 2$, $t>0$, and $x\in\mathbb{R}$.

We will use Lemma 3.2 in order to establish Proposition 3.1 in two steps: An upper bound (see Lemma 3.3) and a “matching” lower bound (see Lemma 3.4). Those results are presented in the sequel, and without further comment.

Lemma 3.3**.**

Let $N$ denote the constant of Lemma 3.2. Then for all $\beta\geqslant 12N$,

[TABLE]

Proof.

Choose and fix a real number $k\geqslant 2$. By Lemma 3.2 and Chebyshev’s inequality, the following holds uniformly for all $t>0$ and $x\in\mathbb{R}$:

[TABLE]

as $t\to\infty$. The asymptotically-optimum choice of $k$ is $\sqrt{\beta/3N}$, which is at least $2$ when $\beta\geqslant 12N$. Plug $k:=\sqrt{\beta/3N}$ in (3.1) to obtain the lemma. ∎

Lemma 3.4**.**

Let $M$ and $N$ denote the constants of Lemma 3.2. Then for all $\beta\geqslant 4M$,

[TABLE]

Proof.

Choose and fix real numbers $q>1$ and $m\geqslant 2$, and define

[TABLE]

By the Paley–Zygmund inequality (apply [21, Lemma 7.3, p. 64] with $n:=qm$),

[TABLE]

where $\|X\|_{m}:=[\mathrm{E}(X^{m})]^{1/m}$ for every $X\in L^{m}(\Omega)$. Therefore, Lemma 3.2 implies that

[TABLE]

where $D_{m,q}:=C_{2,q}(M_{0}/N_{0})^{mq/(q-1)}$.

If $4M\leqslant\beta<m^{2}M$, then

[TABLE]

for all $x\in\mathbb{R}$, and $t$ sufficiently large. Therefore, (3.2) and (3.3) together imply that

[TABLE]

for all $m\geqslant 2$ such that $4M\leqslant\beta<m^{2}M$. In particular,

[TABLE]

for all $m\geqslant 2$ such that $4M\leqslant\beta<m^{2}M$. Let $m$ tend downward to ${\sqrt{\beta/M}}$ in order to see that

[TABLE]

for all $q>1$. We can optimize the right-hand side of this expression over all $q>1$ to complete the derivation. ∎

Proof of Proposition 3.1.

The upper bound in the statement of the proposition follows immediately from Lemma 3.3. In order to deduce the lower bound, we first apply (3.4) with $m:=\sqrt{2\beta/M}$ in order to see that uniformly for all $x\in\mathbb{R}$, $\beta\geqslant 4M$, $q>1$, and large $t>0$,

[TABLE]

In particular,

[TABLE]

These facts and Lemma 3.4 together establish the lower bound of the proposition for $t$ sufficiently large, say $t>t_{0}$ for a sufficiently-large $t_{0}>1$. When $t\in(1\,,t_{0})$, we appeal to (3.2), but adjust the constants in (3.4) suitably. ∎

3.2 Correlation length

The main result of this section is a carefully-stated version of the following (see Theorem 3.9): As $t\to\infty$, the correlation length of $x\mapsto u(t\,,x)$ is at least $at^{2}$ for a suitable constant $a$. The proof relies on a localization idea that was introduced in Conus et al [9].

First, recall [11, 27] that the solution to (1.1) can be written as the unique solution to the stochastic integral equation,

[TABLE]

valid for all $x\in\mathbb{R}$ and $t>0$, where $p_{t}(x)$ denotes the heat kernel; that is,

[TABLE]

Now, let us choose and fix some $c>0$, and define intervals

[TABLE]

for every $x\in\mathbb{R}$ and $t>0$. Define $u^{(c)}_{0}(x):=u_{0}(x)$ for all $x\in\mathbb{R}$, and consider the random integral equation,

[TABLE]

for all $x\in\mathbb{R}$ and $t>0$. This is a “localized form” of the solution $u$ to (1.1).

One can prove that (3.7) has a unique strong solution, in the usual way, using Picard’s iteration; see Proposition 3.7 for a statement. Since we will need to pay close attention to the quantitative details of the argument—see in particular (3.16) below—we work out the details of that argument [together with the requisite estimates] in this section.

Let $u^{(c,0)}(t,x):=u_{0}(x)$ for all $t\geqslant 0$ and $x\in\mathbb{R}$, and then define

[TABLE]

iteratively for all $n\geqslant 0$.

The sequence $u^{(c,1)},u^{(c,2)},\ldots$ is basically the Picard-iteration approximation to the desired solution $u^{(c)}$ of (3.7). Our first lemma estimates the moments of each $u^{(c,n)}$.

Lemma 3.5**.**

There exist positive and finite constants $A,A_{0}$ such that for all real numbers $k\geqslant 2$, $t\geqslant 0$, and $x\in\mathbb{R}$,

[TABLE]

Proof.

Since $p_{t}*u_{0}$ is bounded uniformly by $\|u_{0}\|_{L^{\infty}(\mathbb{R})}$, Minkowski’s inequality yields

[TABLE]

where we have used a special form of the Burkholder–Davis–Gundy inequality [21, Theorem B.1, p. 103] in the first inequality. Because $\sigma$ is Lipschitz and vanishes at zero, $|\sigma(z)|\leqslant\text{\rm Lip}_{\sigma}|z|$ for all $z\in\mathbb{R}$, where $\text{\rm Lip}_{\sigma}:=\sup_{z\in\mathbb{R}/0}|\sigma(z)/z|$ denotes the Lipschitz constant of $\sigma$. Therefore,

[TABLE]

where

[TABLE]

for all space-time random fields $\Phi$, and all real numbers $\alpha$ and $k\in[2\,,\infty)$. Since

[TABLE]

it follows from (3.10) that

[TABLE]

where $Q$ is a finite constant that does not depend on $(c\,,k\,,\alpha\,,n)$. In particular, we can set $\alpha:=16(Qk)^{2}$ to see that

[TABLE]

for all integers $n\geqslant 0$ and reals $k\geqslant 2$. Because

[TABLE]

we iterate (3.13) in order to see that

[TABLE]

Equivalently,

[TABLE]

The lemma follows from this inequality. ∎

Next we wish to show that $\{u^{(c,n)}\}_{n=0}^{\infty}$ forms a Cauchy sequence. En route we will also control carefully the size of the gaps $u^{(c,n+1)}-u^{(c,n)}$ of that Cauchy sequence.

Lemma 3.6**.**

Let $A$ be as in the statement of Lemma 3.5. There exist positive and finite constants $B_{0}$ and $B$ such that for all integers $n\geqslant 1$ and real numbers $k\geqslant 2$, and $t\geqslant 0$,

[TABLE]

Proof.

As in (3.9), we obtain

[TABLE]

Therefore, if we define $\mathcal{N}_{\alpha,k}$ as in (3.11), then

[TABLE]

where $Q$ denotes the same constant that appeared earlier in (3.12). It follows that

[TABLE]

for all $n\geqslant 1$. Lemma 3.5 and its proof together show that both sides of the preceding inequality are finite. Therefore, iteration yelds the following for all $n\geqslant 1$:

[TABLE]

This, (3.14), and (3.15) together yield the following for all $n\geqslant 1$:

[TABLE]

which is more than enough to establish the lemma. ∎

Lemmas 3.5 and 3.6 readily yield the following.

Proposition 3.7**.**

The random integral equation (3.7) admits a predictable solution $u^{(c)}$ that is unique among all solutions that satisfy the inequality

[TABLE]

where $A$ and $A_{0}$ are as in Lemma 3.5. In addition, there exists a finite constant $B_{1}$ such that

[TABLE]

for all $t\geqslant 0$ and $k\geqslant 2$, $B$ was defined in Lemma 3.6.

As was implied earlier, it is a standard fact that the solution $u^{(c)}$ to (3.7) exists and is unique. The key feature of the preceding is the quantitative bound (3.16), which is a byproduct of our particular method.

Next we prove that if $c$ is large enough then $u^{(c)}\approx u$. This is also a natural statement. We are being careful only because we need to be able to control the size of the error $u^{(c)}-u$. The following does that for us.

Lemma 3.8**.**

There exists a positive and finite constant $C_{0}=C_{0}(N\,,N_{0}\,,\text{\rm Lip}_{\sigma})$ such that

[TABLE]

uniformly for all real numbers $c,t>0$ and $k\geqslant 2$.

Proof.

We begin by studying a slightly different problem.

Let us combine (3.7) and (3.5), via the Burkholder–Davis–Gundy inequality in the form given in [21, Proposition 4.4, p. 36] in order to see that for all $x\in\mathbb{R}$, $c,T>0$, $t\in(0\,,T)$, and $k\in[2\,,\infty)$,

[TABLE]

We recall that $|\sigma(z)|\leqslant\text{\rm Lip}_{\sigma}|z|$ and $|\sigma(w)-\sigma(z)|\leqslant\text{\rm Lip}_{\sigma}|z-w|$ for all $w,z\in\mathbb{R}$. Thus,

[TABLE]

Define, for all $0<s<t<T$,

[TABLE]

Lemma 3.2 can now be used to control the size of $\|u(s\,,y)\|_{k}$ as follows:

[TABLE]

Since the right-hand side of (3.18) is independent of $x$, we can rewrite (3.18) as the following self-referential inequality for the process $M$:

[TABLE]

The probability that a standard normal random variable exceeds $q>0$ is at most $\frac{1}{2}\exp(-q^{2}/2)$. Therefore, uniformly for all $t>s>0$,

[TABLE]

It follows from (3.19) that for all $t>0$,

[TABLE]

Note that, if $\Phi:\mathbb{R}_{+}\to\mathbb{R}$ is deterministic, then by (3.11),

[TABLE]

for all $\eta>0$ and $k\geqslant 2$. With this in mind, it follows readily from the inequality (3.20) that, uniformly for all $t>0$,

[TABLE]

We multiply both sides by $\exp\{-\eta t\}$ and optimize over $t>0$ to find that

[TABLE]

Lemma 3.2 and Proposition 3.7 together imply that

[TABLE]

Consequently, $\mathcal{N}_{\eta}(M)\leqslant K^{2/k}<\infty$ regardless of the value of $\eta>0$. We now select $\eta:=\eta_{k}:=(64\text{\rm Lip}_{\sigma}^{4}\vee 2N)k^{2}$ in order to find that

[TABLE]

and hence,

[TABLE]

In particular, for all $t>0$,

[TABLE]

This yields the desired effect. ∎

We conclude this section with its main assertion, which is a carefully-phrased version of the following statement: If $t\to\infty$ and if $|x-y|=\Omega(t^{2})$ with a suitably-large constant, then $u(t\,,x)$ and $u(t\,,y)$ are approximately independent. See Footnote 1 for the notation $f(t)=\Omega(g(t))$. Another way to say this is that the “correlation length” of $x\mapsto u(t\,,x)$ is $\Omega(t^{2})$. It is believed that the said correlation length is $\Theta(t^{3/2})$; to date, our $\Omega(t^{2})$ bound is the best rigorously-known lower bound for the correlation length.

The proof of the above assertion is based on a coupling argument that is similar to the fixed-time coupling of Conus et al [9], though one has to compute the various constants very carefully in the present large-time context.

First, let

[TABLE]

where and $C_{0}$ and $B_{1}$ were defined respectively in Lemma 3.8 and Proposition 3.7. As was the case with the various constants $C_{0},B_{1},N,N_{0},\cdots$ that we have encountered so far, the constant $M_{1}$ depends on various parameters of our SPDE, such as $\text{\rm Lip}_{\sigma}$ and $\|u_{0}\|_{L^{\infty}(\mathbb{R})}$, but is otherwise universal, once we concentrate on a particular set of parameters in our stochastic heat equation (1.1).

Theorem 3.9**.**

There exists a finite constant $m_{0}>0$ such that for all $\mu,t\geqslant 1$ and $k\geqslant 2$, and for all nonrandom points $x_{1},\ldots,x_{m}\in\mathbb{R}$ that satisfy

[TABLE]

there exist independent random variables $Y_{1},\ldots,Y_{m}$ such that for every integer $j=1,\ldots,m$:

1. i)

$Y_{j}$* depends only on $(t\,,x_{j}\,,k)$;* 2. ii)

$Y_{j}\in L^{p}(\Omega)$* for all real numbers $p\geqslant 2$; and* 3. iii)

$\mathrm{E}\left(\left|u(t\,,x_{j})-Y_{j}\right|^{k}\right)\leqslant 2M_{1}^{k}{\rm e}^{-\mu k^{3}t}.$**

Proof.

Proposition 3.7 and Lemma 3.8 together imply that

[TABLE]

simultaneously for all real numbers $k\geqslant 2$, $c,t>0$, $x\in\mathbb{R}$, and all integers $n\geqslant 0$. We can relabel $c\leftrightarrow 2ct$ and set $n:=\lceil ct\rceil$ for constant $c>0$ in order to see that

[TABLE]

where $M_{1}$ was defined in (3.21). We apply this inequality with the following particular choice of the constant $c$:

[TABLE]

We can backtrack through the constants $A,N$, and $B$ in order to see that $c$ is $k^{2}$ times a constant that depends only on $\mu$ and the parameter choices of the SPDE (1.1); it is otherwise universal. Moreover, this choice of $c$ is large enough to ensure that:

(3.23) is applicable; and 2. 2.

$\min(c-(32\text{\rm Lip}_{\sigma}^{4}\vee N)k^{2}\,,Bc-Ak^{2})>\mu k^{2}$.

In particular, (3.23) implies the following:

[TABLE]

simultaneously for all real numbers $t>0$ and $k\geqslant 2$.

Recall that $u^{(2c_{k}t,0)}\equiv u_{0}$ is deterministic and

[TABLE]

see also (3.8). Choose and fix $t>0$, and observe that if $x_{1},\ldots,x_{m}\in\mathbb{R}$ are points that satisfy $|x_{i}-x_{j}|>2t\sqrt{c_{k}}$ when $1\leqslant i\neq j\leqslant m$, then $u^{(2c_{k}t,1)}(t\,,x_{1}),\ldots,u^{(2c_{k}t,1)}(t\,,x_{m})$ are independent. This is because the Wiener integrals $\int\varphi_{1}\,{\rm d}\xi$, …, $\int\varphi_{m}\,{\rm d}\xi$ are independent if $\varphi_{1},\ldots,\varphi_{m}\in L^{2}(\mathbb{R}_{+}\times\mathbb{R})$ have disjoint supports.

Next we observe that $u^{(2c_{k}t,2)}(t\,,x_{1}),\ldots,u^{(2c_{k}t,2)}(t\,,x_{m})$ are independent as long as the points $x_{1},\ldots,x_{m}\in\mathbb{R}$ satisfy $|x_{i}-x_{j}|>4t\sqrt{c_{k}}$ when $1\leqslant i\neq j\leqslant m$. This is because:

1. (i)

With probability one, for every $x\in\mathbb{R}$,

[TABLE] 2. (ii)

If $\Phi_{1},\ldots,\Phi_{m}$ are independent Walsh-integrable space-time random fields, then $\int_{E_{1}}\Phi_{1}\,{\rm d}\xi$, …, $\int_{E_{m}}\Phi_{m}\,{\rm d}\xi$ are independent as long as $E_{1},\ldots,E_{m}$ are disjoint Borel subsets of $\mathbb{R}_{+}\times\mathbb{R}$.

Indeed, (i) follows from the definition of $u^{(c,n)}$; and (ii) holds because $\int_{E_{i}}\Phi_{i}\,{\rm d}\xi$ and $\int_{E_{j}}\Phi_{j}\,{\rm d}\xi$ are uncorrelated jointly Gaussian random variables when $i\neq j$.

An iteration of the preceding argument proves the following: For every integer $\ell\geqslant 1$, the random variables $u^{(2c_{k}t,\ell)}(t\,,x_{1}),\ldots,u^{(2c_{k}t,\ell)}(t\,,x_{m})$ are independent as long as

[TABLE]

We may apply this final observation with $\ell:=\lceil c_{k}t\rceil$, and then use (3.24) in order to deduce the theorem with

[TABLE]

This indeed concludes the theorem because (3.25) implies (3.22).

To prove the final assertion of the theorem, one may notice first that

[TABLE]

because $c_{k}$ and $t$ are both greater than one. Consequently,

[TABLE]

Thus, one can set

[TABLE]

in order to see that indeed (3.25) implies (3.22), which concludes the proof. ∎

3.3 A lower bound

The main result of this section is the following lower bound on the macroscopic Hausdorff dimension of the [$\vartheta$-rescaled] space-time set of tall peaks of level $\beta$. We will prove it shortly.

Theorem 3.10**.**

For all $\beta>b_{1}:=\max\{\log 3\,,b_{0}/2\}$ and $\vartheta\in(0\,,L^{-1}(2\beta)^{-3/2})$, the following holds with probability one:

[TABLE]

Before we present the proof we pause and first establish a certain tail probability inequality. That inequality will play a role in the proof of Theorem 3.10, which is presented afterward.

The following is the desired tail probability bound for the solution to (1.1).

Proposition 3.11**.**

For every real number $\beta>b_{1}$ there exists a finite constant $m_{1}=m_{1}(\beta\,,L\,,M_{1}\,,K_{0})$ such that for all real numbers $t\geqslant 1$, all integers $k\geqslant 2$ and $m\geqslant 3$,

[TABLE]

uniformly for all points $x_{1},\ldots,x_{m}\in\mathbb{R}$ that satisfy the gap condition

[TABLE]

Proof.

Part iii) of Theorem 3.9 and Chebyshev’s inequality together imply that if the sequence $\{x_{i}\}_{i=1}^{m}$ satisfies the gap condition (3.22), then there exist independent random variables $Y_{1},\ldots,Y_{m}$ such that

[TABLE]

for all $z>0$, $\mu\geqslant 1$ and all integers $k\geqslant 2$. Because the $Y_{j}$’s are independent, the preceding implies that

[TABLE]

since $\beta>0$ and $t\geqslant 1$. Now, for every $j=1,\ldots,m$ and $n\geqslant 2$,

[TABLE]

the last line uses also the fact that $\beta>\log 3$ and $t\geqslant 1$ in order to deduce that $3\exp(\beta t)\leqslant\exp(2\beta t)$. Because $2\beta>b_{0}$, Proposition 3.1 yields

[TABLE]

valid for all $j=1,\ldots,m$. Now we choose $\mu\geqslant 1$ such that

[TABLE]

Then for all integers $k\geqslant 2$, we have

[TABLE]

This and (3.28) give the last term in (3.26). Moreover, it follows from (3.30) and (3.29) that

[TABLE]

owing to the elementary real-variable inequality, $1-x\leqslant\exp(-x)$ valid for all $x>0$. Plug (3.31) into (3.28) and appeal to (3.30) once again to deduce that the probability inequality of the proposition is valid uniformly for every sequence $\{x_{i}\}_{i=1}^{m}$ that satisfies the minimum gap condition (3.22). To conclude, we need to verify that (3.27) implies (3.22) for a suitable choice of $m_{1}$ which depends only on the initially-set parameters of (1.1). If $m$ is sufficiently large—how large depends only on $(\beta\,,L\,,M_{1}\,,K_{0})$—then this is clear because $\mu/\ln m$ is bounded above and below by universal constants that depend only on $(\beta,,L\,,M_{1}\,,K_{0})$ in that case. And if $m$ is below the threshold of being sufficiently large, then the proposition is tautologically true, uniformly for any choice of $m_{1}$. ∎

Now we have Proposition 3.11, we conclude this section with the following.

Proof of Theorem 3.10.

Let us choose the parameters $(\beta\,,\vartheta)$ as has been stated in the theorem, and consider the random set

[TABLE]

whose macroscopic dimension is of interest to us. Define

[TABLE]

Since $\mathcal{G}_{0}\subseteq\mathcal{G}$ a.s., it suffices to prove that

[TABLE]

Since $\vartheta L(2\beta)^{3/2}<1$, we may choose and fix an arbitrary number

[TABLE]

Define, for all integers $n\geqslant 0$,

[TABLE]

and

[TABLE]

For all such integers $n$ and $j$, and for every

[TABLE]

we can find points $x_{1},\ldots,x_{m}\in\mathcal{I}_{j,n}(\gamma)$—depending only on $(n\,,\gamma\,,\varepsilon\,,j)$—such that whenever $n>1$:

1. (C.1)

$|x_{i}-x_{l}|\geqslant{\rm e}^{n\varepsilon}$ whenever $1\leqslant i<l\leqslant m$; 2. (C.2)

$\frac{1}{2}{\rm e}^{n(\gamma-\varepsilon)}\leqslant m\leqslant 2{\rm e}^{n(\gamma-\varepsilon)}$.

Note in particular, that for every constant $Q$ there exists an integer $N(Q)\geqslant 1$ such that

[TABLE]

Let $Q:=4m_{1}\vartheta^{2}(\log 2+\gamma)^{3/2}k^{3}$, where $k\geqslant 2$ is an integer such that

[TABLE]

Observe that

[TABLE]

uniformly for all reals $t\in(\exp(n)\,,\exp(n+1)]$ and integers $n\geqslant 1$. Therefore, the minimum gap condition (3.27) is valid with $t$ replaced by $\vartheta\log t$, when $t\in(\exp(n)\,,\exp(n+1)]$ and $n\gg 1$ is sufficiently large, and the integer $k$ in (3.37). In this way we see that Proposition 3.11 applies to yield the following: Uniformly for all sufficiently-large nonnegative integers $n\gg 1$, and for all real numbers $\beta>\max\{\log 3\,,b_{0}/2\}$, $\varepsilon\in(0\,,\gamma)$, and $t\in(\exp(n)\,,\exp(n+1)]$,

[TABLE]

with

[TABLE]

It is possible to explain the meaning of the bound (3.38) in terms of the random set $\mathcal{G}_{0}$ (see (3.32)) as follows:

[TABLE]

for all sufficiently large integers $n\gg 1$.

Of course, $\kappa>0$ and $k^{3}L(2\beta)^{3/2}\vartheta>2-\gamma$; thanks to (3.34), (3.35), (3.37), and (3.39). Therefore, (3.40) and the Borel–Cantelli lemma together imply that

[TABLE]

for all but a finite number of integers $n\geqslant 1$.

Define

[TABLE]

where $\inf\varnothing:=\infty$. The assertion (3.41) ensures that $\eta_{j,n}(t)$ is a well-defined, finite, random variable for all integers $t\in({\rm e}^{n},{\rm e}^{n+1}]$, $0\leqslant j<{\rm e}^{n(1-\gamma)}$, and $n\gg 1$ sufficiently large. In fact, (3.41) can be stated in the following equivalent form:

Conclusion A. *With probability one, $(\eta_{j,n}(s)\,,s)\in\mathcal{G}_{0}$ for all integers $s\in({\rm e}^{n},{\rm e}^{n+1}]$, $0\leqslant j<{\rm e}^{n(1-\gamma)}$, and $n\gg 1$ sufficiently large.

Now let $\mu$ denote a purely atomic random measure on $\mathbb{R}\times(1\,,\infty)$ that is defined shell-by-shell as follows: For all Borel sets $A\subseteq\mathbb{R}\times(1\,,\infty)$ and $n\geqslant 0$,

[TABLE]

where $\bm{1}_{A}$ denotes the customary indicator function of the space-time set $A$, and $\bm{1}_{A}(\infty\,,s):=0$ for all $s\geqslant 0$ and $A\subseteq\mathbb{R}^{2}$.

The $\mu$-mass of $\mathcal{S}_{n}$ is easy to compute when $n$ is large. Indeed, (3.41)—see especially Conclusion A—ensures that with probability one,

[TABLE]

for all $n\gg 1$ sufficiently large.

Choose and fix an integer $n\geqslant 1$. Next consider an arbitrary integer $s\in({\rm e}^{n},{\rm e}^{n+1}]$, and real numbers $x$ and $r\geqslant 1$ such that $(x\,,x+r]\subseteq({\rm e}^{n},{\rm e}^{n+1}]$. We consider separately two cases:

1. (a)

If $r\leqslant{\rm e}^{\gamma n}$, then there are at most two integers $j_{1}$ and $j_{2}$ such that $\eta_{j_{1}}(s)$ and $\eta_{j_{2}}(s)$ are in $(x\,,x+r]$. Consequently, $\mu((x\,,x+r]\times\{s\})\leqslant 2\leqslant 2r^{\rho}$ for every $\rho>0$. Sum this inequality over all $s\in(t\,,t+r]$ to see that

[TABLE]

as long as $(x\,,x+r]\times(t\,,t+r]\subseteq\mathcal{S}_{n}$, $1\leqslant r\leqslant 2{\rm e}^{\gamma n}$, and $\rho>0$. 2. (b)

If ${\rm e}^{\gamma n}<r\leqslant{\rm e}^{n+1}-{\rm e}^{n}$, then every interval of the form $(x\,,x+r]\subseteq({\rm e}^{n},{\rm e}^{n+1}]$ can contain at most $1+r{\rm e}^{-\gamma n}\leqslant 2r{\rm e}^{-\gamma n}$ many points of the form $\eta_{j,n}(s)$. That is, in this case, for every integer $s\in({\rm e}^{n},{\rm e}^{n+1}]$ and all intervals $(x\,,x+r]\subseteq({\rm e}^{n},{\rm e}^{n+1}]$,

[TABLE]

regardless of the value of $\rho>0$. We sum the preceding over all integers $s\in(t\,,t+r]$, and set $\rho:=1-\gamma$, in order to see that, as long as $(x\,,x+r]\times(t\,,t+r]\subseteq\mathcal{S}_{n}$,

[TABLE]

At this stage, we can combine our observations (3.43) and (3.44) in order to see that (3.44) in fact holds in both cases, as long as $r\geqslant 1$, $(x\,,x+r]\times(t\,,t+r]\subseteq\mathcal{S}_{n}$, and $n\gg 1$ is sufficiently large. Because $\mu$ is a measure on $\mathcal{G}_{0}$, the preceding fact and the density theorem of Barlow and Taylor [4, Theorem 4.1] together imply that

[TABLE]

a.s. uniformly for all $n\gg 1$ large, where the last line is deduced from (3.42). Consequently,

[TABLE]

Let $\varepsilon\downarrow 0$ and $\gamma\downarrow L(2\beta)^{3/2}\vartheta$, without violating either (3.34) or (3.35), in order to deduce half of the assertion (3.33), namely that

[TABLE]

This proves (3.33), thus concludes the proof of the theorem. ∎

We end this section with the following remark on cases of $\vartheta,\beta>0$ that may not be covered by the conditions of Theorem 3.10.

Remark 3.12**.**

Theorem 3.10 has been formulated in a way that does not need the following peculiar property. Thus, we include it here as a remark: For all $\vartheta,\beta>0$,

[TABLE]

Indeed, the following stronger statement is valid:

[TABLE]

We will now prove the following equivalent formulation:

[TABLE]

For all $m\geqslant{\rm e}$ and $\gamma>[\log m]^{-2/3}\vartheta\beta$, let

[TABLE]

Let $\mathcal{R}_{m}(\gamma)$ denote the set on the right-hand side of the above expression. Since the set difference between $\mathcal{R}_{m}(\gamma)$ and $\mathcal{R}_{0}(\gamma)$ is a bounded set, $\mathcal{R}_{m}(\gamma)\times\{{\rm e}\}$ and $\mathcal{R}_{0}(\gamma)\times\{{\rm e}\}$ have the same [macroscopic] Hausdorff dimension. Therefore, the theory of Khoshnevisan, Kim, and Xiao [22] implies that there exists a finite and positive constant $A$—independent of $(\gamma\,,m)$—such that with probability one,

[TABLE]

The left-most quantity decreases as $m$ increases, whereas the right-most term increases with $m$. This proves (3.45).

3.4 An upper bound

Theorem 3.10 implies the first inequality of Theorem 1.1; that is the lower bound on the Hausdorff dimension of $S_{\vartheta}(\mathscr{P}(\beta))$. Now we work toward proving a complementary upper bound for the Hausdorff dimension of the same sort of set. This effort begins with the following technical lemma.

Lemma 3.13**.**

There exist positive and finite constants $N_{1}$ and $N_{2}$ such that

[TABLE]

for every real number $a\in\mathbb{R}$, $b>0$, and $k\geqslant 2$.

In order to understand what this lemma says, let us note the following formulation of (1.3), which was mentioned already in the Introduction:

[TABLE]

Thus, Lemma 3.13 asserts that, at cost of having slightly larger constants, we can “put both of the suprema inside the expectation.”

Proof of Lemma 3.13.

Throughout this proof we define

[TABLE]

for all $w,r\in\mathbb{R}$. We may think of $\varrho$ as a “parabolic metric” on space-time $\{(t\,,x):\ t\geqslant 0,\,x\in\mathbb{R}\}$, where “parabolic” loosely refers to a kind of compatibility with the geometric structure of the heat equation.

It is known that there exist finite constants $L_{1}$ and $L_{2}$—independently of $(a\,,b)$—such that for all real numbers $k\geqslant 2$,

[TABLE]

See the proof of Theorem 1.3 of Conus, Joseph, and Khoshnevisan [9], for example.

Define, for all $z\in\mathbb{R}$,

[TABLE]

Then a quantitative form of the Kolmogorov continuity theorem [21, Theorem C.6, p. 114] implies that for all real numbers $k\geqslant 2$, $q\in(0\,,1-(6/k))$, and $\delta\in(q\,,1-(6/k))$,

[TABLE]

where $D$ is a finite constant that does not depend on $(a\,,b\,,k)$. Note that, as long as $k(1-\delta)>12$,

[TABLE]

since $\rho(x\,,t)\leqslant\rho(1\,,1)=2$ for all $(x\,,t)\in(0\,,1]^{2}$. Therefore, Lemma 3.2 ensures that for all $k>12/(1-\delta)$,

[TABLE]

This proves the lemma in the case that $k>12/(1-\delta)$. The conclusion of Lemma 3.13, in the case that $k\in[2\,,12/(1-\delta))$, follows from Jensen’s inequality and the lemma in the case that $k>12/(1-\delta)$. ∎

Next, we present a ready consequence of Lemma 3.13. It might also help to recall the constants $N_{1}$ and $N_{2}$ from Lemma 3.13.

Proposition 3.14**.**

There exist positive and finite constants $N_{3}$—depending only on $N_{1},N_{2},\beta$—and $N_{4}$—depending only on $N_{2}$—such that for every $\beta\geqslant 24N_{2}$ and for all real numbers $a>0$ and $b\geqslant 1$ and integers $\ell_{1},\ell_{2}\geqslant 1$,

[TABLE]

Proposition 3.14 is a “maximal inequality” that corresponds to the pointwise inequality of Proposition 3.1.

Proof of Proposition 3.14.

Let $i$ and $j$ denote two arbitrary integers between [math] and respectively $\ell_{1}$ and $\ell_{2}$. Since $b+(j+1)\leqslant 2(b+j)$, Lemma 3.13 and Chebyshev’s inequality together imply that

[TABLE]

Set $k:=\sqrt{\beta/(6N_{2})}$ in the preceding infimization problem in order to see that

[TABLE]

valid as long as $k=\sqrt{\beta/(6N_{2})}\geqslant 2$. The lemma follows from adding the preceding expression from $j=0$ and $i=0$ to $j=\ell_{2}$ and $i=\ell_{1}$. ∎

Proposition 3.14 paves the way for the main result of this section, which is presented next. The following theorem complements the lower bound of Theorem 3.10 by yielding a corresponding almost-sure upper bound for the macroscopic Hausdorff dimension of $S_{\vartheta}(\mathscr{P}(\beta))$. Recall the universal constant $N_{2}$ from Lemma 3.13.

Theorem 3.15**.**

For all $\beta\geqslant 24N_{2}$ and all $\vartheta>0$,

[TABLE]

Proof.

Clearly, for all $b\geqslant 1$,

[TABLE]

Therefore, Proposition 3.14 implies that for all $a,\vartheta>0$, $b\geqslant 1$, and $\beta\geqslant 24N_{2}$,

[TABLE]

Choose and fix an arbitrary constant $q>1$. The preceding shows that, uniformly for all $b\in({\rm e}^{n/q}\,,{\rm e}^{n+1}]$ and integers $n\geqslant 1$,

[TABLE]

This is the key probability estimate required for the proof.

Let us consider the random set

[TABLE]

Next we study the structure of $\mathcal{G}_{\vartheta,\beta}\cap\mathcal{S}_{n}$ for all sufficiently-large integers $n\geqslant 1$. Let us recall that $q>1$ is a fixed but arbitrary real number, and then observe that for all integers $n\geqslant 1$,

[TABLE]

where

[TABLE]

Eq. (3.47) essentially decomposes $\mathcal{G}_{\vartheta,\beta}\cap\mathcal{S}_{n}$ into a “little” part $\ell_{n}$ and a “large” part $\mathcal{L}_{n}$. Because $q>1$ and

[TABLE]

Corollary 1.4 ensures that

[TABLE]

Owing to (3.47), the inequality (3.48) is good enough to imply that

[TABLE]

We estimate an upper bound for the Hausdorff dimension of $\cup_{n=1}^{\infty}\mathcal{L}_{n}$ as follows: There are at most $O({\rm e}^{2n})$ squares of of the form $(a\,,a+1]\times(b\,,b+1]\subset({\rm e}^{n/q}\,,{\rm e}^{n+1}]^{2}$, uniformly for all integers $n\geqslant 1$. We can cover each $\mathcal{L}_{n}$ with only such squares of the form $(a\,,a+1]\times(b\,,b+1]$ that additionally satisfy

[TABLE]

These remarks and (3.46) together show that, for every integer $n\geqslant 1$ and for all real numbers $\rho>0$,

[TABLE]

where the implied constant depends only on $(\vartheta\,,\beta\,,N_{2})$. In particular,

[TABLE]

provided that $2>q^{-1}N_{4}\beta^{3/2}\vartheta$. This, and the definition of Hausdorff dimension, together imply that

[TABLE]

where $a_{+}:=\max(a\,,0)$ for all real $a$, as is usual. Because $q>1$ is arbitrary, and since the definition of $\mathcal{G}_{\vartheta,\beta}$ does not depend on $q$, eq. (3.49) implies that

[TABLE]

A symmetric argument implies that almost surely,

[TABLE]

Therefore, the preceding two displayed inqualities together imply the theorem since

[TABLE]

as can be checked from first principles or from an example of Barlow and Taylor [3, §4.1]. ∎

3.5 Proof of Theorem 1.1 and Corollary 1.2

We complete this section by first deriving Theorem 1.1 and then its Corollary 1.2, in this order.

Proof of Theorem 1.1.

Because the macroscopic dimension of a set is unaffected by local changes in that set, one can see readily that

[TABLE]

Therefore, we can deduce Theorem 1.1 readily from Theorems 3.10 and 3.15. ∎

Proof of Corollary 1.2.

It accord with Theorem 1.1, for every $\beta_{1}>b$ we can choose $\vartheta_{1}\in(0\,,\varepsilon\beta_{1}^{-3/2})$ such that $1<{\rm Dim}_{{}_{\rm H}}\left[S_{\vartheta_{1}}(\mathscr{P}(\beta_{1}))\right]<2$. Next we first choose $\beta_{2}>\beta_{1}$, sufficiently close to $\beta_{1}$ that $\vartheta_{1}<\varepsilon\beta_{2}^{-3/2}$, and then choose $\vartheta_{2}\in(\vartheta_{1}\,,\varepsilon\beta_{2}^{-3/2})$. Theorem 1.1 implies that $1<{\rm Dim}_{{}_{\rm H}}\left[S_{\vartheta_{i}}(\mathscr{P}(\beta_{j}))\right]<2$ for $i,j\in\{1,2\}.$ Repetitive application of this procedure yields the first conclusion of Corollary 1.2.

To achieve the property in the second conclusion, we choose the sequences $\{\beta_{i}\}_{i=1}^{\infty}$ and $\{\vartheta_{i}\}_{i=1}^{\infty}$ a little more carefully. For example, if we choose $\vartheta_{1}$ small, then we can choose $\beta_{2}$ and $\vartheta_{2}$ such that, in addition to the two properties mentioned above, they also satisfy

[TABLE]

where $A$ and $a$ are the constants in Theorem 1.1. In this way, we find that

[TABLE]

An inductive application of this procedure yields two sequences $\{\beta_{i}\}_{i=1}^{\infty}$ and $\{\vartheta_{i}\}_{i=1}^{\infty}$ such that $n\mapsto{\rm Dim}_{{}_{\rm H}}[S_{\vartheta_{n}}(\mathscr{P}(\beta_{n}))]$ is strictly decreasing. This completes the proof of the corollary. ∎

4 A non-intermittent case

Let us now consider our stochastic PDE (1.1) under the condition that the function $\sigma$ is constant. In particular, $\sigma$ fails to satisfy the intermittency condition (1.2). For simplicity, we consider only the case that the initial function is identically zero and $\sigma\equiv 1$. That is, we are interested in the random field $Z$ that solves

[TABLE]

It is well-known that $Z(t\,,x)$ has the following “mild formulation”:

[TABLE]

See Walsh [27, Chapter 3]. In particular, the process $Z$ is a centered Gaussian process with variance function

[TABLE]

for all $t>0$ and $x\in\mathbb{R}$. Because the moments of $Z(t\,,x)$ are completely described by its variance, it follows that $Z$ is not intermittent in the sense that its moments do not grow exponentially with time. As a result, one does not expect the exceptionally-large peaks of $Z$ to be exponentially large in the time variable. Still, we proved a few years ago [22] that the spatial peaks of $Z(t)$ form a multifractal for every fixed $t>0$.

The main result of this section is the following description of the complex, multifractal nature of the tall spatio-temporal peaks of $Z$. The following is the analogue of Theorem 1.1 for the constant-coefficient, linear SPDE (4.1).

Theorem 4.1**.**

For every $\beta,\varepsilon>0$,

[TABLE]

Remark 4.2**.**

There is no canonical choice of how one can measure the heights of the very tall spatio-temporal peaks of $Z$. However, in light of (4.2), the gauge function $t\mapsto t^{(1/4)+\varepsilon}$ is a natural choice. And Theorem 4.1 basically says that a certain “stretching” $[(x\,,t)\mapsto(x\,,\exp(t^{2\varepsilon}))]$ of the random set

[TABLE]

has dimension $\max\{1\,,2-\beta^{2}/2\}$ a.s. for every $\beta,\varepsilon>0$. By contrast with Theorem 1.1, however, the stretch factor here is not arbitrary and depends on $\varepsilon$. In principle, however, it should be possible to produce similar results for various stretch factors.

We conclude this section by presenting two prefatory results about the solution $Z$ to (4.1). These results will be used in the next 2 subsections in order to complete the proof of Theorem 4.1.

First, we observe the following immediate consequence of the Gaussian nature of the law of $Z$; see also (4.2).

Lemma 4.3**.**

There exists a constant $c>1$ such that

[TABLE]

uniformly for all $t>0$, $x\in\mathbb{R}$ and $\lambda>1$.

Next, we observe, using the semigroup property of the heat kernel, that

[TABLE]

The stationary Gaussian process $x\mapsto Z(t\,,x)$ has the following properties:

1. P1.

For all $x\in\mathbb{R}$ and $\alpha\in(0\,,\nicefrac{{1}}{{2}})$,

[TABLE] 2. P2.

For all $x\in\mathbb{R}$ and $\alpha>\nicefrac{{1}}{{2}}$,

[TABLE]

which rapidly tends to zero as $t\to\infty$.

Thus, we may combine P1 and P2 in order to deduce the well-known informal assertion that the “correlation length” of the stochastic process $x\mapsto Z(t\,,x)$ is $\sqrt{t}$. From this, and the rapid rate of convergence to zero in P2, one might surmise that $Z(t\,,x)$ and $Z(t\,,y)$ are in fact asymptotically independent when $|x-y|\gg\sqrt{t}$. Lemma 4.4 below verifies this by giving a rigorous meaning to “asymptotic independence.”

For all $x\in\mathbb{R}$ and $B>0$, define

[TABLE]

The following lemma is essentially borrowed from Khoshnevisan, Kim, and Xiao [22]; see Eq. (6.20) and Observation 1 of that paper (loc. cit.).

Lemma 4.4**.**

For all $t,B,\lambda>0$,

[TABLE]

In addition, if $x_{1},x_{2},\ldots,x_{m}\in\mathbb{R}$ satisfy $|x_{i}-x_{j}|>2(Bt)^{1/2}$ when $1\leqslant i\neq j\leqslant m$, then the random variables $Z^{(B)}_{t}(x_{1}),\ldots,Z^{(B)}_{t}(x_{m})$ are independent.

Armed with these preliminary facts, we proceed with developing bounds for the macroscopic Hausdorff dimension of the tall spatio-temporal peaks of the random field $Z$.

4.1 A lower bound

The main result of this section is the following lower bound on the macroscopic Hausdorff dimension of the space-time set of extremely-tall peaks of height $f(t)=t^{(1/4)+\varepsilon}$ and level $\beta$. The following is the precise statement that we will prove shortly.

Proposition 4.5**.**

For every $\beta,\varepsilon>0$,

[TABLE]

Clearly, Proposition 4.5 implies half of the content of Theorem 4.1.

Proof.

We use the same procedure as in the proof of Theorem 3.10. Consider the random set

[TABLE]

whose macroscopic dimension is of interest to us. Define

[TABLE]

Since $\mathcal{G}_{0}\subseteq\mathcal{G}$ a.s., it suffices to prove that

[TABLE]

Define, for all reals $\gamma\in(0\,,1)$ and integers $n\geqslant 0$,

[TABLE]

and

[TABLE]

For all such reals $\gamma$ and integers $n$ and $j$, and for every $\varepsilon\in(0\,,1)$, we can find points $x_{1},\ldots,x_{m}\in\mathcal{I}_{j,n}(\gamma)$—depending only on $(n\,,\gamma\,,\varepsilon\,,j)$—such that whenever $n>1$:

$|x_{k}-x_{l}|\geqslant{\rm e}^{n\varepsilon}$ whenever $1\leqslant k<l\leqslant m$; and

• -

$\frac{1}{2}{\rm e}^{n(\gamma-\varepsilon)}\leqslant m\leqslant 2{\rm e}^{n(\gamma-\varepsilon)}$.

We can write

[TABLE]

where

[TABLE]

Thanks to Lemma 4.3 and Lemma 4.4, whenever the condition

[TABLE]

holds, we can deduce that

[TABLE]

[There is nothing special about the fact that the $x$-variable in $Z^{(n)}(t\,,0)$ and $Z(t\,,0)$ is chosen as $x=0$ since $Z^{(n)}(t,x)$ and $Z(t,x)$ are stationary random fields for every fixed $t>0$.] An elementary argument now shows that

[TABLE]

Now, let us plug the bounds (4.5) and (4.6) in (4.3), then replace $t$ by $[\log t]^{1/(2\varepsilon)}$ throughout, and use the previously-mentioned inequality, $\frac{1}{2}{\rm e}^{n(\gamma-\varepsilon)}\leqslant m\leqslant 2{\rm e}^{n(\gamma-\varepsilon)}$, in order to obtain the following:

[TABLE]

A change of variables —from $t$ to $[\log t]^{1/(2\varepsilon)}$—justifies the asymptotic independence that is required for the preceding to hold. More precisely put, we have

[TABLE]

which verifies that (4.4) holds after we replace $t$ by $[\log t]^{1/(2\varepsilon)}$.

In any event, we can deduce from the preceding that, as $n\to\infty$,

[TABLE]

Thus, the Borel–Cantelli lemma implies that, as long as

[TABLE]

the following holds with probability one:

[TABLE]

for all but a finite number of integers $n\geqslant 1$. From here, the remainder of the proof follows exactly the same pattern as the one for its counterpart in Theorem 3.10; see Conclusion A [following shortly after (3.41)] and its justification. We omit the remaining details. ∎

4.2 An upper bound

The main result of this section is the following upper bound on the macroscopic Hausdorff dimension of the space-time set of tall peaks of height $t^{\varepsilon}$ and level $\beta$.

Proposition 4.6**.**

For every $\beta,\varepsilon>0$,

[TABLE]

Theorem 4.1 is manifestly a consequence of Propositions 4.5 and 4.6.

We first consider the following estimate of the tail probability. This estimate is essential to the proof of Proposition 4.6.

Proposition 4.7**.**

Let $f:(1\,,\infty)\to(1\,,\infty)$ be a strictly increasing function that satisfies $\lim_{t\to\infty}f(t)=\infty$. Then, for all $\ell\geqslant 1$ there exists a finite constant $C=C(\ell)>1$ such that

[TABLE]

uniformly for every $a\in\mathbb{R}$ and all sufficiently large $b>1$,

Proof.

It is well-known that there exists a finite constant $C>0$ such that for all $t>0$ and $x,y\in\mathbb{R}$,

[TABLE]

See, for example, §3.3 of Khoshnevisan [21]. Since $Z$ is a Gaussian process, a quantitative form of the Kolmogorov continuity theorem [21, Theorem C.6, p. 114] implies that

[TABLE]

Because of this, and the fact that the random variable $Z(t\,,x)$ has a centered normal distribution with variance $\sqrt{t/\pi}$, we obtain the following: Uniformly for all $a\in\mathbb{R}$, $b>1$, and $\ell\geqslant 1$,

[TABLE]

where $C_{2}:=\max(C_{1}\,,1)$ is a finite constant that is independent of $(a\,,b\,,\ell)$. With (4.7) under way, we can easily complete the proof.

Define

[TABLE]

Then, clearly,

[TABLE]

A standard appeal to the Borell, Sudakov–T’sirelson inequality now yields

[TABLE]

[For a readable account see Adler [1, Chapter II].] As $t\uparrow\infty$, $f(t)$ increases strictly to $\infty$. Therefore, for all sufficiently large $b$,

[TABLE]

Therefore, (4.7) implies the result. ∎

Armed with Proposition 4.7, we next proceed with the proof of Theorem 4.6.

Proof of Proposition 4.6.

We follow the general procedure, and use the same notation, as in the proof of Theorem 3.15. Recall the random set

[TABLE]

From Proposition 4.7, we have that for all sufficiently large $b$,

[TABLE]

where $\tilde{C}$ is a finite and positive constant that depends only on the constant $C$ of Proposition 4.7.

Choose and fix an arbitrary constant $q>1$. The preceding implies that, uniformly for all $a\in\mathbb{R}$, $b\in({\rm e}^{n/q}\,,{\rm e}^{n+1}]$ and sufficiently large integers $n\geqslant 1$,

[TABLE]

This is the key probability estimate required for the proof.

We now follow the same pattern as we did in the proof of Theorem 3.15. Let us recall that $q>1$ is a fixed but arbitrary real number, and then observe that for all integers $n\geqslant 1$,

[TABLE]

where

[TABLE]

As we observed earlier in the proof of Theorem 3.15, it is suffices to prove that

[TABLE]

We estimate an upper bound for the Hausdorff dimension of $\cup_{n=1}^{\infty}\mathcal{L}_{n}$ as follows: There are at most $O({\rm e}^{2n})$ squares of of the form $(a\,,a+1]\times(b\,,b+1]\subset({\rm e}^{n/q},{\rm e}^{n+1}]^{2}$, uniformly for all integers $n\geqslant 1$. We can cover each $\mathcal{L}_{n}$ with only such squares of the form $(a\,,a+1]\times(b\,,b+1]$ that additionally satisfy

[TABLE]

These remarks and (4.8) together show that

[TABLE]

It follows immediately from this bound and the Borel–Cantelli lemma that

[TABLE]

where $a_{+}:=\max(a\,,0)$ for all real $a$, as before. Because $q>1$ is arbitrary, and since the definition of $\mathcal{G}$ does not depend on $q$, Eq. (4.9) implies that

[TABLE]

The remainder of the proof is exactly the same as the one for Theorem 3.15; that is, we use a symmetric argument and a general example of Barlow and Taylor [3, §4.1]. ∎

Acknowledgements. Two of us [K.K. and D.K.] would like to thank the Mathematical Sciences Research Institute [Berkeley, CA] for providing us with a wonderful research environment in October 2015 [D.K.] and the Fall Semester [K.K.] of 2015.

Appendix A A table of universal constants