Fernique-type inequalities and moduli of continuity for anisotropic Gaussian random fields

Abstract

This paper is concerned with sample path properties of anisotropic Gaussian random fields. We establish Fernique-type inequalities and utilize them to study the global and local moduli of continuity for anisotropic Gaussian random fields. Applications to fractional Brownian sheets and to the solutions of stochastic partial differential equations are investigated.

1 Introduction

Many data sets from various areas such as image processing, hydrology, geostatistics and spatial statistics have anisotropic nature in the sense that they have different geometric and statistical characteristics along different directions, hence fractional Brownian motion is not adequate for modelling such phenomena. Many people have proposed to apply anisotropic Gaussian random fields as more realistic models. See, for example, Davies and Hall (1999), Bonami and Estrade (2003), Benson et al. (2006) and Biermé et al. (2007).

Several classes of anisotropic Gaussian random fields have been introduced for theoretical and application purposes. For example, Kamont (1996) introduced fractional Brownian sheets and studied some of their regularity properties. Benassi et al. (1997) and Bonami and Estrade (2003) considered some anisotropic Gaussian random fields with stationary increments. Biermé et al. (2007) have constructed a large class of operator self-similar Gaussian or stable random fields with stationary increments. Anisotropic Gaussian random fields also arise naturally in stochastic partial different equations (see, e.g., Dalang (1999), Mueller and Trible (2002), Øksendal and Zhang (2000), Nualart (2006)); and in studying the most visited sites of symmetric Markov processes (Eisenbaum and Khoshnevisan (2002)). Hence it is of importance in both theory and applications to investigate the probabilistic and statistical properties of anisotropic random fields.

Recently, Xiao (2009) investigated sample path properties of anisotropic Gaussian random fields under general conditions. Typical examples for Gaussian random fields covered by his framework are fractional Brownian sheets, operator-scaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation. In the present paper, we are concerned with the global and local moduli of continuity of general anisotropic Gaussian random fields. Our objective is to characterize the anisotropic nature of Gaussian random fields, from an analytic point of view, in terms of their Hurst parameters explicitly.

For this purpose, we first establish Fernique-type inequalities which may be of interest beyond the scope of the present paper. Then we utilize these inequalities to study the global moduli of continuity and the local moduli of continuity (or the law of the iterated logarithm) for anisotropic Gaussian random fields.

Many authors have investigated moduli of continuity of Gaussian random fields. When $W=\{W(t),t\in {ℝ}_{+}^N\}$ is N -parameter Brownian sheet, Orey and Pruitt (1973) considered increments of W over intervals and points, and established the corresponding global and local moduli of continuity. Benassi et al. (1997) proved, among other things, results on the global and local moduli of continuity for a large class of elliptic Gaussian random fields. Ayache and Xiao (2005) studied the global moduli of continuity for fractional Brownian sheets by using the wavelet method and they obtained a sharp upper bound for the global moduli of continuity for points. Wang (2007) investigated the global and the local moduli of continuity for fractional Brownian sheets for intervals. As an immediate consequence of our results in the present paper, we give a sharp results for the global and local moduli of continuity for fractional Brownian sheets for points.

Throughout this paper, we will use c to denote an unspecified positive and finite constant which may not be the same in each occurrence. More specific constants in Section i are numbered as c_i,1,, c_i,2, ….

2 General assumptions

The parameter space is ℝ^N or ${\mathbb{R}}_{+}^N={[0,\infty )}^N$, endowed with the Euclidean norm || · ||. A typical parameter (“time point”) is t = (t₁, …, t_N), sometimes also written as 〈t_i〉, or 〈c〉, if t₁ = · · · = t_N = c. The inner product of s, t ∈ ℝ^N is denoted by 〈s, t〉. Given two points s = 〈s_i〉, $t=\langle t_i\rangle \in {\mathbb{R}}_{+}^N$, s ≤ t (resp. s < t) means that s_i ≤ t_i (resp. s_i < t_i) for all 1 ≤ i ≤ N. When s ≤ t, we use [s, t] to denote the N-dimensional interval $[s,t]={\times }_{i=1}^N[s_i,t_i]$. For x ≤ ℝ₊, let log x = ln(x ∨ e), log log x = ln(ln(x) ∨ e).

Let X = {X(t), t ∈ ℝ^N} be a centered Gaussian random field with values in ℝ. Let I ⊂ ℝ^N be a fixed compact N-dimensional interval, and our goal is to determine the exact uniform and local moduli of continuity of X(t) when t ∈ I. Typically this paper, we will take I = [0, 1]^N or I = [a, 1]^N, where a ∈ (0, 1) is a fixed constant.

Many sample path properties of the Gaussian random field X can be determined by the function:

$${\sigma }^2(s,t)=\mathbb{E}{(X(s)-X(t))}^2,\forall s,t\in {\mathbb{R}}^N.$$

As shown by Xiao (2009), the following general conditions are useful: Let H = (H₁, …, H_N) ∈ (0, 1]^N be a fixed vector and denote

$$\rho (s,t)=\sum _{j=1}^N{\mid s_j-t_j\mid }^{H_j},\forall s,t\in {\mathbb{R}}^N.$$

• (A1)
  There exist positive and finite constants c_2,1 and c_2,2 such that

  $$c_{2,1}\rho {(s,t)}^2\le {\sigma }^2(s,t)\le c_{2,2}\rho (s,t)$$

  for all s, t ∈ I.

• (A2)
  There exists a constant c_2,3 > 0 such that for all integers n ≥ 1, all u, t1, …, tⁿ ∈ I,

  $$\text{Var}\left(X(u)\mid X(t^1),\dots ,X(t^n)\right)\ge c_{2,3}\sum _{j=1}^N\underset{0\le k\le n}{min}{\mid u_j-t_j^k\mid }^{2H_j},$$

  where $t_j^0=0$ for every j = 1, …, N.

• (A3)
  There exists a constant c_2,4 > 0 such that for all integers n ≥ 1, and all u, t¹, …, tⁿ ∈ I,

  $$\text{Var}\left(X(u)\mid X(t^1),\dots ,X(t^n)\right)\ge c_{2,4}\underset{0\le k\le n}{min}\rho {(u,t^k)}^2,$$

  where t⁰ = 0.

Remark 2.1

The following are some remarks about the above conditions.

• It is helpful to note that ρ(s, t) defined above is a metric on ℝ^N. It is more convenient for studying anisotropic random fields than the Euclidean metric.

• Under condition (A1), X has a version which has continuous sample functions on I almost surely. Henceforth we will assume without loss of generality that the Gaussian random field X has continuous sample paths.

• Pitt (1978) proved that fractional Brownian motion X^α satisfies condition (A3) for all intervals I ∈ ℝ^N with H = 〈α〉. Khoshnevisan and Xiao (2007) proved that, for every ε > 0, the Brownian sheet W satisfies the property (A2) with H = 〈1/2〉 for all intervals I ∈ [ε, ∞)^N. It has been proved in Ayache and Xiao (2005), Wu and Xiao (2007) that fractional Brownian sheets satisfy Conditions (A1) and (A2) for all intervals I ⊂ [ε, ∞)^N.

• Condition (A3) is listed here mainly for comparison purpose. Condition (A3) implies (A2). It is known that the converse does not even hold for the Brownian sheet [see, e.g., Khoshnevisan and Xiao (2007) or Xiao (2009)]. Roughly speaking, when H = 〈α〉, the behavior of a Gaussian random field X satisfying Conditions (A1) and (A3) is comparable to that of a fractional Brownian motion of index α; while a Gaussian random field X satisfying Conditions (A1) and (A2) is comparable to that of a fractional Brownian sheet. Hence, in analogy to terminology respectively for fractional Brownian motion and the Brownian sheet, Condition (A3) will be called the strong local nondeterminism [in metric ρ] and Condition (A2) will be called sectorial local nondeterminism.

In this paper, we establish the global and local moduli of continuity for Gaussian random fields satisfying Conditions (A1) and (A2). The rest of the paper is organized as follows. In Section 3 we state and prove Fernique type inequalities for anisotropic Gaussian random fields, which will be used in latter sections. The global moduli of continuity are discussed in Sections 4 and the local moduli of continuity are investigated in Section 5. Some applications are discussed in Section 6.

3 Fernique-type inequalities

The aim of this section is to establish Fernique-type inequalities for anisotropic Gaussian random fields, which will be used in latter sections and may be of independent interest. We start with the following lemma which is a consequence of the results in Fernique (1974) and Berman (1985).

Lemma 3.1

Suppose that Y = {Y (t), t ∈ ℝ^N} is a centered Gaussian random field with values in ℝ and denote

$$d(s,t):=d_Y(s,t)={\left(\mathbb{E}{\mid Y(t)-Y(s)\mid }^2\right)}^{1/2},s,t\in {\mathbb{R}}^N.$$

Let S be a closed cube in ℝ^N of edge-length δ and let ${\sigma }^2=\underset{t\in S}{sup}\mathbb{E}(Y{(t)}^2)$. For any h > 0, ε > 0, define

$$\gamma (\epsilon )=\underset{s,t\in S,\left|\right|s-t\left|\right|\le \epsilon }{sup}d(s,t)$$

and

$$Q(h)=(2+\sqrt{2}){\int }_1^{\infty }\gamma (h{2}^{-y^2})dy.$$

Then for all x > 0 which satisfy x ≥ (1 + 4N log 2)^1/2(σ + x⁻1),

$$\mathbb{P}\left\{\underset{t\in S}{sup}\mid Y(t)\mid >x\right\}\le 2^{2N+2}{\left(\frac{\delta }{Q^{-1}(1/x)}+1\right)}^N\frac{\sigma +x^{-1}}{x}exp\left(-\frac{x^2}{2{(\sigma +x^{-1})}^2}\right),$$ (3.1)

where Q⁻¹(x) = sup{y : Q(y) ≤ x}. Particularly, from (3.1) it follows that for any ε > 0 there exist positive constants x₀ = x₀(ε, σ) and c_3,1 = c_3,1 (ε, σ,N) such that for any x ≥ x₀,

$$\mathbb{P}\left\{\underset{t\in S}{sup}\mid Y(t)\mid >x\right\}\le c_{3,1}{\left(\frac{\delta }{Q^{-1}(1/x)}+1\right)}^{N}exp\left(-\frac{x^2}{(2+\epsilon ){\sigma }^2}\right).$$ (3.2)

Proof

For every h ∈ (0, δ], S can be covered by (⌊δ/h⌋ + 1)^N closed subcubes {S_i} of side-length h, where ⌊u⌋ denote the largest integer ≤ u. Hence

$$\mathbb{P}\left\{\underset{t\in S}{sup}\mid Y(t)\mid >x\right\}\le {\left(\lfloor \frac{\delta }{h}\rfloor +1\right)}^N\underset{i}{max}\mathbb{P}\left\{\underset{t\in S_i}{sup}\mid Y(t)\mid >x\right\}.$$ (3.3)

Take h = Q⁻¹(1/x) ∧ δ. It follows from the Fernique inequality [with p = 2] in Section 4.1.3 of Fernique (1974) or (4.2) in Berman (1985) that for every subcube S_i, we have

$$\mathbb{P}\left\{\underset{t\in S_i}{sup}\mid Y(t)\mid >x\right\}\le 5\sqrt{2\pi }{2}^{2N-1}\mathrm{\Psi }\left(\frac{x}{\sigma +Q(h)}\right)$$ (3.4)

for all x ≥ (1 + 4N log 2)^1/2(σ + Q(h)), where Ψ(u) = ℙ{N(0, 1) > u} is the tail probability of a standard normal random variable. In deriving (3.4), we have also used the fact that Ψ(u) is decreasing.

Notice that Q(h) ≤ x⁻¹ and $\mathrm{\Psi }(u)\le {(2\pi )}^{-\frac{1}{2}}{u}^{-1}{e}^{-\frac{u^2}{2}}$ for all u > 0, we can write the inequality (3.4) as

$$\mathbb{P}\left\{\underset{t\in S_i}{sup}\mid Y(t)\mid >x\right\}\le 52^{2N-1}\frac{\sigma +x^{-1}}{x}exp\left(-\frac{x^2}{2{(\sigma +x^{-1})}^2}\right),$$ (3.5)

which, together with (3.3), yields (3.1) and thus Lemma 3.1.

For the next lemma, we need the following notation. For every t ∈ ℝ^N, x ∈ ℝ and ℓ = 1, …,N, we denote by (t̂_ℓ, x) the vector in ℝ^N obtained from t by replacing its ℓth coordinate t_ℓ by x. For example, (t̂_N, x) = (t₁, …, t_N−1, x).

Lemma 3.2

Let X = {X(t), t ∈ ℝ^N} be a centered Gaussian random field with values in ℝ and satisfy the upper bound in Condition (A1) with I = [0, 1]^N. Then for any ε > 0 there exist positive constants u₀ = u₀(ε, c_2,2) and c_3,2 = c_3,2 (ε, c_2,2,H,N) such that for all x ∈ [0, 1], 0 < y ≤ z ≤ 1 that satisfy [x, x + y] ⊂ [0, 1], all u ≥ u₀ and 1 ≤ ℓ ≤ N

$$\mathbb{P}\left\{\underset{t\in {[0,1]}^N}{sup}\mid X({\widehat{t}}_{\ell },x+y)-X({\widehat{t}}_{\ell },x)\mid \ge {uz}^{H_{\ell }}\right\}\le c_{3,2}{z}^{-\frac{(N-1)H_{\ell }}{min\{H_i\}}}{u}^{\frac{N-1}{min\{H_i\}}}{e}^{-\frac{u^2}{(2+\epsilon )c_{2,2}}}.$$ (3.6)

Here and in the sequel the minimum of H_i is taken over all 1 ≤ i ≤ N.

Proof

For simplicity of notation, we only prove (3.6) for ℓ = N and write (t̂_N, x) as (t, x), where t ∈ [0, 1]^N−1 or more generally t ∈ ℝ^N−1. For any x ∈ [0, 1] and 0 < y ≤ z, we consider the Gaussian process Y = {Y (t), t ∈ ℝ^N−1} defined by

$$Y(t)=\frac{X(t,x+y)-X(t,x)}{z^{H_N}},\forall t\in {\mathbb{R}}^{N-1}.$$

We now show (3.6) by applying Lemma 3.1 to Y with S = [0, 1]^N−1.

By the Minkowski inequality and Condition (A1), we have

$$\begin{array}{l}d(s,t)\le \frac{1}{z^{H_N}}{\left[\mathbb{E}{(X(t,x+y)-X(s,x+y))}^2+\mathbb{E}{(X(t,x)-X(s,x))}^2\right]}^{1/2}\\ \le \frac{2c_{2,2}^{1/2}}{z^{H_N}}\sum _{j=1}^{N-1}{\mid s_j-t_j\mid }^{H_j}\end{array}$$

for all s, t ∈ S. By Jensen’s inequality we derive

$$\begin{array}{l}\sum _{j=1}^{N-1}{\mid t_j-s_j\mid }^{2H_j}\le {(N-1)}^{1-min\{H_i\}}{\left(\sum _{j=1}^{N-1}{\mid t_j-s_j\mid }^{2H_j/min\{H_i\}}\right)}^{min\{H_i\}}\\ \le {(N-1)}^{1-min\{H_i\}}{\left(\sum _{j=1}^{N-1}{\mid t_j-s_j\mid }^2\right)}^{min\{H_i\}}\\ ={(N-1)}^{1-min\{H_i\}}{\left|\right|t-s\left|\right|}^{2min\{H_i\}}.\end{array}$$

Thus

$$d(s,t)\le \frac{2c_{2,2}^{1/2}}{z^{H_N}}{(N-1)}^{(1-min\{H_i\})/2}{\left|\right|t-s\left|\right|}^{min\{H_i\}}.$$

It follows that

$$\gamma (\epsilon )=\underset{s,t\in S,\left|\right|s-t\left|\right|\le \epsilon }{sup}d(s,t)\le \frac{2c_{2,2}^{1/2}}{z^{H_N}}{(N-1)}^{(1-min\{H_i\})/2}{\epsilon }^{min\{H_i\}}.$$

This implies that

$$Q(h)\le c{h}^{min\{H_i\}}/z^{H_N}$$

and the inverse function of Q satisfies

$$Q^{-1}(r)\ge c{r}^{1/min\{H_i\}}{z}^{H_N/min\{H_i\}}.$$

Since ${\sigma }^2=\underset{t\in S}{sup}\mathbb{E}(Y{(t)}^2)\le c_{2,2}$ by Condition (A1), we use (3.2) to derive that for any ε > 0, there exists a positive constant u₀ = u₀(ε, c_2,2) such that for all u ≥ u₀,

$$\mathbb{P}\left\{\underset{t\in {[0,1]}^{N-1}}{sup}\mid X(t,x+y)-X(t,x)\mid \ge {uz}^{H_N}\right\}\le c{u}^{(N-1)/min\{H_i\}}{z}^{-(N-1)H_N/min\{H_i\}}{e}^{-\frac{u^2}{(2+\epsilon )c_{2,2}}}.$$

This yields (3.6) for ℓ = N.

The following is the main result of this section, which is a generalization of Lemma 2.1 of Csáki et al. (1992) for Gaussian random fields.

Proposition 3.3

Let {X(t), t ∈ ℝ^N} be a centered Gaussian random field in ℝ satisfying the upper bound in Condition (A1) for I = [0, 1]^N. For any τ > 0 and any ε > 0 there exist positive constants u^* = u^* (ε, c_2,2) and c_3,3 = c(τ, ε, c_2,2,H,N) such that

$$\begin{array}{l}\mathbb{P}\left\{\underset{\langle x_i\rangle \le t\le \langle x_i+T_i\rangle }{sup}\underset{\langle 0\rangle \le s\le \langle a_i\rangle }{sup}\mid X(t+s)-X(t)\mid \ge ((1+\tau )u+\tau )\sum _{j=1}^N{a}_j^{H_j}\right\}\\ \le c_{3,3}\left[\sum _{j=1}^N\left(\frac{T_j}{a_j}+1\right)·{\left(\frac{1}{a_j}\right)}^{\frac{(N-1)H_j}{min\{H_i\}}}\right]\left(u^{\frac{N-1}{min\{H_i\}}}+1\right)e^{-\frac{u^2}{(2+\epsilon )c_{2,2}}}\end{array}$$ (3.7)

for all u ≥ u^*, x_i, T_i ∈ [0, 1] and a_i ∈ (0, 1] (i = 1, …,N) which satisfy [〈x_i〉, 〈x_i + T_i + a_i〉] ⊂ [0, 1]^N.

Proof

We prove (3.7) by using induction on N. If N = 1, (3.7) is an immediate consequence of Lemma 2.1 of Csáki et al. (1992).

Now we consider the case N ≥ 2. It is easy to see that for any s, t ∈ ℝ^N,

$$X(t+s)-X(t)=\sum _{j=1}^N(X(t_1,\dots ,t_{j-1},t_j+s_j,t_{j+1}+s_{j+1},\dots ,t_N+s_N)-X(t_1,\dots ,t_{j-1},t_j,t_{j+1}+s_{j+1},\dots ,t_N+s_N))$$ (3.8)

with the convention that X(t₁, …, t_j−1, t_j + s_j, t_j+1 + s_j+1, …, t_N + s_N) = X(t + s) if j = 1 and X(t₁, …, t_j−1, t_j, t_j+1 + s_j+1, …, t_N + s_N) = X(t) if j = N. By (3.8), we have for each N ≥ 2

$$\underset{\langle x_i\rangle \le t\le \langle x_i+T_i\rangle }{sup}\underset{\langle 0\rangle \le s\le \langle a_i\rangle }{sup}\mid X(t+s)-X(t)\mid \le \sum _{j=1}^N\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{0\le s_j\le a_j}{sup}\mid X(t,t_j+s_j)-X(t,t_j)\mid .$$

Here and in the rest of the proof, for t ∈ ℝ^N−1, we write (t, t_j) ∈ ℝ^N for the point whose jth coordinate is t_j and so (t, t_j) = (t₁, …, t_N).

Thus for any τ > 0 and u > 0

$$\begin{array}{l}\mathbb{P}\left\{\underset{\langle x_i\rangle \le t\le \langle x_i+T_i\rangle }{sup}\underset{\langle 0\rangle \le s\le \langle a_i\rangle }{sup}\mid X(t+s)-X(t)\mid \ge ((1+\tau )u+\tau )\sum _{j=1}^N{a}_j^{H_j}\right\}\\ \le \sum _{j=1}^N\mathbb{P}\left\{\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{0\le s_j\le a_j}{sup}\mid X(t,t_j+s_j)-X(t,t_j)\mid \ge ((1+\tau )u+\tau )a_j^{H_j}\right\}\\ =:\sum _{j=1}^N{P}_j.\end{array}$$ (3.9)

Let 1 ≤ j ≤ N be fixed. We follow the proof of Lemma 2.1 of Csáki et al. (1992) to estimate P_j (1 ≤ j ≤ N) by using a chaining argument. For any positive real number r and n ≥ 3, denote ${(r)}_n=a_j\lfloor r\frac{2^n}{a_j}\rfloor /2^n$, where ⌊x⌋ is the largest integer ≤ x. Clearly (r)_n → r as n → ∞.

Let k ≥ 3 be a fixed integer whose value will be specified later. By the triangle inequality, we have

$$\begin{array}{l}\mid X(t,t_j+s_j)-X(t,t_j)\mid \\ \le \mid X(t,{(t_j+s_j)}_k)-X(t,{(t_j)}_k)\mid +\mid X(t,t_j+s_j)-X(t,{(t_j+s_j)}_k)\mid +\mid X(t,t_j)-X(t,{(t_j)}_k)\mid \\ \le \mid X(t,{(t_j+s_j)}_k)-X(t,{(t_j)}_k)\mid +\sum _{l=0}^{\infty }\mid X(t,{(t_j+s_j)}_{k+l+1})-X(t,{(t_j+s_j)}_{k+l})\mid +\sum _{l=0}^{\infty }\mid X(t,{(t_j)}_{k+l+1})-X(t,{(t_j)}_{k+l})\mid .\end{array}$$ (3.10)

In order to use the above inequality to estimate P_j, we make some preparation first. For l ≥ 0, put

$$u_l^2=u^2+c_{2,2}\left(2+\frac{2(N-1)H_j}{min\{H_i\}}+\frac{N-1}{min\{H_i\}}\right)(k+l+1).$$

Then

$$u_l\le u+c_{2,2}^{1/2}{\left(2+\frac{2(N-1)H_j}{min\{H_i\}}+\frac{N-1}{min\{H_i\}}\right)}^{1/2}{(k+l+1)}^{1/2}.$$

It follows that

$$\begin{array}{l}{ua}_j^{H_j}+u{\left(\frac{2a_j}{2^k}\right)}^{H_j}+2\sum _{l=0}^{\infty }{u}_l{\left(\frac{a_j}{2^{k+l}}\right)}^{H_j}\\ \le {ua}_j^{H_j}+u\left[{\left(\frac{2a_j}{2^k}\right)}^{H_j}+\sum _{l=0}^{\infty }{\left(\frac{a_j}{2^{k+l}}\right)}^{H_j}\right]+c_{2,2}^{1/2}{\left(2+\frac{2(N-1)H_j}{min\{H_i\}}+\frac{N-1}{min\{H_i\}}\right)}^{1/2}\sum _{l=0}^{\infty }{(k+l+1)}^{1/2}{\left(\frac{a_j}{2^{k+l}}\right)}^{H_j}\\ ={ua}_j^{H_j}+{ua}_j^{H_j}\frac{2^{H_j}+{(1-2^{-H_j})}^{-1}}{2^{{kH}_j}}+c_{2,2}^{1/2}{a}_j^{H_j}{\left(2+\frac{2(N-1)H_j}{min\{H_i\}}+\frac{N-1}{min\{H_i\}}\right)}^{1/2}\sum _{l=0}^{\infty }\frac{{(k+l+1)}^{1/2}}{2^{H_j(k+l)}}\\ \le {ua}_j^{H_j}+\tau {ua}_j^{H_j}+\tau a_j^{H_j}=((1+\tau )u+\tau )a_j^{H_j}.\end{array}$$ (3.11)

In deriving the last inequality we have chosen k large enough such that

$$\frac{2^{H_j}+{(1-2^{-H_j})}^{-1}}{2^{{kH}_j}}\le \tau$$

and

$$c_{2,2}^{1/2}{\left(2+\frac{2(N-1)H_j}{min\{H_i\}}+\frac{N-1}{min\{H_i\}}\right)}^{1/2}\sum _{l=k}^{\infty }\frac{{(l+1)}^{1/2}}{2^{H_{j}l}}\le \tau .$$

It follows from (3.10) and (3.11) that

$$\begin{array}{l}{P}_j\le \mathbb{P}\left\{\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{0\le s_j\le a_j}{sup}\mid X(t,{(t_j+s_j)}_k)-X(t,{(t_j)}_k)\mid \ge (1+2^{-k+1}){ua}_j^{H_j}\right\}\\ +\sum _{l=0}^{\infty }\mathbb{P}\left\{\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{0\le s_j\le a_j}{sup}\mid X(t,{(t_j+s_j)}_{k+l+1})-X(t,{(t_j+s_j)}_{k+l})\mid \ge u_l{\left(\frac{a_j}{2^{k+l}}\right)}^{H_j}\right\}\\ +\sum _{l=0}^{\infty }\mathbb{P}\left\{\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\mid X(t,{(t_j)}_{k+l+1})-X(t,{(t_j)}_{k+l})\mid \ge u_l{\left(\frac{a_j}{2^{k+l}}\right)}^{H_j}\right\}\\ =:P_{j,1}+P_{j,2}+P_{j,3}.\end{array}$$

We estimate the terms P_j,1, P_j,2 and P_j,3 separately. In order to estimate P_j,1, we use the triangle inequality again to write

$$\begin{array}{l}\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{0\le s_j\le a_j}{sup}\mid X(t,{(t_j+s_j)}_k)-X(t,{(t_j)}_k)\mid \\ \le \underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{0\le s_j\le a_j(1-2^{-k})}{sup}\mid X(t,{(t_j+s_j)}_k)-X(t,{(t_j)}_k)\mid \\ +\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{a_j(1-2^{-k})\le s_j\le a_j}{sup}\mid X(t,{(t_j+s_j)}_k)-X(t,{(t_j+a_j(1-2^{-k}))}_k)\mid .\end{array}$$ (3.12)

Since

$$\underset{0\le t_j\le T_j}{sup}\underset{0\le s_j\le a_j(1-2^{-k})}{sup}\mid {(t_j+s_j)}_k-{(t_j)}_k\mid \le a_j,$$

and there are at most $2^k(\frac{T_j}{a_j}+1)$ different points (t_j)_k and at most 2^k different (t_j + s_j)_k, we derive from Lemma 3.2 with z = a_j that for each u ≥ u^*,

$$\begin{array}{l}\mathbb{P}\left\{\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{0\le s_j\le a_j(1-2^{-k})}{sup}\mid X(t,{(t_j+s_j)}_k)-X(t,{(t_j)}_k)\mid \ge {ua}_j^{H_j}\right\}\\ \le c_{3,2}{2}^{2k}\left(\frac{T_j}{a_j}+1\right)·{\left(\frac{1}{a_j}\right)}^{\frac{(N-1)H_j}{min\{H_i\}}}{u}^{\frac{N-1}{min\{H_i\}}}exp\left(-\frac{u^2}{(2+\epsilon )c_{2,2}}\right).\end{array}$$ (3.13)

Similarly, because

$$\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{a_j(1-2^{-k})\le s_j\le a_j}{sup}\mid {(t_j+s_j)}_k-{(t_j+a_j(1-2^{-k}))}_k\mid \le 2a_j·2^{-k},$$

we derive from Lemma 3.2 that for each u ≥ u^*,

$$\begin{array}{l}\mathbb{P}\left\{\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{a_j(1-2^{-k})\le s_j\le a_j}{sup}\mid X(t,{(t_j+s_j)}_k)-X(t,{(t_j+a_j(1-2^{-k}))}_k)\mid \ge u{(2^{-k+1}{a}_j)}^{H_j}\right\}\\ \le c_{3,2}{2}^k\left(\frac{T_j}{a_j}+1\right)·{\left(\frac{2^{k+1}}{a_j}\right)}^{\frac{(N-1)H_j}{min\{H_i\}}}{u}^{\frac{N-1}{min\{H_i\}}}exp\left(-\frac{u^2}{(2+\epsilon )c_{2,2}}\right).\end{array}$$ (3.14)

Combining (3.12)–(3.14), we obtain

$$P_{j,1}\le c_{3,4}\left(\frac{T_j}{a_j}+1\right)·{\left(\frac{1}{a_j}\right)}^{\frac{(N-1)H_j}{min\{H_i\}}}{u}^{\frac{N-1}{min\{H_i\}}}exp\left(-\frac{u^2}{(2+\epsilon )c_{2,2}}\right).$$ (3.15)

In the same way, note that

$$\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{0\le s_j\le a_j}{sup}\mid {(t_j+s_j)}_{k+l+1}-{(t_j+s_j)}_{k+l}\mid \le a_j·2^{-(k+l)},$$

we apply Lemma 3.2 to derive

$$\begin{array}{l}\mathbb{P}\left\{\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\underset{0\le s_j\le a_j}{sup}\mid X(t,{(t_j+s_j)}_{k+l+1})-X(t,{(t_j+s_j)}_{k+l})\mid \ge u_l{(a_j{2}^{-(k+l)})}^{H_j}\right\}\\ \le c_{3,2}{2}^{k+l+1}\left(\frac{T_j}{a_j}+1\right)·{\left(\frac{2^{k+l}}{a_j}\right)}^{\frac{(N-1)H_j}{min\{H_i\}}}{u}_l^{\frac{N-1}{min\{H_i\}}}exp\left(-\frac{u_l^2}{(2+\epsilon )c_{2,2}}\right)\end{array}$$ (3.16)

and

$$\begin{array}{l}P\{\underset{t\in {[0,1]}^{N-1}}{sup}\underset{x_j\le t_j\le x_j+T_j}{sup}\mid X(t,{(t_j)}_{k+l+1})-X(t,{(t_j)}_{k+l})\mid \ge u_l{(a_j{2}^{-(k+l)})}^{H_j}\}\\ \le c_{3,2}{2}^{k+l+1}\left(\frac{T_j}{a_j}+1\right)·{\left(\frac{2^{k+l}}{a_j}\right)}^{\frac{(N-1)H_j}{min\{H_i\}}}{u}_l^{\frac{N-1}{min\{H_i\}}}exp\left(-\frac{u_l^2}{(2+\epsilon )c_{2,2}}\right).\end{array}$$ (3.17)

By the definition of u_l, we have

$$u_l^{\frac{N-1}{min\{H_i\}}}\le 2^{\frac{N-1}{min\{H_i\}}-1}\left[u^{\frac{N-1}{min\{H_i\}}}{c}_{2,2}^{\frac{N-1}{2min\{H_i\}}}{\left(2+\frac{(N-1)(2H_j+1)}{min\{H_i\}}\right)}^{\frac{N-1}{2min\{H_i\}}}(k+l+1)T\frac{N-1}{2min\{H_i\}}\right]$$

and

$$\begin{array}{l}\sum _{l=0}^{\infty }{2}^{\left(1+\frac{(N-1)H_j}{min\{H_i\}}\right)(k+l+1)}{(k+l+1)}^{\frac{N-1}{2min\{H_i\}}}exp\left(-\frac{u_l^2}{(2+\epsilon )c_{2,2}}\right)\\ \le \sum _{l=0}^{\infty }{2}^{\left(1+\frac{(N-1)H_j}{min\{H_i\}}+\frac{N-1}{2min\{H_i\}}\right)(k+l+1)}exp\left(-\frac{u_l^2}{(2-\epsilon )c_{2,2}}\right)\\ =\sum _{l=0}^{\infty }{2}^{\left(1+\frac{(N-1)H_j}{min\{H_i\}}+\frac{N-1}{2min\{H_i\}}\right)(k+l+1)}exp\left(-\left(1+\frac{(N-1)H_j}{min\{H_i\}}+\frac{N-1}{2min\{H_i\}}\right)(k+l+1)\right)\times exp\left(-\frac{u^2}{(2+\epsilon )c_{2,2}}\right)\\ \le exp\left(-\frac{u^2}{(2+\epsilon )c_{2,2}}\right).\end{array}$$

Hence, (3.16) and (3.17) yield

$$P_{j,2}+P_{j,3}\le c_{3,5}\left(\frac{T_j}{a_j}+1\right)·{\left(\frac{1}{a_j}\right)}^{\frac{(N-1)H_j}{min\{H_i\}}}\left(u^{\frac{N-1}{min\{H_i\}}}+1\right)exp\left(-\frac{u^2}{(2+\epsilon )c_{2,2}}\right).$$

Combining the above inequality with (3.15) shows that for each 1 ≤ j ≤ N

$$P_j\le c_{3,5}\left(\frac{T_j}{a_j}+1\right)·{\left(\frac{1}{a_j}\right)}^{\frac{(N-1)H_j}{min\{H_i\}}}\left(u^{\frac{N-1}{min\{H_i\}}}+1\right)exp\left(-\frac{u^2}{(2+\epsilon )c_{2,2}}\right).$$

This, together with (3.9), implies (3.7). The proof is now completed.

4 Global modulus of continuity

In this section we investigate the global modulus of continuity for anisotropic Gaussian random fields. As a consequence of our result, we establish a sharp result for the global modulus of continuity of fractional Brownian sheets for points, which extend Theorem 1 of Ayache and Xiao (2005) and Theorem 3.2 of Wang (2007) (see Section 6 below).

Theorem 4.1

Let {X(t), t ∈ ℝ^N} be a centered Gaussian random field with values in ℝ and satisfy Conditions (A1) and (A2). Put

$$\beta (s,t)=\sigma (s,t)\sqrt{log(1+\sigma {(s,t)}^{-1})},s,t\in I.$$ (4.1)

Then

$$\underset{\epsilon \to 0+}{lim}\underset{s,t\in I,\sigma (s,t)\le \epsilon }{sup}\frac{\mid X(t)-X(s)\mid }{\beta (s,t)}={\kappa }_{1}a.s.,$$ (4.2)

where κ₁ is a positive constant satisfying

$$\sqrt{\frac{2c_{2,3}}{c_{2,2}min\{H_i\}}}\le {\kappa }_1\le \sqrt{\frac{2N^3{c}_{2,2}}{c_{2,1}min\{H_i\}}}.$$ (4.3)

Proof

For simplicity of notation, we assume I = [a, 1]^N, where a ∈ [0, 1) is a constant. For any ε > 0, put

$$J(\epsilon )=\underset{s,t\in I,\sigma (s,t)\le \epsilon }{sup}\frac{\mid X(t)-X(s)\mid }{\beta (s,t)},$$

then ε ↦ → J(ε) is non-decreasing. Hence the limit in the left hand side of (4.2) exists almost surely. We claim that

$$\underset{\epsilon \to 0+}{lim}J(\epsilon )\le c_{4,2}\text{a}.\text{s}.$$ (4.4)

and

$$\underset{\epsilon \to 0+}{lim}J(\epsilon )\le c_{4,1}\text{a}.\text{s}.,$$ (4.5)

where

$$c_{4,1}=\sqrt{\frac{2c_{2,3}}{c_{2,2}min\{H_i\}}}\text{and}{c}_{4,2}=\sqrt{\frac{2N^3{c}_{2,2}}{c_{2,1}min\{H_i\}}}.$$

Before proving (4.4) and (4.5), let us notice that, (4.4) and the proof of Lemma 7.1.1 in Marcus and Rosen (2006) imply (4.2) and the constant κ₁ ∈ [c_4,1, c_4,2].

Hence, it only remains to verify (4.4) and (4.5). We show (4.4) first. Proofs of (4.4) with a generic constant by using general Gaussian principles such as majorizing measure or isoperimetric inequality are available [see, e.g., Marcus and Rosen (2006, Chapter 7) or Xiao (2009)]. Here we apply Proposition 3.3 to provide a more careful deviation to connect this constant with the constants in (A1). Since the function $x\mapsto x\sqrt{log(1+1/x)}$ is increasing for x ∈ (0, 1) small and

$$\sigma {(s,t)}^2\ge c_{2,1}max\{{\mid t_j-s_j\mid }^{2H_j}\},$$

we have

$$\beta (s,t)\ge c_{2,1}^{1/2}{\mid t_j-s_j\mid }^{H_j}\sqrt{log(1+c_{2,1}^{-1/2}{\mid t_j-s_j\mid }^{-H_j})}$$

for every 1 ≤ j ≤ N. Thus by (3.8) we have

$$\begin{array}{l}J(\epsilon )\le \sum _{j=1}^N\underset{\begin{array}{c}t\in {[a,1]}^{N-1},s_j,t_j\in [a,1]\\ \sigma (s,t)\le \epsilon \end{array}}{sup}\frac{\mid X(t,t_j)-X(t,s_j)\mid }{\beta (s,t)}\\ \le \sum _{j=1}^N\underset{\begin{array}{c}t\in {[a,1]}^{N-1},s_j,t_j\in [a,1]\\ c_{2,1}^{1/2}{\mid t_j-s_j\mid }^{H_j}\le \epsilon \end{array}}{sup}\frac{\mid X(t,t_j)-X(t,s_j)\mid }{\beta (s,t)}\\ \le \sum _{j=1}^N\underset{\begin{array}{c}t\in {[a,1]}^{N-1},s_j,t_j\in [a,1]\\ c_{2,1}^{1/2}{\mid t_j-s_j\mid }^{H_j}\le \epsilon \end{array}}{sup}\frac{\mid X(t,t_j)-X(t,s_j)\mid }{c_{2,1}^{1/2}{\mid t_j-s_j\mid }^{H_j}\sqrt{log(1+c_{2,1}^{-1/2}{\mid t_j-s_j\mid }^{-H_j})}}\\ =:\sum _{j=1}^N{J}_j(\epsilon ).\end{array}$$ (4.6)

We now show that for each 1 ≤ j ≤ N,

$$\underset{\epsilon \to 0+}{limsup}{J}_j(\epsilon )\le \frac{c_{4,2}}{N}\text{a}.\text{s}.$$ (4.7)

For every 1 ≤ j ≤ N and μ > 0, define the event

$$E_j(l,k,n)=\left\{sup\frac{\mid X(t,t_j)-X(t,s_j)\mid }{c_{2,1}^{1/2}{\mid t_j-s_j\mid }^{H_j}\sqrt{log(1+c_{2,1}^{-1/2}{\mid t_j-s_j\mid }^{-H_j})}}\ge \frac{(1+\mu )c_{4,2}}{N}\right\},$$

where the supremum is taken over t ∈ [a, 1]^N−1 and all s_j, t_j which satisfy

$$\frac{l-1}{2^n}\le s_j<\frac{l}{2^n},\frac{l+k}{2^n}\le t_j<\frac{l+k+1}{2^n}.$$ (4.8)

Let

$$A=c_{2,1}^{1/2}{k}^{H_j}{2}^{-{nH}_j}\text{and}B=c_{2,1}^{1/2}{(k+2)}^{H_j}{2}^{-{nH}_j}.$$

Then A is the infimum of $c_{2,1}^{1/2}{\mid t_j-s_j\mid }^{H_j}$ taken over s_j, t_j satisfy (4.8), and B is the corresponding supremum. The parameters k and l will be restricted to the following ranges:

$$a{2}^n+1\le l\le 2^n,\frac{1}{4}n\le k\le n$$ (4.9)

for n = 3, 4, …. It is easy to check that

$$0\le 1-{AB}^{-1}\le c{n}^{-1}\to 0.$$

By taking $x_j=\frac{l-1}{2^n},T_j=a_j=\frac{k+2}{2^n}$ and $u=\frac{(1+\mu )c_{4,2}{c}_{2,1}^{1/2}}{N}\sqrt{log(1+B^{-1})}$ in (3.7), we obtain that for n large enough

$$\begin{array}{l}\mathbb{P}\left(E_j(l,k,n)\right)\le c{\left(\frac{2^n}{k+2}\right)}^{\frac{(N-1)H_j}{min\{H_i\}}}exp\left(-\frac{{(1+\mu /2)}^2{c}_{4,2}^2{c}_{2,1}log(1+B^{-1})}{2N^2{c}_{2,2}}\right)\\ \le c{\left(\frac{n}{2^n}\right)}^{\frac{{(1+\mu /2)}^2{c}_{4,2}^2{c}_{2,1}{H}_j}{2N^2{c}_{2,2}}-\frac{(N-1)H_j}{min\{H_i\}}}.\end{array}$$

Thus

$$\sum _{n=1}^{\infty }\sum _{l,k}\mathbb{P}(E_j(l,k,n))\le c\sum _{n=1}^{\infty }n{2}^n{\left(\frac{n}{2^n}\right)}^{\frac{{(1+\mu /2)}^2{c}_{4,2}^2{c_{2,1}}^{H_j}}{2N^2{c}_{2,2}}-\frac{(N-1)H_j}{min\{H_i\}}}<\infty ,$$

where the summation Σ_l,k is taken over integers l, k that satisfy (4.9) and, for obtaining the last inequality, we have used the definition of c_4,2. Thus, by the Borel-Cantelli lemma, there exists a.s. an integer n₀ = n₀(ω) such that none of the events E_j(l, k, n) occur for n ≥ n₀ and l, k satisfying (4.9).

Let [s_j, t_j ] be an interval with t_j − s_j ≤ n₀2^−n₀. Now we first choose n ≥ n₀ so that

$$(n+1)2^{-n-1}<t_j-s_j\le n{2}^{-n}$$

and then l and k so that

$$(l-1)2^{-n}\le s_j<l{2}^{-n},(l+k)2^{-n}\le t_j<(l+k+1)2^{-n}.$$

It is now easy to check that if [s_j, t_j] ⊆ [a, 1] then the indices l, k satisfy (4.9) and [s_j, t_j] is one of the intervals in the event E_j(l, k, n). Hence $\underset{\epsilon \to 0+}{limsup}{J}_j(\epsilon )\le (1+\mu )c_{4,2}/N$ a.s. Letting μ ↓ 0 yields (4.7) for every 1 ≤ j ≤ N. Combining (4.6) and (4.7) we obtain (4.4) immediately.

Now we show (4.5). Let 1 < θ < 2 be a constant which will be specified later and, for all n ≥ 1, let

$${\epsilon }_n=\sqrt{c_{2,2}\sum _{j=1}^N{\theta }^{-2H_{jn}}.}$$

Since for any 0 < ε < 1 there is an integer n ≥ 2 such that ε_n < ε ≤ ε_n−1, by the monotonicity of J(ε) and Condition (A1), we have

$$\begin{array}{l}\underset{\epsilon \to 0+}{lim}J(\epsilon )=\underset{n\to \infty }{lim}\underset{s,t\in I,\sigma (s,t)\le {\epsilon }_n}{sup}\frac{\mid X(t)-X(s)\mid }{\beta (s,t)}\\ \ge \underset{n\to \infty }{liminf}\underset{a{\theta }^n/2\le i\le ({\theta }^n-1)/2}{max}\frac{\mid X(\langle (2i+1){\theta }^{-n}\rangle )-X(\langle 2i{\theta }^{-n}\rangle )\mid }{{\epsilon }_n\sqrt{log(1+{\epsilon }_n^{-1})}}\\ =:\underset{n\to \infty }{liminf}{J}_n.\end{array}$$ (4.10)

Recall that 〈c(i)〉 means the N-dimensional vector (c(i), …, c(i)).

For any μ ∈ (0, 1), we have

$$\begin{array}{l}\mathbb{P}\left(J_n\le (1-\mu )c_{4,1}\right)\\ \le \mathbb{P}\left(\left\{\frac{\mid X(\langle 1\rangle )-X(\langle 1-{\theta }^{-n}\rangle )\mid }{{\epsilon }_n\sqrt{log(1+{\epsilon }_n^{-1})}}\le (1-\mu )c_{4,1}\right\}\cap \left\{\underset{a{\theta }^n/2\le i\le ({\theta }^n-1)/2-1}{max}\frac{\mid X(\langle (2i+1){\theta }^{-n}\rangle )-X(\langle 2i{\theta }^{-n}\rangle )\mid }{{\epsilon }_n\sqrt{log(1+{\epsilon }_n^{-1})}}\le (1-\mu )c_{4,1}\right\}\right)\\ =P_1(n)·P_2(n),\end{array}$$

where

$$P_1(n)=\mathbb{P}\left(\underset{a{\theta }^n/2\le i\le ({\theta }^n-1)/2-1}{max}\frac{\mid X(\langle (2i+1){\theta }^{-n}\rangle )-X(\langle 2i{\theta }^{-n}\rangle )\mid }{{\epsilon }_n\sqrt{log(1+{\epsilon }_n^{-1})}}\le (1-\mu )c_{4,1}\right)$$

and

$$P_2(n)=\mathbb{P}(\frac{\mid X(\langle 1\rangle -X(\langle 1-{\theta }^{-n}\rangle )\mid }{{\epsilon }_n\sqrt{log(1+{\epsilon }_n^{-1})}}\le (1-\mu )c_{4,1}|X(\langle 1-{\theta }^{-n}\rangle );X(\langle (2i+1){\theta }^{-n}\rangle ),X(\langle 2i{\theta }^{-n}\rangle ),a{\theta }^n/2\le i\le ({\theta }^n-1)/2-1).$$

By Condition (A2), we have

$$\text{Var}(X(\langle 1\rangle )-X(\langle 1-{\theta }^{-n}\rangle )|X(\langle 1-{\theta }^{-n}\rangle );X(\langle (2i+1){\theta }^{-n}\rangle ),X(\langle 2i{\theta }^{-n}\rangle ),a{\theta }^n/2\le i\le ({\theta }^n-1)/2-1)\ge c_{2,3}\sum _{j=1}^N{\theta }^{-2H_{j}n}=\frac{c_{2,3}}{c_{2,2}}{\epsilon }_n^2.$$

Thus, by the fact that the conditional distributions of the Gaussian process is still Gaussian and Anderson’s inequality [see Anderson (1955)], we derive

$$P_2(n)\le \mathbb{P}\left(N(0,1)\le (1-\mu )c_{4,1}\sqrt{(c_{2,2}/c_{2,3})log(1+{\epsilon }_n^{-1})}\right),$$

where N(0, 1) denotes a standard normal random variable. By using the following well known in-equality

$${(2\pi )}^{-\frac{1}{2}}(1-x^{-2})x^{-1}{e}^{-\frac{x^2}{2}}\le \mathbb{P}(N(0,1)>x)\le {(2\pi )}^{-\frac{1}{2}}{x}^{-1}{e}^{-\frac{x^2}{2}},\forall x>0,$$ (4.11)

we derive that for all n large enough

$$\begin{array}{l}{P}_2(n)=1-\mathbb{P}\left(N(0,1)>(1-\mu )c_{4,1}\sqrt{(c_{2,2}/c_{2,3})log(1+{\epsilon }_n^{-1})}\right)\\ \le 1-{\epsilon }_n^{\frac{{(1-\mu /2)}^2{c}_{2,2}2c_{4,1}^2}{2c_{2,3}}}\\ \le exp\left(-{\epsilon }_n^{\frac{{(1-\mu /2)}^2{c}_{2,2}{c}_{4,1}^2}{2c_{2,3}}}\right),\end{array}$$

where for obtaining the last inequality we have used the elementary inequality ∀x, 1−x ≤ e^−x. Hence

$$\mathbb{P}\left(J_n\le (1-\mu )c_{4,1}\right)\le exp\left(-{\epsilon }_n^{\frac{{(1-\mu /2)}^2{c}_{2,2}{c}_{4,1}^2}{2c_{2,3}}}\right)·P_1(n).$$

By repeating the above argument, we derive that

$$\begin{array}{l}\mathbb{P}\left(J_n\le (1-\mu )c_{4,1}\right)\le exp\left(-\frac{(1-a){\theta }^n-1}{2}{\epsilon }_n^{\frac{{(1-\mu /2)}^2{c}_{2,2}{c}_{4,1}^2}{2c_{2,3}}}\right)\\ \le exp\left(-c{\theta }^n{\theta }^{-\frac{min\{H_i\}{(1-\mu /2)}^2{c}_{2,2}{c}_{4,1}^{2}n}{2c_{2,3}}}\right),\end{array}$$

where the last inequality follows from the estimate: ${\epsilon }_n^2\ge c_{2,2}{\theta }^{-2min\{H_i\}n}$. Thus, by the definition of c_4,1, we get

$$\sum _{n=1}^{\infty }\mathbb{P}\left(J_n\le (1-\mu )c_{4,1}\right)<\infty .$$

Thus, the Borel-Cantelli lemma implies

$$\underset{n\to \infty }{liminf}{J}_n\ge (1-\mu )c_{4,1}\text{a}.\text{s}.$$ (4.12)

Letting μ ↓ 0 along a sequence, by (4.10) and (4.12) we obtain (4.5). The proof of Theorem 4.1 is completed.

5 Local moduli of continuity, laws of the iterated logarithm

In this section we investigate the local moduli of continuity for anisotropic random fields with stationary increments. Let X = {X(t), t ∈ ℝ^N} be a real-valued, centered Gaussian random field with X(〈0〉) = 0. We assume that the covariance function R(s, t) = [X(s)X(t)] is continuous and X has stationary increments. The latter means that for any h ∈ ℝ^N,

$$\{X(t+h)-X(h),t\in {\mathbb{R}}^N\}\stackrel{d}{=}\{X(t),t\in {\mathbb{R}}^N\},$$

where $\stackrel{d}{=}$ means equality in finite dimensional distributions. According to Yaglom (1957), R(s, t) can be represented as

$$R(s,t)={\int }_{{\mathbb{R}}^N}(e^{i(s,\xi )}-1)(e^{i\langle t,\xi \rangle }-1)\mathrm{\Delta }(d\xi )+\langle s,Qt\rangle ,$$ (5.1)

where Q = (q_ij) is an N × N non-negative definite matrix and Δ(dξ) is a nonnegative symmetric measure on ℝ^N \ {0} satisfying

$${\int }_{{\mathbb{R}}^N}\frac{{\mid \xi \mid }^2}{1+{\mid \xi \mid }^2}\mathrm{\Delta }(d\xi )<\infty .$$ (5.2)

The measure Δ and its density f(ξ) (if it exists) are called the spectral measure and spectral density of X, respectively.

It follows from (5.1) that X has the following stochastic integral representation:

$$X(t)={\int }_{{\mathbb{R}}^N}(e^{i\langle t,\xi \rangle }-1)\mathcal{M}(d\xi )+\langle \mathbf{Y},t\rangle ,$$ (5.3)

where Y is an N-dimensional Gaussian random vector with mean zero and (dξ) is a centered complex-valued Gaussian random measure which is independent of Y and satisfies

$$\mathbb{E}\left[\mathcal{M}(A)\overline{\mathcal{M}(B)}\right]=\mathrm{\Delta }(A\cap B)\text{and}\mathcal{M}(-A)=\overline{\mathcal{M}(A)}$$

for all Borel sets A,B ⊂ℝ^N with finite Δ-measure. The spectral measure Δ is called the control measure of . Since the linear term 〈Y, t〉 in (5.3) will not have any effect on the problems considered in this paper, we will from now on assume Y = 0. This is equivalent to assuming Q = 0 in (5.1). Consequently, we have

$${\sigma }^2(h)=\mathbb{E}[{(X(t+h)-X(t))}^2]=2{\int }_{{\mathbb{R}}^N}(1-cos\langle h,\xi \rangle )\mathrm{\Delta }(d\xi ).$$ (5.4)

For t₀ ∈ ℝ^N and a family of neighborhoods {O(δ), δ > 0} of 〈0〉 ∈ ℝ^N whose diameters go to 0 as δ → 0, we consider the corresponding local moduli of continuity of X at t₀

$$\omega (t_0,\delta )=\underset{s\in O(\delta )}{sup}\mid X(t_0+s)-X(t_0)\mid .$$

It can be seen that, if X is anisotropic, then the rate at which ω(t₀, δ) goes to 0 as δ → 0 depends on the shape of O(δ).

In the following, we consider two kinds of local moduli of continuity for X. Theorem 5.1 is concerned with the local modulus of continuity measured in the most general way. Theorem 5.6 provides the local modulus of continuity in the metric σ. It should be noticed that the logarithmic factors in these two theorems are quite different, since (A1) implies $\sigma (s)\asymp {\sum }_{j=1}^N{\mid s_j\mid }^{H_j}$ as s → 0 (ratio remains bounded away from zero and infinity) in Theorem 5.6, and the corresponding term ${\prod }_{j=1}^N{\mid s_j\mid }^{H_j}$ in Theorem 5.1 is much smaller.

Theorem 5.1

Let {X(t), t ∈ ℝ^N} be a real valued, centered Gaussian random field with stationary increments and X(〈0〉) = 0. If X satisfies Condition (A1) for I = [0, 1]^N, then there is a positive constant κ₂ such that for every t₀ ∈ ℝ^N we have

$$\underset{\left|\right|\epsilon \left|\right|\to 0+}{limsup}\underset{\langle \mid s_j\mid \rangle \le \langle {\epsilon }_j\rangle }{sup}\frac{\mid X(t_0+s)-X(t_0)\mid }{\gamma (s)}={\kappa }_{2}a.s.,$$ (5.5)

where

$$\gamma (s)=\sigma (s){\left[loglog\left(1+\frac{1}{{\prod }_{j=1}^N{\mid s_j\mid }^{H_j}}\right)\right]}^{\frac{1}{2}},\forall s\in {\mathbb{R}}^N.$$ (5.6)

Remark 5.2

Eq. (5.5) means that, for any η > 0, there exists a.s. δ₀ = δ₀(ω) > 0 such that for all ε ∈ (0, 1)^N which satisfies ||ε|| ≤ δ₀, we have

$$\underset{\langle \mid s_j\mid \rangle \le \langle {\epsilon }_j\rangle }{sup}\frac{\mid X(t_0+s)-X(t_0)\mid }{\gamma (s)}<{\kappa }_2+\eta .$$

Moreover for any η > 0 there exists a sequence ε(n) ∈ (0, 1)^N such that ||ε(n)|| → 0 and

$$\underset{\langle \mid s_j\mid \rangle \le \epsilon (n)}{sup}\frac{\mid X(t_0+s)-X(t_0)\mid }{\gamma (s)}<{\kappa }_2-\eta$$

for n large enough.

In order to show Theorem 5.1, we will make use of the following lemmas. The first one (Lemma 5.3) is taken from Talagrand (1995). Let {Z(t), t ∈ S} be a centered Gaussian process with values in ℝ. The index set S is equipped with the pseudometric d(s, t) = [ (Z(t) − Z(s))²]^1/2. We denote by N_d(S, ε) the smallest number of (open) d-balls of radius ε needed to cover S and we denote by D the diameter of S, that is, D = sup{d(s, t) : s, t ∈ S}.

Lemma 5.3

Given x > 0, we have

$$\mathbb{P}\left(\underset{s,t\in S}{sup}\mid Z(t)-Z(s)\mid \ge c_{5,1}\left(x+{\int }_0^D\sqrt{log{N}_d(S,\epsilon )}d\epsilon \right)\right)\le exp\left(-\frac{x^2}{D^2}\right),$$ (5.7)

where c_5,1 is a positive and finite constant.

Lemma 5.4

Let {X(t), t ∈ ℝ^N} be a real valued, centered Gaussian random field satisfying the upper bound in Condition (A1). Then there exist positive and finite constants u₀ and c_5,2 such that for all t₀ ∈ I and u ≥ u₀

$$\mathbb{P}\left(\underset{\langle \mid s_j\mid \rangle \le \langle a_j\rangle }{sup}\mid X(t_0+s)-X(t_0)\mid \ge u\sum _{j=1}^N{a}_j^{H_j}\right)\le e^{-c_{5,2}{u}^2}$$ (5.8)

for all 〈a_j〉 ∈ (0, 1]^N such that t₀ − 〈a_j〉 ∈ I and t₀ + 〈a_j〉 ∈ I.

Proof

We will use Lemma 5.3 to prove this lemma. Consider the Gaussian random field {Z(t), t ∈ S} defined by Z(t) = X(t₀ + t) − X(t₀), where S = {t ∈ ℝ^N : |t_j| ≤ a_j}. By Condition (A1), we have

$$d(s,t)=d_Z(s,t)\le c_{2,2}\sum _{j=1}^N{\mid s_j-t_j\mid }^{H_j},\forall s,t\in S.$$

Thus

$$N_d(S,\epsilon )\le c\left(\prod _{j=1}^N{a}_j\right){\epsilon }^{-{\sum }_{j=1}^N\frac{1}{H_j}}$$

and the diameter D of S is at most $c{\sum }_{j=1}^N{a}_j^{H_j}$. Let j₀ be the index such that $a_{j_0}^{H_{j_0}}=max\{a_j^{H_j},1\le j\le N\}$. It is elementary to verify that

$$\begin{array}{l}{\int }_0^D\sqrt{log{N}_d(S,\epsilon )}d\epsilon \le c{\int }_0^D\sqrt{log\left(\prod _{j=1}^N\frac{a_j}{{\epsilon }^{1/H_j}}\right)}d\epsilon \\ \le c{\int }_0^{cN{a}_{j_0}^{H_{j_0}}}\sqrt{log{\left(\frac{a_{j_0}^{H_{j_0}}}{\epsilon }\right)}^{{\sum }_{j=1}^N\frac{1}{H_j}}}d\epsilon \\ =c{a}_{j_0}^{H_{j_0}}{\int }_0^{cN}\sqrt{log\left(\frac{1}{\eta }\right)}d\eta \\ \le c\sum _{j=1}^N{a}_j^{H_j}.\end{array}$$

Hence the conclusion of the lemma follows from (5.7).

The following truncation inequalities are extensions of those in Loève (1977, p. 209) for N = 1 and (3.4) and (3.5) in Xiao (1996) for N > 1 and ρ being replaced by the Euclidean metric. In the current form, they are proved in Luan and Xiao (2010).

Lemma 5.5

Let Δ be a nonnegative symmetric Borel measure on ℝ^N\{0} which satisfies (5.2). Then for any u > 0 and any t ∈ ℝ^N with ρ(0, t)u ≤ 1/N we have

$${\int }_{\{\xi :\rho (0,\xi )<u\}}{\langle t,\xi \rangle }^2\mathrm{\Delta }(d\xi )\le c{\int }_{{\mathbb{R}}^N}(1-cos\langle t,\xi \rangle )\mathrm{\Delta }(d\xi )$$ (5.9)

and for all u > 0

$${\int }_{\{\xi :\rho (0,\xi )\ge u\}}\mathrm{\Delta }(d\xi )\le c{u}^Q{\int }_{\{v:\rho (0,v)\le 1/u\}}dv{\int }_{{\mathbb{R}}^N}(1-cos\langle v,\xi \rangle )\mathrm{\Delta }(d\xi ),$$ (5.10)

where $Q={\sum }_{j=1}^N\frac{1}{H_j}$.

We are in position to prove Theorem 5.1. Due to the stationarity of increments of X, it is sufficient to consider t₀ = 〈0〉. However, we will keep writing t₀ because the method for proving Theorem 5.1 remains valid as long as X has an appropriate stochastic integral representation. In particular, we will see in Section 6 that the proof below can be modified to obtain local moduli of continuity for fractional Brownian sheets, which do not have stationary increments in the usual sense.

Proof of Theorem 5.1

For any ε = 〈ε_j〉 ∈ (0, 1)^N, put

$$M(\epsilon )=\underset{\langle \mid s_j\mid \rangle \le \langle {\epsilon }_j\rangle }{sup}\frac{\mid X(t_0+s)-X(t_0)\mid }{\gamma (s)}.$$

Note that, even though M(ε) is a nondecreasing function of ε ∈ (0, 1)^N in the partial order ≤, it is in general not monotone in ||ε||. We claim that

$$\underset{\left|\right|\epsilon \left|\right|\to 0+}{limsup}M(\epsilon )\le c_{5,3}\text{a}.\text{s}.$$ (5.11)

for some constant c_5,3 > 0 and

$$\underset{\left|\right|\epsilon \left|\right|\to 0+}{limsup}M(\epsilon )\ge \sqrt{2}\text{a}.\text{s}.$$ (5.12)

Before proving (5.11) and (5.12), let us notice again that, (5.11) and the proof of Lemma 7.1.1 in Marcus and Rosen (2006) imply (5.5) and the constant ${\kappa }_2\in [\sqrt{2},c_{5,3}]$.

Hence, it is enough to verify (5.11) and (5.12). We show (5.11) first. For any n = (n₁, …, n_N) ∈ ℕ^N, let h_n = 〈2^−n_j〉. Let δ > 0 be a constant whose value will be determined later. Define the event

$$F_{\mathbf{n}}=\left\{\underset{h_{\mathbf{n}}\le \langle \mid s_j\mid \rangle \le h_{\mathbf{n}-(1)}}{sup}\gamma {(s)}^{-1}\mid X(t_0+s)-X(t_0)\mid \ge \delta \right\}.$$

By Condition (A1), we see that for any s ∈ ℝ^N that satisfies h_n ≤ 〈|s_j|〉 ≤ h_n−〈1〉 we have

$$\gamma (s)\ge c_{2,1}\left(\sum _{j=1}^N{2}^{-n_j{H}_j}\right)\sqrt{loglog\left(1+\prod _{j=1}^N{2}^{(n_j-1)H_j}\right)}.$$

This and Lemma 5.4 imply

$$\begin{array}{l}\mathbb{P}(F_{\mathbf{n}})\le exp\left(c_{5,4}{\delta }^{2}loglog\left(1+{\prod }_{j=1}^N{2}^{(n_j-1)H_j}\right)\right)\\ \le c{\left(\sum _{j=1}^N{n}_j\right)}^{-c_{5,4}{\delta }^2},\end{array}$$

where $c_{5,4}=c_{5,2}{c}_{2,1}^2$. By taking δ large enough such that c_5,4 δ² > N, we see that

$$\sum _{\mathbf{n}\in {\mathbb{N}}^N}\mathbb{P}(F_{\mathbf{n}})\le c\sum _{\mathbf{n}\in {\mathbb{N}}^N}{\left|\right|\mathbf{n}\left|\right|}^{-c_{5,4}{\delta }^2}<\infty .$$

Thus, by the Borel-Cantelli lemma, a.s. only finitely many of the events F_n occur. This implies

$$\underset{\left|\right|\mathbf{n}\left|\right|\to \infty }{limsup}\underset{h_{\mathbf{n}}\le \langle \mid s_j\mid \rangle \le h_{\mathbf{n}-\langle 1\rangle }}{sup}\frac{\mid X(t_0+s)-X(t_0)\mid }{\gamma (s)},\le \delta \text{a}.\text{s}.$$ (5.13)

Let s ∈ ℝ^N be a point such that, for some n⁰ ∈ ℕ^N we have $\mid s_j\mid \le 2^{-n_j^0}$ for every 1 ≤ j ≤ N. Now we choose n ∈ ℕ^N so that

$$2^{-n_j}<\mid s_j\mid \le 2^{-n_j+1}$$

for every 1 ≤ j ≤ N. This and (5.13) yield (5.11).

Now we show that (5.12) holds. For this purpose, it is sufficient to provide a sequence $\langle {\epsilon }_j^{(n)}\rangle \in {(0,1)}^N$ such that $\left|\right|\langle {\epsilon }_j^{(n)}\rangle \left|\right|\to 0$ and

$$\underset{n\to \infty }{limsup}\frac{\mid X(t_0+\langle {\epsilon }_j^{(n)}\rangle )-X(t_0)\mid }{\gamma (\langle {\epsilon }_j^{(n)}\rangle )}\ge \sqrt{2}\text{a}.\text{s}.$$ (5.14)

To this end we will use the spectral representation (5.3) of X to create independence among the random variables. This argument is a modification of those in Monrad and Rootzen (1995), Talagrand (1995) or Li and Shao (2001) so that it adapts to the anisotropy of X.

For any 0 < μ < 1 and n ≥ 1, we define $\langle {\epsilon }_j^{(n)}\rangle =({\epsilon }_1^{(n)},\dots ,{\epsilon }_N^{(n)})$ by

$${\epsilon }_j^{(n)}=exp(-H_j^{-1}{n}^{1+\mu })(j=1,\dots ,N).$$

Then $\rho (0,\langle {\epsilon }_j^{(n)}\rangle )=Nexp(-n^{1+\mu })$.

For every integer n ≥ 1, we denote

$$d_n=exp(n^{1+\mu }+n^{\mu })$$

and define the following Gaussian random fields

$${\stackrel{\sim }{X}}_n(t)={\int }_{\{\rho (0,\xi )\notin (d_{n-1},d_n]\}}(e^{i\langle t,\xi \rangle }-1)\mathcal{M}(d\xi ),\forall t\in {\mathbb{R}}^N$$ (5.15)

and

$$X_n(t)={\int }_{\{\rho (0,\xi )\in (d_{n-1},d_n]\}}(e^{i\langle t,\xi \rangle }-1)\mathcal{M}(d\xi ),\forall t\in {\mathbb{R}}^N.$$ (5.16)

Then {X̃_n(t), t ∈ ℝ^N} and {X_n(t), t ∈ ℝ^N} are independent and X(t) = X_n(t)+ X̃_n(t) for all t ∈ ℝ^N. Moreover, the random fields {X_n(t), t ∈ ℝ^N}, n = 1, 2, … are independent. Notice that

$$\begin{array}{l}\underset{n\to \infty }{limsup}\frac{\mid X(t_0+\langle {\epsilon }_j^{(n)}\rangle )-X(t_0)\mid }{\gamma (\langle {\epsilon }_j^{(n)}\rangle )}\ge \underset{n\to \infty }{limsup}\frac{\mid X_n(t_0+\langle {\epsilon }_j^{(n)}\rangle )-X_n(t_0)\mid }{\gamma (\langle {\epsilon }_j^{(n)}\rangle )}-\underset{n\to \infty }{limsup}\frac{\mid {\stackrel{\sim }{X}}_n(t_0+\langle {\epsilon }_j^{(n)}\rangle )-{\stackrel{\sim }{X}}_n(t_0)\mid }{\gamma (\langle {\epsilon }_j^{(n)}\rangle )}\\ =:\underset{n\to \infty }{limsup}{I}_1(n)-\underset{n\to \infty }{limsup}{I}_2(n).\end{array}$$ (5.17)

By the definition of X̃_n(t) we have

$$\begin{array}{l}\mathbb{E}{({\stackrel{\sim }{X}}_n(t_0+\langle {\epsilon }_j^{(n)}\rangle )-{\stackrel{\sim }{X}}_n(t_0))}^2=2({\int }_{\rho (0,\xi )\le d_{n-1}}+{\int }_{\rho (0,\xi )>d_n})(1-cos\langle \langle {\epsilon }_j^{(n)}\rangle ,\xi \rangle )\mathrm{\Delta }(d\xi )\\ \le {\int }_{\rho (0,\xi )\le d_{n-1}}{\langle {\epsilon }_j^{(n)},\xi \rangle }^2\mathrm{\Delta }(d\xi )+4{\int }_{\rho (0,\xi )>d_n}\mathrm{\Delta }(d\xi ).\end{array}$$

To derive the above inequality, we bound 1 − cos〈t, x〉 by 〈t, x〉²/2 and by 2, respectively.

Now we estimate the last two integrals separately. Denote U = exp (μ(n − 1)^μ. Notice that

$$\begin{array}{l}\rho (0,\langle {\epsilon }_j^{(n)}U\rangle )d_{n-1}\le N{U}^{max\{H_i\}}exp(-n^{1+\mu }+{(n-1)}^{1+\mu }+{(n-1)}^{\mu })\\ \le Nexp(-\mu (1-max\{H_i\}){(n-1)}^{\mu }),\end{array}$$

which is smaller than 1/N for n large. It follows from (5.9) that

$$\begin{array}{l}{\int }_{\rho (0,\xi )\le d_{n-1}}{\langle \langle {\epsilon }_j^{(n)}\rangle ,\xi \rangle }^2\mathrm{\Delta }(d\xi )=U^{-2}{\int }_{\rho (0,\xi )\le d_{n-1}}{\langle \langle {\epsilon }_j^{(n)}U\rangle ,\xi \rangle }^2\mathrm{\Delta }(d\xi )\\ \le c{U}^{-2}{\sigma }^2(\langle {\epsilon }_j^{(n)}U\rangle )\\ \le c{U}^{-2(1-max\{H_i\})}\rho {(0,\langle {\epsilon }_j^{(n)}\rangle )}^2,\end{array}$$ (5.18)

where the last inequality follows from condition (A1).

On the other hand, (5.10) and condition (A1) imply that

$$\begin{array}{l}{\int }_{\rho (0,\xi )>d_n}\mathrm{\Delta }(d\xi )\le c{d}_n^Q{\int }_{v:\rho (0,v)\le d_n^{-1}}{\sigma }^2(v)dv\\ \le c{d}_n^{-2}=c\rho {(0,\langle {\epsilon }_j^{(n)}\rangle )}^{2}exp(-2n^{\mu }).\end{array}$$ (5.19)

Combining (5.18) and (5.19) we obtain

$$\begin{array}{l}\mathbb{E}{({\stackrel{\sim }{X}}_n(t_0+\langle {\epsilon }_j^{(n)}\rangle )-{\stackrel{\sim }{X}}_n(t_0))}^2\le c(U^{-2(1-max\{H_i\})}\rho {(0,\langle {\epsilon }_j^{(n)}\rangle )}^2+d_n^{-2})\\ \le c\rho {(0,\langle {\epsilon }_j^{(n)}\rangle )}^{2}exp(-2(1-max\{H_i\})\mu n^{\mu }).\end{array}$$ (5.20)

Hence for any η > 0 we have

$$\begin{array}{l}\mathbb{P}(I_2(n)\ge \eta )\le \mathbb{P}\left(\mid {\stackrel{\sim }{X}}_n(t_0+\langle {\epsilon }_j^{(n)}\rangle )-{\stackrel{\sim }{X}}_n(t_0)\mid \ge \eta \gamma (\langle {\epsilon }_j^{(n)}\rangle )\right)\\ \le \mathbb{P}\left(\mid N(0,1)\mid \ge \eta \sqrt{logn}exp((1-max\{H_i\})\mu n^{\mu })\right)\\ \le n^{-2}\end{array}$$

for all n large enough. Thus ${\displaystyle \sum _{n=1}^{\infty }}\mathbb{P}(I_2(n)\ge \eta )<\infty$. By the Borel-Cantelli lemma and the arbitrariness of η, we obtain

$$\underset{n\to \infty }{limsup}{I}_2(n)=0\text{a}.\text{s}.$$ (5.21)

In order to estimate $\underset{n\to \infty }{limsup}{I}_1(n)$, notice that

$$\mathbb{E}{\left(X_n(t_0+\langle {\epsilon }_j^{(n)}\rangle )-X_n(t_0)\right)}^2\le {\sigma }^2(0,\langle {\epsilon }_j^{(n)}\rangle ).$$

It follows from this and (4.11) that for any 0 < η < 1,

$$\begin{array}{l}\mathbb{P}\left(\mid X_n(t_0+\langle {\epsilon }_j^{(n)}\rangle )-X_n(t_0)\mid \ge (1-\eta )\sqrt{2}\gamma (\langle {\epsilon }_j^{(n)}\rangle )\right)\ge \mathbb{P}\left(\mid N(0,1)\mid \ge (1-\eta )\sqrt{2loglog(1+exp(N{n}^{1+\mu }))}\right)\\ \ge c{n}^{-{(1-\eta )}^2(1+\mu )}.\end{array}$$

Now we choose μ > 0 small such that (1−η)²(1+μ) < 1 and consequently ${\displaystyle \sum _{n=1}^{\infty }}\mathbb{P}(I_1(n)\ge (1-\eta )\sqrt{2})=\infty$. Since the events $\{I_1(n)\ge (1-\eta )\sqrt{2}\}$ are independent, the Borel-Cantelli lemma and the arbitrariness of η yield

$$\underset{n\to \infty }{limsup}{I}_1(n)\ge \sqrt{2}\text{a}.\text{s}.$$ (5.22)

Hence (5.12) follows from (5.17), (5.21) and (5.22). The proof of Theorem 5.1 is now completed.

Combining Theorem 5.1 and Lemma 7.1.1 in Marcus and Rosen (2006) we derive the following local modulus of continuity.

Theorem 5.6

Let X = {X(t), t ∈ ℝ^N} be a real-valued, centered Gaussian random field with stationary increments and X(〈0〉) = 0. If X satisfies Condition (A1) for I = [0, 1]^N, then there is a positive and finite constant κ₃ such that for every t₀ ∈ ℝ^N we have

$$\underset{\epsilon \to 0+}{lim}\underset{s:\sigma (s)\le \epsilon }{sup}\frac{\mid X(t_0+s)-X(t_0)\mid }{\sigma (s)\sqrt{loglog(1+\sigma {(s)}^{-1})}}={\kappa }_{3}a.s.$$ (5.23)

Proof

It suffices to prove that there exists a finite constant c such that

$$\underset{\epsilon \to 0+}{lim}\underset{s:\sigma (s)\le \epsilon }{sup}\frac{\mid X(t_0+s)-X(t_0)\mid }{\sigma (s)\sqrt{loglog(1+\sigma {(s)}^{-1})}}=c\text{a}.\text{s}.$$ (5.24)

and

$$\underset{\epsilon \to 0+}{lim}\underset{s:\sigma (s)\le \epsilon }{sup}\frac{\mid X(t_0+s)-X(t_0)\mid }{\sigma (s)\sqrt{loglog(1+\sigma {(s)}^{-1})}}\ge \sqrt{2}.$$ (5.25)

Eq. (5.24) can be proved by using Lemma 5.4 and the Borel-Cantelli lemma which is simpler than the proof of (5.11). We omit the details.

On the other hand, we notice that for 〈 ${\epsilon }_j^{(n)}$ 〉 as in the proof of Theorem 5.1, we have

$$\sigma (\langle {\epsilon }_j^{(n)}\rangle )\sqrt{loglog(1+\sigma {(\langle {\epsilon }_j^{(n)}\rangle )}^{-1})}~\gamma (\langle {\epsilon }^{(n)}\rangle )$$

as n → ∞. It follows from (5.14) that

$$\underset{n\to \infty }{lim}\frac{\mid X(t_0+\langle {\epsilon }_j^n\rangle )-X(t_0)\mid }{\sigma (\langle {\epsilon }_j^{(n)}\rangle )\sqrt{loglog(1+\sigma {(\langle {\epsilon }_j^{(n)}\rangle )}^{-1})}}\ge \sqrt{2},$$

which implies (5.25). The proof is finished.

6 Applications

In this section we concern with some applications of our results. In particular we concern with the applications to fractional Brownian sheets and to the solutions of stochastic partial differential equations.

6.1. Applications to fractional Brownian sheets

Fractional Brownian sheets were first introduced by Kamont (1996) who also studied some of their regularly properties. As applications of our results, we establish the global and local moduli of continuity for fractional Brownian sheet for points. Our results extend the related results of Orey and Pruitt (1973), Ayache and Xiao (2005) and Wang (2007). For a given vector H = (H₁, …,H_N) (0 < H_j < 1 for j = 1, …,N), a one-dimensional fractional Brownian sheet B^H = {B^H(t), t ∈ ℝ^N} with Hurst index H is a real-valued, centered Gaussian random field with covariance function given by

$$\mathbb{E}\left[B^H(s)B^H(t)\right]=\prod _{j=1}^N\frac{1}{2}\left({\mid s_j\mid }^{2H_j}+{\mid t_j\mid }^{2H_j}-{\mid s_j-t_j\mid }^{2H_j}\right),s,t\in {\mathbb{R}}^N.$$ (6.1)

It follows from (6.1) that B^H is an anisotropic Gaussian random field and B^H = 0 a.s. for every t ∈ ℝ^N with at least one zero coordinate. Ayache and Xiao (2005) and Wu and Xiao (2007) showed that for every ε ∈ (0, 1), fractional Brownian sheets satisfy Conditions (A1) and (A2) for all I ⊂ [ε,∞)^N. Therefore, by Theorem 4.1, we have the following global modulus of continuity for fractional Brownian sheets. The bound is sharp. Ayache and Xiao (2005) established a sharp upper bound for the global modulus of continuity of fractional Brownian sheets by using the wavelet method. Theorem 6.1 below not only gives its sharp lower bound, but also improves the upper bound of Ayache and Xiao (2005). Wang (2007) established a lower bound for the modulus of continuity of fractional Brownian sheets, but the bound is not as sharp as that given by Theorem 6.1 below. Theorem 6.2 below is the local moduli of continuity or laws of the iterated logarithm for fractional Brownian sheets, which is complementary to Theorem 2 and Proposition 1 in Ayache and Xiao (2005).

Theorem 6.1

Let {B^H(t), t ∈ ℝ^N} be a fractional Brownian sheet with index H = (H₁, …,H_N) ∈ (0, 1)^N and let I = [a, 1]^N, where a ∈ (0, 1) is a constant. Then

$$\underset{\epsilon \to 0+}{lim}\underset{s,t\in I,\sigma (s,t)\le \epsilon }{sup}\frac{\mid B^H(t)-B^H(s)\mid }{\beta (s,t)}={\kappa }_{4}a.s.,$$ (6.2)

where β(s, t) is defined as in (4.1) and κ₄ is a positive and finite constant.

Even though Theorems 5.1 and 5.6 can not be applied directly to B^H because it does not have stationary increments in the ordinary sense, one can apply the harmonizable representation of B^H and modify the proof of Theorem 5.6 to prove the following Theorem 6.2.

Theorem 6.2

Let {B^H(t), t ∈ ℝ^N} be a fractional Brownian sheet with index H = (H₁, …,H_N) ∈ (0, 1)^N and let a ∈ (0, 1) be a constant. Then for every t₀ ∈ [a, 1]^N there exist positive and finite constants κ₅ and κ₆ such that

$$\underset{\left|\right|\epsilon \left|\right|\to 0+}{limsup}\underset{\langle \mid s_j\mid \rangle \le \langle {\epsilon }_j\rangle }{sup}\frac{\mid B^H(t_0+s)-B^H(t_0)\mid }{\rho (0,s)\sqrt{loglog(1+{\prod }_{j=1}^N{\mid s_j\mid }^{-H_j})}}={\kappa }_{5}a.s.$$ (6.3)

and

$$\underset{\epsilon \to 0+}{lim}\underset{s:\sigma (s)\le \epsilon }{sup}\frac{\mid B^H(t_0+s)-B^H(t_0)\mid }{\rho (0,s)\sqrt{loglog(1+\rho {(0,s)}^{-1})}}={\kappa }_{6}a.s.$$ (6.4)

Remark 6.3

The constants κ₅ and κ₆ may depend on t₀, but they are bounded from above and below by positive constants which only depend on H, a and N.

Proof of Theorem 6.2

The upper bounds in (6.3) and (6.4) follows respectively from the proofs of the upper bounds in Theorems 5.1 and 5.6, which only rely on condition (A1). For proving the lower bounds in (6.3) and (6.4), we need to modify the proofs of the lower bound in Theorems 5.1. Instead of using (5.3), we will make use of the following harmonizable representation for B^H:

$$B^H(t)=K_H^{-1}{\int }_{{\mathbb{R}}^N}\prod _{j=1}^N\frac{e^{{it}_j{\xi }_j}-1}{{\mid {\xi }_j\mid }^{H_j+\frac{1}{2}}}\stackrel{\sim }{W}(d\xi ),$$ (6.5)

where K_H > 0 is a normalizing constant and W̃ is a centered complex-valued Gaussian random measure in ℝ^N with Lebesgue control measure. Or one may use the stochastic integral representation given by (2.6) in Wang (2007).

Let 〈 ${\epsilon }_j^{(n)}$〉 and d_n be defined as in the proof of Theorem 5.1. Similarly to (5.15) and (5.16), we define

$${\stackrel{\sim }{B}}_n^H(t)={\int }_{\rho (0,\xi )\notin (d_{n-1},d_n]}\prod _{j=1}^N\frac{e^{{it}_j{\xi }_j}-1}{{\mid {\xi }_j\mid }^{H_j+\frac{1}{2}}}\stackrel{\sim }{W}(d\xi ),\forall t\in {\mathbb{R}}^N$$

and

$$B_n^H(t)={\int }_{\rho (0,\xi )\in (d_{n-1},d_n]}\prod _{j=1}^N\frac{e^{{it}_j{\xi }_j}-1}{{\mid {\xi }_j\mid }^{H_j+\frac{1}{2}}}\stackrel{\sim }{W}(d\xi ),\forall t\in {\mathbb{R}}^N.$$

Then the random fields ${\stackrel{\sim }{B}}_n^H$ and $B_n^H$ are independent. Moreover

$$\begin{array}{l}\mathbb{E}{({\stackrel{\sim }{B}}_n^H(t_0+\langle {\epsilon }_j^{(n)}\rangle )-{\stackrel{\sim }{B}}_n^H(t_0))}^2\\ \le {\int }_{\rho (0,\xi )\le d_{n-1}}\left|\prod _{j=1}^N(e^{i(t_j+{\epsilon }_j^{(n)}){\xi }_j}-1)-\prod _{j=1}^N(e^{{it}_j{\xi }_j}-1)\right|\frac{d\xi }{{\prod }_{j=1}^N{\mid {\xi }_j\mid }^{2H_j+1}}\\ +{\int }_{\rho (0,\xi )>d_n}\left|\prod _{j=1}^N(e^{i(t_j+{\epsilon }_j^{(n)}){\xi }_j}-1)-\prod _{j=1}^N(e^{{it}_j{\xi }_j}-1)\right|\frac{d\xi }{{\prod }_{j=1}^N{\mid {\xi }_j\mid }^{2H_j+1}}\\ =J_1+J_2.\end{array}$$ (6.6)

By using the triangle inequality and the fact that t₀ ∈ [a, 1]^N, we derive directly that

$$\begin{array}{l}{J}_1\le c\sum _{j=1}^N{\int }_{{\mid {\xi }_j\mid }^{H_j}\le d_{n-1}}(1-cos({\epsilon }_j^{(n)}{\xi }_j))\frac{d{\xi }_j}{{\mid {\xi }_j\mid }^{2H_j+1}}\\ \le c{U}^{-2(1-H_N)}\rho {(0,\langle {\epsilon }_j^{(n)}\rangle )}^2.\end{array}$$ (6.7)

To bound J₂, notice that ρ(0, ξ) > d_n implies |ξ_j0|^H_j0 > d_n/N for some j₀ ∈ {1, …,N}. For simplicity of notation, we assume j₀ = 1. Then

$$J_2\le c{\int }_{{\mid {\xi }_1\mid }^{H_1}>d_n/N}\frac{d{\xi }_1}{{\mid {\xi }_1\mid }^{2H_1+1}}=c{d}_n^{-2}.$$ (6.8)

Combining (6.6), (6.7) and (6.8) we obtain

$$\mathbb{E}{({\stackrel{\sim }{B}}_n^H(t_0+\langle {\epsilon }_j^{(n)}\rangle )-{\stackrel{\sim }{B}}_n^H(t_0))}^2\le c\rho {(0,\langle {\epsilon }_j^{(n)}\rangle )}^{2}exp(-2(1-H_N)\mu n^{\mu }),$$

which is the same as (5.20). Now the same proof as that of Theorem 5.1 shows that for every t ∈ [a, 1]^N

$$\underset{n\to \infty }{limsup}\frac{\mid B^H(t_0+\langle {\epsilon }_j^{(n)}\rangle )-B^H(t_0)\mid }{\rho (0,\langle {\epsilon }_j^{(n)}\rangle )\sqrt{loglog(1+{\prod }_{j=1}^N{\mid {\epsilon }_j^{(n)}\mid }^{-H_j})}}\ge c_{6,1}\text{a}.\text{s}.$$

This proves the lower bounds in (6.3) and (6.4). Finally, by using Lemma 7.1.1 of Marcus and Rosen (2006) [one may also apply the wavelet expansion for B^H in Ayache and Xiao (2005) and Kolmorogov’s 0–1 law], we derive from the above that (6.3) and (6.4) hold.

6.2. Applications to the solutions of stochastic partial differential equations

Gaussian random fields arise naturally as solutions to stochastic partial differential equations. In this subsection, as applications of our results, we establish the global and local moduli of continuity of the solutions of the stochastic heat equation.

Funaki’s model for random string in ℝ is specified by the following stochastic heat equation:

$$\frac{\partial u(t,x)}{\partial t}=\frac{{\partial }^{2}u(t,x)}{\partial x^2}+\stackrel{.}{W},$$ (6.9)

where Ẇ (t, x) is an ℝ-valued space-time white noise, which is assumed to be adapted with respect to a filtered probability space (Ω, , , ℙ), where is complete and the filtration { , t ≥ 0} is right continuous; see Funaki (1983) and Mueller and Tribe (2002) for more information. Recall from Mueller and Tribe (2002) that a solution of (6.9) is defined as an -adapted, continuous random field {u(t, x), t ∈ ℝ₊, x ∈ ℝ} with values in ℝ satisfying the following properties:

1. u(0, ·) ∈ almost surely and is adapted to , where = ∪_λ>0 and

   $${\mathcal{E}}_{\lambda }=\left\{f\in C(\mathbb{R}):\mid f(x)\mid e^{-\lambda \mid x\mid }\to 0\text{as}\mid x\mid \to \infty \right\};$$

2. For every t > 0, there exists λ > 0 such that u(s, ·) ∈ for all s ≤ t, almost surely;

3. For every t > 0 and x ∈ ℝ, the following Green’s function representation holds

   $$u(t,x)={\int }_{\mathbb{R}}G(t,x-y)u(0,y)dy+{\int }_0^{t}G(t-r,x-y)W(dydr),$$ (6.10)

   where $G(t,x)=\frac{1}{\sqrt{4\pi t}}{e}^{-\frac{x^2}{4t}}$ is the fundamental solution of the heat equation.

We call each solution {u(t, x), t ∈ ℝ₊, x ∈ ℝ} of (6.9) a random string process with values in ℝ, or simply a random string as in Mueller and Tribe (2002). Note that, in general, a random string may not be Gaussian, a powerful step in the proofs of Mueller and Tribe (2002) is to reduce the problems about a general random string process to those of the stationary pinned string U₀ = {U₀(t, x), t ∈ ℝ₊, x ∈ ℝ}, obtained by taking the initial function u(0, ·) in (6.10) to be defined by

$$u(0,x)={\int }_0^{\infty }{\int }_{\mathbb{R}}(G(r,x-z)-G(r,z))\stackrel{\sim }{W}(dzdr),$$

where W̃ is a space-time white noise independent of the white noise Ẇ. Consequently, the stationary pinned string is a continuous version of the following Gaussian field

$$U_0(t,x)={\int }_0^{\infty }{\int }_{\mathbb{R}}\left(G(t+r,x-z)-G(t+r,z)\right)\stackrel{\sim }{W}(dzdr)+{\int }_0^t{\int }_{\mathbb{R}}G(r,x-z)W(dzdr),$$

Mueller and Tribe (2002) proved that the Gaussian field U₀ = {U₀(t, x), t ∈ ℝ₊, x ∈ ℝ} has stationary increments and satisfies the Condition (A1) [with H₁ = 1/2 and H₂ = 1]. Let U₁, …,U_d be independent copies of U₀, and consider the Gaussian random field U = {U(t, x), t ∈ ℝ₊, x ∈ ℝ} with values in ℝ^d defined by U(t, x) = (U₁(t, x), …,U_d(t, x)). Mueller and Tribe (2002) found necessary and sufficient conditions [in terms of the dimension d] for U to hit points or to have double points of various types. They have also studied the question of recurrence and transience for {U(t, x), t ∈ ℝ₊, x ∈ ℝ}. Wu and Xiao (2007) studied the fractal properties of various random sets generated by the random string processes. Further results on hitting probabilities of random string process can be found in Dalang et al. (2006). In this subsection we establish the following law of the iterated logarithm for the Gaussian random field U₀.

Theorem 6.4

Let t₀ ∈ [a, 1]² ⊂ ℝ^N with 0 ≤ a ≤ 1. Then

$$\underset{\left|\right|\epsilon \left|\right|\to 0+}{lim}\underset{(\mid t\mid ,\mid x\mid )\le \langle {\epsilon }_j\rangle }{sup}\frac{\mid U_0({\mathbf{t}}_0+(t,x))-U_0({\mathbf{t}}_0)\mid }{\gamma ({\mathbf{t}}_0,{\mathbf{t}}_0+(t,x))}={\kappa }_{7}a.s.,$$

where γ(s, t) is defined as in Theorem 5.1, and

$$\underset{\epsilon \to 0+}{lim}\underset{s:\sigma (s)\le \epsilon }{sup}\frac{\mid U_0({\mathbf{t}}_0+s)-U_0({\mathbf{t}}_0)\mid }{\sigma (s)\sqrt{loglog(1+\sigma {(s)}^{-1})}}={\kappa }_{8}a.s.$$

Here, κ₇ and κ₈ are positive and finite constants.

On the other hand, Pitt and Robeva (2004, Proposition 10) showed that the Gaussian random field

$$u_0(t,x)=\frac{1}{2\pi }{\int }_{{\mathbb{R}}^2}\frac{e^{i({\xi }_{1}t+{\xi }_{2}x)}-1}{i{\xi }_1+{\xi }_2^2}\stackrel{\sim }{\mathcal{M}}(d{\xi }_1,d{\xi }_2),\forall t\in {\mathbb{R}}_{+},x\in \mathbb{R}$$

is another solution to (6.9) satisfying u₀(0, 0) = 0. Here is a complex Gaussian white noise in ℝ². his Gaussian random field has stationary increments with spectral density

$$f(\xi )=\frac{1}{{\xi }_1^2+{\xi }_2^4}.$$

Hence, one can verify that the Gaussian random field {u₀(t, x), t ∈ ℝ₊, x ∈ ℝ} satisfies the Conditions (A1) [with H₁ = 1/4 and H₂ = 1/2] and (A3). Therefore, as applications of Theorems 4.1 and 5.1, we establish the following global and local moduli of continuity for the Gaussian field {u₀(t, x), t ∈ ℝ₊, x ∈ ℝ}.

Theorem 6.5

We have

$$\underset{\epsilon \to 0+}{lim}\underset{\begin{array}{c}(t_1,x_1),(t_2,x_2)\in I\\ \sigma ((t_1,x_1),(t_2,x_2))\le \epsilon \end{array}}{sup}\frac{\mid u_0(t_1,x_1)-u_0(t_2,x_2)\mid }{\beta ((t_1,x_1),(t_2,x_2))}={\kappa }_{9}a.s.,$$

where β(·, ·) is defined as in (4.1) and κ₉ is a positive constant satisfying (4.3) [with H₁ = 1/4 and H₂ = 1/2].

Theorem 6.6

Let t₀ ∈ [a, 1]² ⊂ ℝ^N with 0 ≤ a ≤ 1. Then, there exist positive and finite constants κ₁₀ and κ₁₁ such that

$$\underset{\left|\right|\epsilon \left|\right|\to 0+}{lim}\underset{(\mid t\mid ,\mid x\mid )\le \langle {\epsilon }_j\rangle }{sup}\frac{\mid u_0({\mathbf{t}}_0+(t,x))-u_0({\mathbf{t}}_0)\mid }{\gamma ({\mathbf{t}}_0,{\mathbf{t}}_0+(t,x))}={\kappa }_{10}a.s.,$$

where γ(s, t) is defined as in Theorem 5.1, and

$$\underset{\epsilon \to 0+}{lim}\underset{s:\sigma (s)\le \epsilon }{sup}\frac{\mid u_0({\mathbf{t}}_0+s)-u_0({\mathbf{t}}_0)\mid }{\sigma (s)\sqrt{loglog(1+\sigma {(s)}^{-1})}}={\kappa }_{11}a.s\mathrm{..}$$