Sample path properties of generalized random sheets with operator scaling

We consider operator scaling $\alpha$-stable random sheets, which were introduced in [12]. The idea behind such fields is to combine the properties of operator scaling $\alpha$-stable random fields introduced in [6] and fractional Brownian sheets introduced in [14]. We establish a general uniform modulus of continuity of such fields in terms of the polar coordinates introduced [6]. Based on this, we determine the box-counting dimension and the Hausdorff dimension of the graph of a trajectory of such fields over a non-degenerate cube $I \subset Rd$.


Introduction
In this paper, we consider a harmonizable operator scaling α-stable random sheet as introduced in [11]. The main idea is to combine the properties of operator scaling αstable random fields and fractional Brownian sheets in order to obtain a more general class of random fields. Let us recall that a scalar valued random field {X (x) : x ∈ R d } is said to be operator scaling for some matrix E ∈ R d×d and some H > 0 if where f.d.
= means equality of all finite-dimensional marginal distributions and, as usual, (log c) k k! E k is the matrix exponential. These fields can be regarded as an anisotropic generalization of self-similar random fields (see, e.g., [8]), whereas the fractional Brownian sheet {B H 1 ,...,H d (x) : x ∈ R d } with Hurst indices 0 < H 1 , . . . , H d < 1 can be seen as an anisotropic generalization of the well-known fractional Brownian field (see, e.g., [13]) and satisfies the scaling property for all constants c 1 , . . . , c d > 0. See [3,10,27] and the references therein for more information on the fractional Brownian sheet. Throughout this paper, let d = m j=1 d j for some m ∈ N andẼ j ∈ R d j ×d j , j = 1, . . . , m be matrices with positive real parts of their eigenvalues. We define matrices E 1 , . . . , E m ∈ R d×d as Further, we define the block diagonal matrix E ∈ R d×d as In analogy to the terminology in [ for all c > 0 and j = 1, . . . , m. Note that, by applying (1.2) iteratively, any operator scaling stable random sheet is also operator scaling for the matrix E and the exponent H = m j=1 H j in the sense of (1.1). Further, note that this definition is indeed a generalization of operator scaling random fields, since for m = 1, d = d 1 and E = E 1 =Ẽ 1 (1.2) coincides with the definition introduced in [4]. Another example of a random field satisfying (1.2) is given by the fractional Brownian sheet, where E j = d j = 1 for j = 1, . . . , m in this case. Operator scaling stable random sheets have been proven to be quite flexible in modeling physical phenomena and can be applied in order to extend the well-known Cahn-Hilliard phase-field model. See [1] and the references therein for more information.
Random fields satisfying a scaling property such as (1.1) or (1.2) are very popular in modeling, see [14,22] and the references in [5] for some applications. Most of these fields are Gaussian. However, Gaussian fields are not always flexible for example in modeling heavy tail phenomena. For this purpose, α-stable random fields have been introduced. See [17] for a good introduction to α-stable random fields.
Using a moving average and a harmonizable representation, the authors in [4] defined and analyzed two different classes of symmetric α-stable random fields satisfying (1.1). Following the outline in [4,5], these two classes were generalized to random fields satisfying (1.2) in [11]. The fields constructed in [4] have stationary increments, i.e., they satisfy This property has been proven to be quite useful in studying the sample path properties. However, the property of stationary increments is no more true for the fields constructed in [11]. The absence of this property is one of the challenging difficulties we face in determining results about their sample paths.
Another main tool in studying sample paths of operator scaling stable random sheets are polar coordinates with respect to the matrices E j , j = 1, . . . , m, introduced in [16] and used in [4,5]. If {X (x) : x ∈ R d } is an operator scaling symmetric α-stable random sheet with α = 2, using (1.2), one can write the variance of X (x), x ∈ R d , as where H = m j=1 H j and τ E (x) is the radial part of x with respect to E and l E (x) is its polar part. Therefore, if the random field has stationary increments in the Gaussian case information about the behavior of the polar coordinates τẼ j (x), lẼ j (x) contains information about the sample path regularity. This property also holds in the stable case α ∈ (0, 2). Moreover, this also remains to be true for operator scaling random sheets which do not have stationary increments but satisfy a slightly weaker property, see Corollary 3.3 below.
This paper is organized as follows. In Sect. 2, we introduce the main tools we need for the study in this paper. Section 2.1 is devoted to a spectral decomposition result from [16]. Section 2.2 is about the change to polar coordinates with respect to scaling matrices and we establish a relation between the radius τ E (x) and the radii τẼ j (x), 1 ≤ j ≤ m, in Lemma 2.2 below. In Sect. 3, we present the results in [11] about the existence of harmonizable and moving average representations of operator scaling α-stable random sheets. Here, we will only focus on a harmonizable representation. Moreover, we prove that these random sheets fulfill a generalized type of modulus of continuity, which is deduced by showing the applicability of results in [5,6]. Based on this and generalizing a combination of methods used in [2,4,5,24], in Sect. 4 we present our results on the Hausdorff dimension and box-counting dimension of the graph of harmonizable operator scaling stable random sheets.

Spectral Decomposition
Let A ∈ R d×d be a matrix with p distinct positive real parts of its eigenvalues 0 < a 1 < · · · < a p for some p ≤ d. Factor the minimal polynomial of A into f 1 , . . . , f p , where all roots of f i have real part equal to a i , and define V i = Ker f i (A) . Then, by [16, Theorem 2.1.14], is a direct sum decomposition, i.e. we can write any x ∈ R d uniquely as Further, we can choose an inner product on R d such that the subspaces V 1 , . . . , V p are mutually orthogonal. Throughout this paper, for any x ∈ R d we will choose x = x, x 1/2 as the corresponding Euclidean norm. In view of our methods this will entail no loss of generality, since all norms are equivalent.

Polar Coordinates
We now recall the results about the change to polar coordinates used in [4,5]. As before, let A ∈ R d×d be a matrix with distinct positive real parts of its eigenvalues 0 < a 1 < · · · < a p for some p ≤ d. According to [4,Sect. 2] there exists a norm · A on R d such that for the unit sphere To be more precise, the norm · A is defined by Thus, we can write any x ∈ R d \ {0} uniquely as We remark that the bounds on the growth rate of τ A (·) have been improved in [5,Proposition 3.3], but the bounds given in Lemma 2.1 suffice for our purposes.
The following Lemma will be needed in the next section in order to give an upper bound on the modulus of continuity.
be a subspace and note that Without loss of generality assume i = 1 and for simplicity in this proof let us assume that m = 2. Thus, for any vector But on the other hand one can write yielding that Further noting that and taking into account the definition of the norm · Ẽ 1 given in (2.1) we obtain Thus, by the uniqueness of the representation we have τẼ This concludes the proof. Corollary 2.3 Let E,Ẽ 1 , . . . ,Ẽ m be as above. Then, there exists a constant C ≥ 1 such that

Harmonizable Operator Scaling Random Sheets
We consider harmonizable operator scaling stable random sheets defined in [11] and present some related results established in [11]. Most of these will also follow from the results derived in [4,5]. Throughout this paper, for j = 1, . . . , m assume that the real parts of the eigenvalues ofẼ j are given by 0 < a Moreover, we assume that ψ j (x) = 0 for x = 0. See [4,5] for various examples of such functions.

exists and is stochastically continuous if and only if H
Proof This result has been proven in detail in [11], but it also follows as an easy consequence of [4,Theorem 4.1]. By the definition of stable integrals (see [17]), Note that from (3.1) it follows that X α (x) = 0 for all x = (x 1 , . . . , x m ) ∈ R d 1 × · · · × R d m = R d such that x j = 0 for at least one j ∈ {1, . . . , m}.
The following result has been established in [11,Corollary 4.2.1]. The proof is carried out as the proof of [4, Corollary 4.2 (a)] via characteristic functions of stable integrals and by noting that

Corollary 3.2 Under the conditions of Theorem
As we shall see below, fractional Brownian sheets fall into the class of random fields given by (3.1). It is known that a fractional Brownian sheet does not have stationary increments. Thus, in general, a random field given by (3.1) does not possess stationary increments. But it satisfies a slightly weaker property, as the following statement shows.

Corollary 3.3 Under the conditions of Theorem 3.1, for any h
where we used the notation x = (x 1 , . . . , x m Proof This result has been established in [11,Corollary 4.2.2] and is proven similarly to [4, Corollary 4.2 (b)].
As an easy consequence of the results in this paper, we will derive global Hölder critical exponents of the random fields defined in (3.1). Following [7,Definition 5], β ∈ (0, 1) is said to be the Hölder critical exponent of the random field {X (x) : x ∈ R d }, if there exists a modification X * of X such that for any s ∈ (0, β) the sample paths of X * satisfy almost surely a uniform Hölder condition of order s on any compact set I ⊂ R d , i.e., there exists a positive and finite random variable Z such that almost surely for some positive and finite constants c j 1 , c j 2 as noted in [6, Remark 5.1], without loss of generality we will assume H j = 1 < a 1 j in the proof of the following statement. We will make this assumption for notational convenience.
In particular, for any 0 < γ < H j and x = (x 1 , . . . , x m ), y = (y 1 , . . . , y m ) ∈ G d one can find a positive and finite constant C such that holds almost surely.
Proof Let us first assume that α = 2. In the following let · p denote the p-norm for p ≥ 1, c an unspecified positive constant, G d ⊂ R d an arbitrary compact set, r > 0 and we denote the canonical metric associated to X 2 . We first show for x, y ∈ G d that By the equivalence of norms one can find a constant c such that for any u ∈ R m . Further let us remark that by definition the variance of the centered Gaussian random variable X 2 (x) in (3.1) is given by Note that for all 1 ≤ j ≤ m and x = ( where θ ∈ R d j with τẼ j (θ ) = 1. Using all this and the elementary inequality . . . , x i−1 , y i , y i+1 , . . . , y m )| with the convention that for i = 1 and for i = m we get for all x = (x 1 , . . . , x m ) and y = (y 1 , . . . , y m where we used Corollary 3.3 in the equality and the equivalence of norms in the last inequality. Using the operator scaling property and the generalized polar coordinates for x i − y i we can further get an upper estimate of the last expression by where we used Corollary 2.3 with H = 2 in the last inequality, which proves (3.5). Now define an auxiliary Gaussian random field Then, using (3.5) it is easy to see that D ≤ cr for some positive constant c. Using the latter inequality, by the arguments made in the proof of [15,Theorem 4.2] if N (ε) denotes the smallest number of open d Y -balls of radius ε > 0 needed to cover G d × B E (r ) we obtain that Therefore, by a standard Borel-Cantelli argument we conclude for a continuous modification X * 2 of X 2 , which by Lemma 2.2 is equivalent to Let us now assume that α ∈ (0, 2). In this case, the proof is a slight modification and extension of the proof of [6, Proposition 5.1] and the idea is to check the assumptions (i), (ii) and (iii) of Proposition 4.3 of the latter reference. Throughout this proof, we let c be a universal unspecified positive and finite constant and in the following let As in [6, Example 5.1] one checks that for all ξ ∈ R d , ξ j = 0, and, in particular there exist constants A j > 0, 1 ≤ j ≤ m, such that (3.7) holds for all ξ j > A j . For ζ > 0 chosen arbitrarily small we consider the functionμ on R d given byμ(ξ ) = m j=1μ j (ξ j ) with We observe thatμ is positive on R d \ {0} and, similarly to the calculations made in the proof of [6, Proposition 5.1], we obtain that Define μ j =μ j c . Moreover, note that Hence, μ =μ c is well defined and now, as in the proof of [6, Proposition 5.1], we are going to check the assumptions (i), (ii) and (iii) of [6,Proposition 4.3] for where u ∈ R d and is assumed to be a random vector on R d with density μ.
Note that this is possible. Then, it follows for the quasi-metric ρ on R d defined by Hence, we have so that we precisely recover assumption (i) in [6,Proposition 4.3] for the random field G = g(h, ξ) h∈[0,∞) . Moreover, assumption (ii) immediately follows as in the proof of [6, Proposition 5.1] from the definition of the norms · Ẽ T j and by noting that the product of monotonic functions again is monotonic. It remains to prove assumption (iii) in [6,Proposition 4.3]. To this end we write Using equality (3.7) similarly as shown in the calculations made in the proof of [6, Proposition 5.1] we obtain that which yields that so that assumption (iii) in [6,Proposition 4.3] with pm instead of p is fulfilled. Following the lines of the proof of [6, Proposition 5.1], we obtain that there exists a modification X * α of X α such that for any ε > 0 and any non-empty compact set G d ⊂ R d , which by Lemma 2.2 is equivalent to This completes the proof. We remark that Corollary 3.5 is not a statement about critical Hölder exponents. However, as a consequence of Theorem 4.1 below, we will see that any continuous version of X α admits min 1≤ j≤m H j a j p j as the critical exponent.

Hausdorff Dimension
We now state our main result on the Hausdorff and box-counting dimension of the graph of X α defined in (3.1). In the following, for a set B ⊂ R d we denote by dim B B, dim B B and dim H B its lower, upper box-counting and Hausdorff dimension, respectively. We refer to [9] for a definition of these objects.
Using the characteristic function of the symmetric α-stable random field X α , as in the proof of [5,Proposition 5.7], it can be shown that there is a constant C 1 > 0 such that Combining this with Theorem 4.2 below we get for someC 1 > 0 and Q 1 = [ 1 2 , 1] d 1 . With this inequality the assertion readily follows from the proof of [4,Theorem 5.6].
The following Theorem is crucial for proving Theorem 4.1 and its proof is based on [23,Theorem 1]. See also [24][25][26]. Let us remark that a similar method of the following proof has been applied in [24,Theorem 3.4] for certain α-stable random fields if 1 ≤ α ≤ 2. In the following, we are able to extend this method for 0 < α < 1 and, in particular this shows that the statement of [24, Theorem 3.5] can be formulated for 0 < α < 1 as well.