Hölder regularity for operator scaling stable random fields

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Hölder regularity for operator scaling stable random fields

Hermine Biermé, Céline Lacaux

To cite this version:

Hermine Biermé, Céline Lacaux. Hölder regularity for operator scaling stable random fields. Stochastic Processes and their Applications, Elsevier, 2009, 119 (7), pp.2222-2248. �hal-00128730v2�

H ¨OLDER REGULARITY FOR OPERATOR SCALING STABLE RANDOM FIELDS

HERMINE BIERM ´E AND C´ELINE LACAUX

Abstract. We investigate the sample paths regularity of operator scaling α-stable random fields.

Such fields were introduced in [6] as anisotropic generalizations of self-similar fields and satisfy the scaling property{X(c^Ex);x∈R^d}^{(f dd)}= {c^HX(x);x∈R^d}whereEis ad×dreal matrix andH >0.

In the case of harmonizable operator scaling random fields, the sample paths are locally H¨olderian and their H¨older regularity is characterized by the eigen decomposition ofR^dwith respect toE. In particular, the directional H¨older regularity may vary and is given by the eigenvalues ofE. In the case of moving average operator scaling randomα-stable random fields, with α∈(0,2) andd≥2, the sample paths are almost surely discontinous.

1. Introduction

In this paper we consider operator scaling stable random fields as introduced in [6]. More precisely, for E a real d×d matrix with positive real parts of the eigenvalues, a scalar valued random field (X(x))_x∈Rd is called operator scaling forE and H >0 if for every c >0

(1) {X(c^Ex);x∈R^d}^{(f dd)}= {c^HX(x);x∈R^d},

where ^{(f dd)}= means equality of finite dimensional distributions and as usual c^E = exp(Elogc) with exp(A) =P∞

k=0A^k

k! is the matrix exponential. Let us remark that up to consider the matrix E/H, we may assume, without loss of generality, that H= 1.

These fields can be seen as anisotropic generalizations of self-similar random fields. Let us recall that a scalar valued random field (X(x))_x∈Rd is said to beH-self-similar withH ∈Rif

{X(cx);x∈R^d}^{(f dd)}= {c^HX(x);x∈R^d}

for everyc >0. Then aH-self-similar field is also an operator scaling field for the matrixE =I_d/H, whereI_d is the identity matrix of sized×d. Self-similar random fields are used in various applications such as internet traffic modelling [29], ground water modelling and mathematical finance, just to mention a few. Various examples can be found for instance in the books [18, 1, 27]. A very important class of such fields are given by Gaussian random fields and especially by the fractional Brownian fieldB_H, where H ∈(0,1) is the so-called Hurst parameter. The random field B_H is H-self-similar and has stationary increments, e.g. {B_H(x+h)−B_H(h);x ∈ R^d} ^{(f dd)}= {B_H(x);x ∈ R^d} for any h ∈ R^d. It is an isotropic generalization of the famous fractional Brownian motion, implicitely introduced in [12] and defined in [19].

Nevertheless, Gaussian random fields are not convenient for some heavy tails phenomena mod- elling. For this purpose,α-stable random fields have been introduced. A scalar valued random field {X(x);x ∈ R^d} is said to be symmetric α-stable (SαS), for α ∈ (0,2), if any linear combination

n

P

k=1

a_kX(x_k) is SαS. We address to the book [27] for a well understanding of such fields. The self- similar α-stable fields with stationary increments have been extensively used to propose alternative to Gaussian modelling (see [23, 29] for instance) and are isotropic.

However, certain applications (see, e.g., [5, 7] and references therein) require that the random field is anisotropic and satisfies a scaling relation. This scaling relation should have different Hurst

2000 Mathematics Subject Classification. Primary: 60617, 60G18; Secondary: 60G60, 60G52, 60G15.

Keywords: Operator scaling random fields, Stable and Gaussian laws, H¨older regularity, Hausdorff dimension.

Date: February 5, 2007.

1

indices in different directions and these directions should not necessarily be orthogonal. To reach this goal the authors of [6] propose a new class of random fields with an anisotropic behavior driven by a d×d matrix E. More precisely, they introduceα-stable random fields which have stationary increments and satisfy the operator scaling property (1). Two different classes of such fields are defined and analyzed, using a moving average representation as well as an harmonizable one. In the Gaussian case α = 2, according to [6] there exist modifications of these fields which are almost surely H¨older-continuous of certain indices. We give similar results here in the stable caseα∈(0,2) for harmonizable operator scaling stable random fields. Actually, we obtain their critical global and directional H¨older exponents, which are given by the eigenvalues of E. In general, such fields are anisotropic and their sample paths properties varies with the direction. In particular, in the case whereEis diagonalizable, for any eigenvectorθ_j associated with the real eigenvalueλ_j, harmonizable operator scaling stable random field admits Hj = 1/λj as critical H¨older exponent in direction θj. Let us point out that we establish an accurrate upper bound for the modulus of continuity. Such upper bound has already been given in the case of real harmonizable fractional stable motions, which are self-similar, in [14] and in the case of some Gaussian random processes in [13]. Then, in this paper, we generalize these results to any dimensiond and any harmonizable operator scaling stable fields. From this upper bound, we can deduce the H¨older sample paths regularity. Let us point out that we also obtain this upper bound in the case of Gaussian operator scaling random field, which improves the sample paths properties establishes in [6].

Furthermore, whereas in the Gaussian caseα = 2, moving average and harmonizable fields have the same kind of regularity properties, this is no more true in the case α ∈ (0,2). In particular, we show that for d ≥ 2, a moving average operator scaling stable random field does not admit any continuous modification. Remark that if d = 1, the sample paths regularity properties are already known since the processes studied are self-similar moving average stable processes, see for example [27, 14, 28].

One of the main tool for the study of sample paths of operator scaling random fields is the change of polar coordinates with respect to the matrixEintroduced in [22] and used in [6]. ForX a Gaussian operator scaling random field with stationary increments, using (1), we can write the variogramme ofX as

v²(x) =E X²(x)

=τ_E(x)^2HE X²(`_E(x)) ,

where τ_E(x) is the radial part of x with respect to E and `<sub>E</sub>(x) is its polar part. Therefore, in the Gaussian case, the sample paths regularity depends on the behavior of the polar coordinates (τ<sub>E</sub>(x), `_E(x)) around x = 0. Such property also holds in the stable case α ∈ (0,2). The H¨older regularity of the sample paths follows from estimates ofτ_E(x) compared tokxk. These estimates are given in Section 3 and their proof are postponed to the Appendix.

Furthermore, to study the sample paths in the stable case, the other main tool we use is a series representation of harmonizable operator scaling α-stable random fields. Such representations in series of infinitely divisible laws have been studied in [17, 16, 25, 24]. As in [14], our study is based on a LePage series representation. Actually, the main idea is to choose a representation which is a conditionnally Gaussian series.

In Section 2, we recall the definition of harmonizable operator scaling random fields. Then, Sec- tions 3 and 4 are devoted to the main tools we need for the study of the sample paths of these fields. More precisely, Section 3 deals with the polar coordinates with respect to a matrix E and Section 4 gives the LePage series representation. In Section 5, the sample paths properties and the Hausdorff dimension of the graph are given. Section 6 is concerned with moving average operator scaling random fields.

2. Harmonizable representation

Let us recall the definition of harmonizable operator scaling stable random fields, given by [6].

LetE be a real d×d matrix. Letλ₁, . . . , λ_d be the complex eigenvalues of E and a_j =<(λ_j) for each j= 1, . . . , d. We assume that

(2) min

1≤j≤da_j >1.

Letψ:R^d →[0,∞) be a continuous, E^t-homogeneous function, which means according to Defini- tion 2.6 of [6] that

ψ(c^Etx) =cψ(x) for all c >0.

We assume moreover thatψ(x)6= 0 forx6= 0. Such functions were studied in detail in [22], Chapter 5 and various examples are given in Theorem 2.11 and Corollary 2.12 of [6].

Let 0< α≤2 andW_α(dξ) be a complex isotropic α-stable random measure onR^d with Lebesgue control measure (see [27] p.281). Ifα= 2,W_α(dξ) is a complex isotropic Gaussian random measure.

Letq= trace (E).

Definition 2.1. Since (2) is fulfilled, the random field

(3) X_α(x) =<

Z

R^d

e^ihx,ξi−1

ψ(ξ)^−1−q/αW_α(dξ) , x∈R^d, is well defined and called harmonizable operator scaling stable random field.

From Theorem 4.1 and Corollary 4.2 of [6],X_α is stochastically continuous, has stationary incre- ments and satisfies the following operator scaling property

(4) ∀c >0, n

X_α(c^Ex);x∈R^d o_{(f dd)}

= n

cX_α(x);x∈R^d o

. For notational sake of simplicity we denote the kernel function by

(5) f(x, ξ) =

e^ihx,ξi−1

ψ(ξ)^−1−q/α.

Let us recall that f(x,·) ∈ L^α(R^d) for any x ∈ R^d, which is a necessary and sufficient condition forX_α to be defined.

Now, let us give some examples of such random fields.

Example 2.1. LetI_d be the identity matrix of size d×d, E =I_d/H (with 0< H <1) andψ(x) = kxk^H withk·kthe Euclidean norm. Then the random field defined by (3) is a real harmonizable stable random field (see [27] for details on such fields). In this case,X_α is self-similar with exponentH, i.e.

∀c >0, {X_α(cx);x∈R^d}^{(f dd)}= {c^HX_α(x);x∈R^d}.

Let us quote that, ifα= 2,X_αis a fractional Brownian field and its critical H¨older exponent is given by its Hurst indexH (see Theorem 8.3.2 of [2] for instance).

Example 2.2. Assume that E is diagonalizable. Then, all the eigenvalues of E are real given by (a_j)_1≤j≤d. Let (θ_j)_1≤j≤d be a basis of some corresponding eigenvectors and consider the function ψdefined by

ψ(x) =





d

X

j=1

|hx, θ_ji|^2/aj





1/2

, x∈R^d.

The functionψis clearly continuous and non negative onR^d. Moreover, sincehc^Etx, θ_ji=hx, c^Eθ_ji= c^ajhx, θ_ji, it is alsoE^t-homogeneous. Finally, since (θ_j)_1≤j≤d is a basis of R^d we have that ψ(x) = 0 if and only ifx= 0. Then we can defineX_α by (3) and in this case the operator scaling property (4) implies that

∀j= 1, . . . , d, ∀c >0, {X_α(ctθ_j);t∈R}^{(f dd)}= n

c^1/ajX_α(tθ_j);t∈R o

.

The random fieldX_α is self-similar with Hurst indexH_j = 1/a_j in the directionθ_j. In particular, in the Gaussian case (α= 2), (X₂(tθ_j))_t∈R is a fractional Brownian motion with Hurst indexH_j. Then, in this case, its critical H¨older exponent is equal toH_j.

The main tool in the study of operator scaling random fields is the change of coordinates in a kind of polar coordinates with respect to the matrix E. Then, before we study the sample paths of X_α, we recall in the next section the definition of these coordinates and give some estimates of the radial part.

3. Polar coordinates

According to Chapter 6 of [22], since E is a real d×d matrix with positive real parts of the eigenvalues there exists a normk · kE onR^d such that for the unit sphereS_E ={x∈R^d: kxkE = 1} the map

ΨE : (0,∞)×SE −→ R^d\ {0} (r, θ) 7−→ r^Eθ

is a homeomorphism. Hence we can write any x ∈ R^d\{0} uniquely as x = τE(x)^E`E(x) for some radial part τ<sub>E</sub>(x) > 0 and some direction `_E(x) ∈ S_E such that x 7→ τ_E(x) and x 7→ `_E(x) are continuous. Observe thatS_E ={x∈R^d :τ_E(x) = 1} is compact and set

(6) m_E = min

SE kxk and M_E = max

SE kxk.

We know that τ_E(x) → ∞ as x → ∞ and τ_E(x) → 0 as x → 0. Hence we can extend τ_E(·) continuously by setting τE(0) = 0.

Let us recall the formula of integration inpolar coordinates established in [6].

Proposition 3.1. There exists a unique finite Radon measure σ_E on S_E such that for all f ∈ L¹(R^d, dx) we have

Z

R^d

f(x)dx= Z _∞

0

Z

SE

f(r^Eθ)σ_E(dθ)r^q−1dr.

In the Gaussian case (α= 2), the variogramme of X₂ can be rewritten as follows v²(x) =τ_E(x)²E

X₂(`_E(x))² ,

using the operator scaling property (4). Then, the H¨older regularity of X₂ follows from estimates of τ_E(x) compared to kxk around x = 0, e.g. the H¨older regularity of τ_E around 0, see [6]. Then, in order to get an upper bound for the modulus of continuity (for anyα), we need precise estimates ofτ_E(x).

As done in [21] for the study of operator-self-similar Gaussian random fields we use the Jordan decomposition of the matrix E to get estimates of τE. From the Jordan decomposition’s theorem (see [10] p. 129 for instance), there exists a real invertible d×d matrixP such that D =P⁻¹EP is of the real canonical form, which means that D is composed of diagonal blocks which are either Jordan cell matrix of the form











λ 1 . . . 0 0 λ . .. ...

... . .. ... 1 0 . . . 0 λ











withλa real eigenvalue ofE or blocks of the form

(7)

















Λ I2 0 . . . 0 0 Λ I₂ . .. ...

... . .. ... ... 0 ... . .. ... ... I₂ 0 . . . 0 Λ

















with Λ =

a −b b a

and I₂ =

1 0 0 1

,

where the complex numbersa±ib(b6= 0) are complex conjugated eigenvalues ofE.

Let us denote byk · k the subordinated norm of the Euclidean norm on the matrix space. Precise estimates ofτ_E follow from the next lemma.

Lemma 3.2. Let J be either a Jordan cell matrix of size l or a block of the form (7) of size 2l associated with the eigenvalue λ=a+ib. Then, for any t∈(0, e⁻¹]∪[e,+∞)

t^a≤ kt^Jk ≤√

2l e t^a|logt|^l−1.

Proof. see the Appendix.

Let us be more precise on the Jordan decomposition of E. Let us recall that the eigenvalues of E are denoted by λ_j, j = 1. . . d and that a_j =<(λ_j) >1 for j = 1, . . . , d. There exist J₁, . . . , J_m, where each J_j is either a Jordan cell matrix or a block of the form (7), andP a reald×d invertible matrix such that

E =P











J₁ 0 . . . 0 0 J₂ 0 ... ... . .. ... 0 0 . . . 0 J_m









 P⁻¹.

We can assume that eachJ_j is associated with the eigenvalueλ_j ofE and that 1< a1≤ · · · ≤am.

We also setHj =a⁻¹_j and have

(8) 0< Hm≤ · · · ≤H1 <1.

Ifλ_j ∈R, J_j is a Jordan cell matrix of size ˜l_j =l_j ∈N\{0}. Ifλ_j ∈C\R,J_j is a block of the form (7) of size ˜lj = 2lj ∈2N\{0}. Then for anyt >0,

t^E =P













t^J1 0 . . . 0 0 t^J2 0 ... ... . .. ... 0 0 . . . 0 t^Jm











 P⁻¹

We denote by (e₁, . . . , e_d) the canonical basis of R^d and set f_j = P e_j for every j = 1, . . . , d.

Hence, (f₁, . . . , f_d) is a basis of R^d. For all j= 1, . . . , m, let

(9) W_j = Vect f_k ;

j−1

X

i=1

l˜_i+ 1≤k ≤

j

X

i=1

l˜_j

! . Then, each W_j is aE-invariant set and E=Lm

j=1W_j.

The following result gives bounds on the growth rate of τ_E(x) in terms of the real parts of the eigenvalues ofE.

Proposition 3.3. For any r∈(0,1) there exist c1, c2>0 such that for every 1≤j0 ≤j ≤m, c₁kxk^Hj0 |logkxk|^{−(pj0,j−1)Hj0} ≤τ_E(x)≤c₂kxk^Hj|logkxk|^{(pj0,j−1)Hj}

holds for any x∈ ⊕^jk=j0W_k with kxk ≤r and pj0,j = max

j0≤k≤jl_k.

Proof. See the Appendix.

Then, we easily deduce the following corollary.

Corollary 3.4. For any r ∈(0,1) there exist c₁, c₂ >0 such that for any x∈W_j with kxk ≤r (10) c₁kxk^Hj |logkxk|^{−(lj−1)Hj} ≤τ_E(x)≤c₂kxk^Hj|logkxk|^(lj−1)Hj

and for any x∈R^d with kxk ≤r

(11) c₁kxk^H1|logkxk|^−(l−1)H1 ≤τ_E(x)≤c₂kxk^Hm|logkxk|^(l−1)Hm

where l= max

1≤j≤ml_j.

Therefore we have precise estimates for the H¨older regularity of the radial part. Let us remark that we improve the first statement of Lemma 2.1 of [6] and that the second one can also be improved in a similar way. From these estimates we deduce the H¨older regularity ofX_α in Section 5. As already mentionned, the study of the sample paths is based on a series representation. Then, before we state regularity properties, it remains to give the LePage series representation of harmonizable operator scaling random fieldsX_α defined by (3).

4. Representation as a LePage series

An overview on series representations of infinitely divisible random variable without Gaussian part can be found for example in [24, 26] and references therein. In particular, LePage series represen- tation ([17, 16]) have been used in [15, 14] to study the sample paths regularity of some self-similar α-stable random motions with α ∈(0,2). Here, this representation is also the main representation we use in the caseα∈(0,2). Actually, in the Gaussian caseα= 2, such representation does not hold.

Let us now introduce some notations that will be used throughout the paper. Letµbe an arbitrary probability measure equivalent to the Lebesgue measure on R^d and let m be its Radon-Nikodym derivative that isµ(dξ) =m(ξ)dξ.

Notation Let (T_n)_n≥1, (g_n)_n≥1 and (ξ_n)_n≥1 be independent sequences of random variables.

• T_n is thenth arrival time of a Poisson process with intensity 1.

• (gn)_n≥1 is a sequence of i.i.d. isotropic complex random variables so that gn

(d)= e^iθgn for any θ∈R. We also assume that 0<E(|gn|^α)<+∞.

• (ξ_n)_n≥1 is a sequence of i.i.d. random variables with common law µ(dξ) =m(ξ)dξ.

According to Chapter 3 and Chapter 4 of [27], stochastic integrals with respect to an α-stable random measure Λ can be represented as a LePage series as soon as the control measure of Λ is a finite measure. The next proposition generalizes this to Wα whose control measure is the Lebesgue measure. It is a consequence of Lemma 4.1 of [15], which is a correction of Lemma 1.4 of [20]. This proposition can also be deduced from [24, 25], concerned with series representations of stochastic integrals with respect to infinitely divisible random measures.

Proposition 4.1. Assume that α∈(0,2). Then, for every complex valued function h∈L^α R^d , the series

(12) Y^h =

+∞

X

n=1

T_n^−1/αm(ξn)^−1/αh(ξn)gn

converges almost surely. Furthermore,

C_αY^{h (d)}= Z

R^d

h(ξ)W_α(dξ) with

(13) C_α=E(|<(g₁)|^α)^−1/α 1

2π Z _π

0 |cos (x)|^αdx

1/αZ +∞

0

sin (x) x^α dx

−1/α

. Remark 4.1. According to Proposition 4.1, taking α∈(0,2), the random measure

Λ_α(dξ) =C_α

+∞

X

n=1

T_n^−1/αm(ξ_n)^−1/αg_nδ_ξn(dξ) is a complex isotropic α-stable random measure.

Proof. Let Vn=m(ξn)^−1/αh(ξn)gn. Then, Vn, n≥1, are i.i.d. isotropic complex random variables.

By Lemma 4.1 in [15], Y^h converges almost surely and

∀z∈C, E

exp i<

¯ zY^h

= exp (−σ^α|z|^α) with

σ^α =E(|<(V1)|^α) Z +∞

0

sin (x) x^α dx.

Sinceg₁ is invariant by rotation and independent with ξ₁, E(|<(V₁)|^α) =E

m(ξ₁)⁻¹|h(ξ₁)|^α

E(|<(g₁)|^α) =E(|<(g₁)|^α) Z

R^d|h(ξ)|^αdξ.

Moreover, by definition of an isotropicα-stable random stable measure (see [27]),

∀z∈C,E

exp

i<

¯ z Z

R^d

h(ξ)M(dξ)

= exp (−c^α_α(h)|z|^α) withc^α_α(h) =

1 2π

Z _π

0 |cos (x)|^αdx

R

R^d|h(ξ)|^αdξ.Therefore, we have CαY^{h (d)}=

Z

R^d

h(ξ)Wα(dξ)

whereC_α is defined by (13), which concludes the proof.

From the previous proposition, we deduce the following statement which is the main series repre- sentation we use in our investigation.

Proposition 4.2. Let α∈(0,2). For everyx∈R^d, the series

(14) Y_α(x) =C_α<

+∞

X

n=1

T_n^−1/αm(ξ_n)^−1/αf(x, ξ_n)g_n

! , where C_α is defined by (13), converges almost surely. Furthermore,

n

Yα(x) :x∈R^do_{(f dd)}

= n

Xα(x) :x∈R^do .

Proof. From Proposition 4.1, for anyx∈R^d, the convergence of the series follows from the fact that f(x,·) ∈L^α R^d

. The equality in distribution between Xα and Yα is obtained by linearity of both

fields.

Using LePage representation (14) of Xα and the estimates given in Section 3, we give an upper bound for the modulus of continuity of X_α and obtain the critical H¨older regularity of its sample paths in the next section.

5. H¨older regularity and Hausdorff dimension

Throughout this section we fixK a compact set ofR^d and investigate the H¨older regularity of the sample paths of X_α on K, with X_α the harmonizable operator scaling stable random field defined by (3).

Let us recall that for the Gaussian caseα= 2, according to Theorem 5.4 of [6], the H¨older regularity of X₂ depends on the subspaces (W_j)_1≤j≤m defined by (9) and associated to the eigenvalues of E.

More precisely, Theorem 5.4 of [6] implies that, when restricted to the subspace W_j, the Gaussian random field {X₂(x);x∈W_j} admits H_j as critical H¨older exponent. This follows from the fact that the regularity of X₂ is determined by the regularity of τ_E around 0, which is given by H_j according to (10). Here, we give an upper bound for the modulus of continuity ofX_α in the general caseα∈(0,2]. Then we prove that the critical H¨older exponents are the same than in the Gaussian caseα= 2. Let us state our main result.

Theorem 5.1. Let α ∈(0,2). There exists a modification X_α^∗ of X_α on K such that

(15) lim

δ↓0 sup

x,y∈K

kx−yk≤δ

|X_α^∗(x)−X_α^∗(y)|

τ_E(x−y)|logτ_E(x−y)|^1/α+1/2+ε = 0 a.s.

for any ε >0.

This result was proved in the case of harmonizable self-similar processes in [14], e.g. in the case of Example 2.1 withd= 1. The main idea is to use the LePage series representation (14) whereg_n, n ≥ 1, are Gaussian complex isotropic random variables. Furthermore, it remains to choose the density distributionm of the ξ_n. In [14], the authors choose

m(ξ) = cη

|ξ||log|ξ||^1+η, ξ∈R\{0}

wherec_η >0. A straightforward generalization in higher dimensiond leads to choose m(ξ) = c_η

kξk^d|logkξk|^1+η, ξ∈R^d\{0}.

Remark that in this case (e.g. Example 2.1) the matrix E = I_d/H = E^t and that we can choose k · kE^t =k · k. Then, using classical polar coordinates, we obtain that forx6= 0,

τ_E^t(x) =kxk^H and `_E^t(x) = x kxk and therefore

m(ξ) = c_η

τ_E^t(ξ)^q|logτ_E^t(ξ)|^1+η

sinceq= trace(E) =d/H. Then we choose this density in the general case. The advantage is thatm only depends on the radial partτ_E^t.

Proof of Theorem 5.1. We can assume without loss of generality that K = [0,1]^d. According to Proposition 4.2 for everyx∈R^d

Y_α(x) =C_α<

+∞

X

n=1

T_n^−1/αm(ξ_n)^−1/αf(x, ξ_n)g_n

!

converges almost surely and Y_α ^{(f dd)}= X_α. As already mentionned, we assume that g_n, n ≥ 1 are Gaussian complex isotropic random variables and that the density distribution ofξnis defined by

m(ξ) = c_η

τ_E^t(ξ)^q|logτ_E^t(ξ)|^1+η, ξ∈R^d\{0}, whereη >0 andc_η is chosen such thatR

R^dm(ξ)dξ= 1. In particular, conditionally to (T_n, ξ_n)_n,Y_α(x) is a real-valued Gaussian random variable.

As in the proof of Kolmogorov-Centsov Theorem (see [11]), we first prove that almost surely forτE(x_k−x_k⁰) small enough,

|Y_α(x_k)−Y_α(x_k⁰)| ≤Cτ_E(x_k−x_k⁰)|logτ_E(x_k−x_k⁰)|^1/α+1/2+ε.

where (x_k)_k is some countable dense sequence of K. Then, X_α satisfies the same property.

Finally, we give a modification X_α^∗ of Xα for which (15) holds. In the first step, we construct the sequence (x_k)_k we use.

Step 1. Fork∈N\{0} let us chooseν_k∈N\{0}the smallest integer such that c₂d^Hm/22^−νkHm|ν_klog 2|^(l−1)Hm ≤2^−k,

wherec2 and l are given by Corollary 3.4 forr = 1/2. Remark that by definition, ν_k≤ν_k+1. Up to change c₂ in Proposition 3.4, we can assume that

c₂d^Hm/22^−Hm|log 2|^(l−1)Hm >1,

which implies thatν_k>1 for every k. Then, since 1≤ν_k−1≤ν_k, c₂d^Hm/22^{−(νk−1)Hm}|(ν_k−1) log 2|^(l−1)Hm >2^−k and we have

(16) 2^−k

2√ d−Hm

c⁻¹₂ <

2^−νk(ν_klog 2)^l−1Hm

≤2^−k√ d−Hm

c⁻¹₂ .

Let us remark that (ν_k)_k≥1 is an increasing sequence and then thatν_k≥k for everyk. Furthermore, taking the logarithm of (16), one easily proves that

k→+∞lim k

ν_k =H_m, which impliesν_k =O(k).

For everyk ∈N\{0} and j= (j₁, . . . , j_d)∈Z^d we set x_k,j = j

2^νk, Dk=n

x_k,j :j∈Z^d∩[0,2^νk]^do

and N_k = cardDk= (2^νk+ 1)^d.

Then Dk is a 2^−k net of K forτ_E in the sense that for anyx∈K one can findx_k,j ∈ Dk such that τ_E(x−x_k,j)≤2^−k. Actually, by Proposition 3.4, it is sufficient to choosejsuch thatj_i≤2^νkx_i < j_i+1 for 1≤i≤d.

Let us remark that the sequence (Dk)_k is increasing and setD=

+∞

[

k=1

Dk. Step 2. Almost surely, for anyx, y∈ D

Y_α(x)−Y_α(y) =C_α<

+∞

X

n=1

T_n^−1/αm(ξ_n)^−1/α(f(x, ξ_n)−f(y, ξ_n))g_n

! ,

whereC_α is defined by (13). Since the sequences (T_n)_n, (ξ_n)_n and (g_n)_n are independent and (g_n)_n is a sequence of i.i.d. Gaussian complex isotropic random variables

R(x, y) =

+∞

X

n=1

T_n^−1/αm(ξn)^−1/α(f(x, ξn)−f(y, ξn))gn

is a Gaussian isotropic complex random variable conditionally to (T_n, ξ_n)_n. Remark that Y_α(x)− Y_α(y) = C_α<(R(x, y)) almost surely. Therefore, conditionally to (T_n, ξ_n)_n, Y_α(x)−Y_α(y) is a real centered Gaussian random variable with variance

v²((x, y)|(Tn, ξn)_n) = C_α² 2 E

|R(x, y)|²|(Tn, ξn)_n

= C_α² 2 E

|g₁|²X^+∞

n=1

T_n^−2/αm(ξ_n)^−2/α|f(x−y, ξ_n)|², since |f(x, ξ_n)−f(y, ξ_n)|=|f(x−y, ξ_n)|.

We consider the set

E_i,j^k ={ω:|Y_α(x_k,i)−Y_α(x_k,j)|> v((x_k,i, x_k,j)|(T_n, ξ_n)_n)ϕ(τ_E(x_k,i−x_k,j))}, where, as in [13],

ϕ(t) = r

2C_ϕdlog1

t, t >0 and C_ϕ >0 will be chosen later. Then, for every (k, i, j),

P

E_i,j^k

=E

E

1_Ek

i,j|(T_n, ξ_n)_n .

We give in the following an upper bound of this probability for well chosen (k, i, j). Note that ifZ is a real centered Gaussian random variable with variance 1, we have

E 1_Ek

i,j|(Tn, ξn)_n

=P(|Z|> ϕ(τE(x_k,i−x_k,j))) almost surely.

Let us choose δ∈(0,1) and set for k∈N\{0}, δ_k= 2^−(1−δ)k and I_k=n

(i, j)∈Z^d∩[0,2^νk]^d :τ_E(x_k,i−x_k,j)≤δ_ko . For every (i, j)∈I_k, since ϕis a decreasing function

P(|Z|> ϕ(τ_E(x_k,i−x_k,j)))≤P(|Z|> ϕ(δ_k)).

Then, we recall that

∀u≥0,P(Z > u)≤ e^−u2/2

√2πu. Therefore, for everyk ∈N\{0}and (i, j)∈I_k,

P

E^k_i,j

≤ r2

π

e^{−ϕ2(δk)/2}

ϕ(δ_k) = 2^{−(1−δ)kCϕd} p2C_ϕd(1−δ)klog 2. Hence,

∞

X

k=1

X

(i,j)∈Ik

P

E_i,j^k

≤ 1

p2C_ϕd(1−δ) log 2

+∞

X

k=1

2^{−(1−δ)kCϕd}card I_k. Sinceν_k=O(k), the lower bounds of (16) and Corollary 3.4 leads to

cardn

j∈Z^d∩[0,2^νk]^d : (i, j)∈I_ko

=O

δ_k^d/H12^dk/Hmk^2d(l−1) , for any i∈Z^d∩[0,2^νk]^d. Then one can find a finite constantC >0 such that

∞

X

k=1

X

(i,j)∈Ik

P

E_i,j^k

≤ C

pC_ϕ(1−δ)

∞

X

k=1

k^3d(l−1)2^−kd

−_Hm² +^1−δ_H

1+(1−δ)Cϕ

,

which is finite forC_ϕ > _H²

m −_H¹₁ and δ small enough. By the Borel-Cantelli Lemma, almost surely there exists an integerk^∗(ω) such that for everyk ≥k^∗(ω),

|Y_α(x)−Y_α(y)| ≤v((x, y)|(T_n, ξ_n)_n)ϕ(τ_E(x−y)) as soon as x, y∈ Dk withτE(x−y)≤δ_k.

Step 3. As in [14] let us give an upper bound of v²((x, y)| (T_n, ξ_n)_n) = C_α²

2 E

|g₁|^2+∞X

n=1

T_n^−2/αm(ξ_n)^−2/α|f(x−y, ξ_n)|² with respect to τ_E(x−y). By definition off

v²((x, y)| (T_n, ξ_n)_n)≤ C_α² 2 E

|g₁|²

σ²(τ_E(x−y)), where for all h >0

σ²(h) =

+∞

X

n=1

T_n^−2/αm(ξ_n)^−2/αmin M_E

h^Etξ_n

,22

ψ(ξ)^−2−2q/α,

with M_E given by (6). For the sake of clearness we postpone the proof of the control of σ²(h) in Appendix and state it in the following lemma.

Lemma 5.2. For any γ ∈(0,1) there exists a finite constant c >0 such that E σ²(h)|(T_n)_n

≤c

+∞

X

n=1

T_n^−2/αh²|logh|(1+η)(2/α−1) almost surely as soon as h≤1−γ.

Following [14] let us denote

b(h) =h|logh|^(1+η)/α. Then by Lemma 5.2,

E

+∞

X

k=1

σ²(2^−k) b²(2^−k)

(T_n)_n

!

<+∞. Therefore by independence of (Tn)_n and (ξn)_n, almost surely

k→+∞lim

σ²(2^−k) b²(2^−k) = 0.

Up to change the Euclidean normk · k by the equivalent normk · kE^t defined in Lemma 6.1.5 of [22]

the maph7→

h^Etξ

is increasing and so isσ². Also, one can conclude, as in [14], that almost surely

h→0lim σ²(h) b²(h) = 0.

Therefore, up to change k^∗ one can assume that for every k≥k^∗(ω), for every x, y∈ Dk, (17) |Y_α(x)−Y_α(y)| ≤p

2dC_ϕτ_E(x−y)|logτ_E(x−y)|(1+η)/α+1/2. as soon as τ_E(x−y)≤δ_k. Let

Ω^∗=

+∞

[

n=1

\

k≥n

\

x,y∈Dk τE(x−y)≤δk

n

|X_α(x)−X_α(y)| ≤p

2dC_ϕτ_E(x−y)|logτ_E(x−y)|(1+η)/α+1/2o

SinceX_α and Y_α have the same finite dimensional marginsP(Ω^∗) = 1.

Step 4. Let ω ∈ Ω^∗. By Step 3 there exists k^∗(ω) ≥ 1 such that X_α satisfies (17) for k≥k^∗(ω),x, y∈ Dk and τ_E(x−y)≤δ_k.

Let us recall that by Lemma 2.2 of [6], there existsK_E ≥1 such that for all x, y∈R^d τ_E(x+y)≤K_E(τ_E(x) +τ_E(y)).

Let us denote F(h) = p

2dC_ϕh|logh|(1+η)/α+1/2 and choose k₀ ∈ N such that 2^k0δ_k0+1 > 3K_E² and F is increasing on (0, δ_k0]. Up to change k^∗(ω), we can assume thatk^∗(ω)≥k₀.

Letx, y∈ D withx 6=y such that 3K_E²τ_E(x−y)≤δ_k∗(ω). Then there exists a unique k≥k^∗(ω) such that δ_k+1 <3K_E²τ_E(x−y) ≤ δ_k. Since x, y ∈ D, there exists n≥ k+ 1 such that x, y ∈ Dn. Moreover, by Step 1, forj=k, . . . , n−1, we can choose x^(j), y^(j)∈ Dj such that

τ_E

x−x^(j)

≤2^−j and τ_E

y−y^(j)

≤2^−j. By construction τ_E x^(k)−y^(k)

≤ K_E² τ_E(x−y) + 22^−k

. Let us point out that since k ≥ k₀, 2^kδ_k+1≥2^k0δ_k0+1>3K_E². Therefore, one easily sees that

τE

x^(k)−y^(k)

≤3K_E²τE(x−y).

Since 3K_E²τ_E(x−y)≤δ_k we obtain by Step 3 on the one hand that

X_α

x^(k)

−X_α y^(k)

≤F

τ_E

x^(k)−y^(k) . On the other hand we can write

X_α(x)−X_α x^(k)

=

n−1

X

j=k

X_α

x^(j+1)

−X_α x^(j)

with τ_E x^(j+1)−x^(j)

≤ 3K_E²2^−(j+1) ≤ δ_j+1 since j ≥k₀. Moreover, note that x^(j) ∈ Dj ⊂ Dj+1

and then by Step 3

Xα(x)−Xα

x^(k)

≤

n−1

X

j=k

F τE

x^(j+1)−x^(j)

≤CF(δ_k+1),

where C =

+∞

X

j=0

(j+ 1)(1+η)/α+1/2δ_j < +∞. Using similar computations for X_α(y)−X_α y^(k) , we obtain that

|X_α(x)−X_α(y)| ≤ F τ_E

x^(k)−y^(k)

+ 2CF(δ_k+1)

≤ (1 + 2C)F 3K_E²τ_E(x−y) . Then one can find a constant C >0 such that for 3K_E²τ_E(x−y)≤δ_k^∗_(ω) (18) |X_α(x)−X_α(y)| ≤Cτ_E(x−y)|logτ_E(x−y)|(1+η)/α+1/2. We now give a modification of X_α. For x∈ D, we set

X_α^∗(x)(ω) =X_α(x)(ω).

For x ∈K let x⁽ⁿ⁾ ∈ D such that limn→+∞x⁽ⁿ⁾ =x. In view of (18), X_α^∗ x⁽ⁿ⁾ (ω)

n is a Cauchy sequence and then converges. We set

X_α^∗(x)(ω) = lim

n→+∞X_α^∗ x⁽ⁿ⁾

(ω).

Remark that this limit does not depend on the choice of x⁽ⁿ⁾

. Moreover, sinceX_α is stochastically continuous,X_α^∗ is a modification of X_α.

By continuity ofτ_E, we easily see that

X_α^∗(x)(ω)−X_α^∗ x⁰ (ω)

≤Cτ_E(x−y)|logτ_E(x−y)|(1+η)/α+1/2

as soon as 3K_E²τ_E(x−y)< δ_k^∗_(ω), which concludes the proof.

Following the same lines as the proof of Theorem 5.1 we obtain a similar result in the Gaussian case (α = 2) for more general fields. Let us remark that Y_α is not defined for α = 2. However, in Step 2 of the proof, let us replaceY_αbyX a centered Gaussian random field andv²((x, y)|(T_n, ξ_n)_n) by the variance ofX(x)−X(y)

v²((x, y)) =E

(X(x)−X(y))² .

Furthermore let us replace Step 3 by the assumption that for some β ∈R and δ > 0 there exists a finite constantC >0 such that for x, y∈K withτ_E(x−y)≤δ

(19) E

(X(x)−X(y))²

≤Cτ_E(x−y)²|logτ_E(x−y)|^β. Then Step 1, Step 2 and Step 4 yields the following proposition.

Proposition 5.3. Let X be a centered Gaussian random field satisfying (19). There exists a modi- fication X^∗ of X on K such that

(20) lim

δ↓0 sup

x,y∈K

kx−yk≤δ

|X^∗(x)−X^∗(y)|

τ_E(x−y)|logτ_E(x−y)|^1/2+β+ε = 0 a.s.

for any ε >0.

Let us point out that ifX₂ is an operator scaling Gaussian random field as defined in [6], then E

(X₂(x)−X₂(y))²

=τ_E(x−y)²E

X₂(`_E(x−y))² ,

and X₂ satisfies (19) with β= 0 by (5.2) of [6]. Therefore this result is more precise than one could expect from the Theorem 5.1, replacingα by 2.

Let us also mention that Marianne Clausel gives a different proof of a similar result for some Gaussian operator scaling random fields with stationary increments in [8].

Now, as in [6], we are looking for global and directional H¨older critical exponents of the harmo- nizable stable random fieldX_α. These exponents have been introduced in [7] in the Gaussian realm but can be defined for any random field. Following Definition 5.1 of [6], H∈(0,1) is said to be the H¨older critical exponent of a random field (X(x))_x∈Rd if there exists a modification X^∗ of X such that for any s∈(0, H), the sample paths of X^∗ satisfy almost surely a uniform H¨older condition of orders onK, that is there exists a finite positive random variable A such that almost surely (21) |X^∗(x)−X^∗(y)| ≤Akx−yk^s for allx, y∈K

while for any s ∈(H,1), almost surely (21) fails. Note that the H¨older critical exponent, if exists, is well defined since any continuous modification of X and X^∗ are indistinguishable. Moreover, according to Definition 5.3 of [6] we say that X admits H(u) as directional regularity in direction u ∈ S^d−1, with S^d−1 the Euclidean unit sphere, if the process (X(tu))_t∈R admits H(u) as H¨older critical exponent onK∩Ru.

For all j= 1, . . . , mwe set K_j =K∩Lj

k=1W_k. Let us remark thatK_m =K.

Corollary 5.4. Let α ∈ (0,2]. There exists a modification X_α^∗ of X_α on K such that for all j = 1, . . . , m

limδ↓0 sup

x,y∈Kj kx−yk≤δ

|X_α^∗(x)−X_α^∗(y)|

kx−yk^Hj|logkx−yk|^Hj(pj−1)+β+1/2+ε = 0 a.s.

for anyε >0, where p_j = max

1≤k≤jl_k, β = 1/α ifα6= 2 andβ = 0 if α= 2.

Proof. It follows from Theorem 5.1 and Corollary 3.3, sinceaj ≤a_d for anyj = 1. . . d.

Corollary 5.5. Let α ∈(0,2]. The random field X_α^∗ has locally H-H¨older sample paths on R^d for every H∈(0, Hm).

Now let us give the directional and global H¨older critical exponents ofX_α. Proposition 5.6. The random field X_α admits H_m as H¨older critical exponent.

Moreover, for any j = 1, . . . , m, for any direction u ∈ W_j, the field X_α admits H_j as directional regularity in the direction u.

Proof. For Z a real SαS random variable we let

kZkα = (−log (E(exp (iZ))))^1/α. Then, for anyx, y∈R^d,

kX_α^∗(x)−X_α^∗(y)kα=C_α(`_E(x−y))τ_E(x−y) where for all θ∈S_E

C_α(θ) =

c_α Z

R^d

e^ihθ,ξi−1

αψ(ξ)^−α−qdξ 1/α

and c_α= 1 2π

Z π

0 |cos(t)|^αdt.

From Lebesgue’s Theorem, the functionC_α is continuous on the compact setS_E with positive values.

Let us denotem_α = min

θ∈SE

C_α(θ)>0. According to (10) in Corollary 3.4, for anyj = 1, . . . , mand u a direction in W_j,

kX_α^∗(tu)−X_α^∗(su)kα ≥m_αc₁|t−s|^Hj|log|t−s||^{−(lj−1)Hj}.