# Fractional Stable random fields on the Sierpiński gasket

Fabrice Baudoin111Partly funded by NSF DMS-2247117 and Simons Foundation,
Céline Lacaux

###### Abstract

We define and study fractional stable random fields on the Sierpiński gasket.
Such fields are formally defined as $(-\Delta)^{-s}W_{K,\alpha}$, where
$\Delta$ is the Laplace operator on the gasket and $W_{K,\alpha}$ is a stable
random measure. Both Neumann and Dirichlet boundary conditions for $\Delta$
are considered. Sample paths regularity and scaling properties are obtained.
The techniques we develop are general and extend to the more general setting
of the Barlow fractional spaces.

###### Contents

  1. 1 Introduction
  2. 2 Sierpiński Gasket and Neumann Fractional Riesz kernels
    1. 2.1 Definition of the gasket
    2. 2.2 Canonical Dirichlet form and heat kernel estimates
    3. 2.3 Neumann Fractional Laplacian
    4. 2.4 Neumann Fractional Riesz kernels
  3. 3 Neumann Fractional Stable random fields
    1. 3.1 Background on stable random measures and LePage Series
    2. 3.2 Definition of the Neumann Fractional Stable Field
    3. 3.3 Density field
    4. 3.4 Sample paths regularity
    5. 3.5 Symmetries and self-similarity properties
      1. 3.5.1 Invariance by reflections
      2. 3.5.2 Invariance by scaling
  4. 4 Dirichlet Fractional Stable random fields
    1. 4.1 Dirichlet fractional Riesz kernels
    2. 4.2 Dirichlet fractional stable fields
  5. A Proof of Lemma 2.13

##  1 Introduction

Fractional random fields are widely studied since the last decades, especially
in the Euclidean framework that is when the field is indexed by
$\mathbb{R}^{d}$. The Fractional Brownian field
$B_{H}=(B_{H}(x))_{x\in\mathbb{R}^{d}}$ with Hurst index $H\in(0,1)$,
introduced in [40, 34], is certainly the most popular fractional Gaussian
field. It admits several representation, as harmonizable and moving average
ones: this has led to several generalizations aiming to relax some its
properties, such as self-similarity or isotropy, but also allowed to introduce
non-Gaussian fractional random fields. Being far from exhaustive, we mention
[26, 33, 16, 17] as examples of non-Gaussian Euclidean fractional random
fields, [7, 25, 24, 8] as examples of anisotropic fractional (Gaussian or not)
fields and [18, 41, 37, 23, 5] as examples of multifractional (Gaussian or
not) fields. Recently, [39] has also named fractional Gaussian field $X$ a
Gaussian field defined as solution, in a distribution sense of

$X=(-\Delta)^{-s}W,$

where $s\geq 0$, $W$ is a white noise and $\Delta$ is the Euclidean Laplacian
on $\mathbb{R}^{d}$. This class extends Fractional Brownian motions, including
smoother random fields especially and some other classical fields such as
white noise, log-correlated Gaussian random field and the Gaussian free field.
Moreover, this approach has been considered before in [18] in terms of
reproducing kernel to define elliptic Gaussian processes. More precisely, the
reproducing kernel of an elliptic Gaussian process and then its covariance
function is characterized by an elliptic symmetric positive pseudodifferential
operator $A$ and in particular choosing for $A=(-\Delta)^{s}$ leads to the
particular case of fractional Euclidean Gaussian fields.

Moreover, since several years, the interest in random fields with heavy-tails,
such as $\alpha$-stable random fields, is increasing and many papers focus on
their sample path regularity. Most often in the literature, the fractional
Euclidean fields studied are defined via an integral representation, such as
moving average or harmonizable one. As far as we know, [19] is the only work
that has considered a class of stable random fields defined via an elliptic
operator and then followed the approach of [18]. For a Gaussian field
($\alpha=2$), it is well-known that the smoothness is linked to the behavior
of the second order moment

$\mathbb{E}\left(X(x)-X(y)\right)^{2}$

as $x\to y$. Based on this behavior and on a chaining argument, the entropy
method allows to obtain quite fine controls of the modulus of continuity for
Gaussian random fields (see e.g. [1]). When $\alpha<2$, the $\alpha$-stable
random field is not anymore square integrable and then the method to study its
smoothness properties differ from the Gaussian case. For example, [47] has
obtained a uniform modulus of continuity of stable random fields using a
chaining argument and the notion of maximal moment index. An other technique
proposed in [35] and followed by [20, 23], relies on a chaining argument and a
representation as a conditionally Gaussian LePage random series. Considering a
more general framework, [22] has also given some sufficient conditions to
derive an upper bound for the modulus of continuity of Euclidean stable random
fields. Other studies, such as [4, 6, 9, 19], are based on a wavelet series
representation of the random field under study.

Non-Euclidean fractional fields have also been considered quite recently: [30]
has defined the fractional Brownian field on manifolds and metric spaces
giving its covariance function, replacing the Euclidean norm by a distance $d$
and [31] has considered some moving average random fields. Using the
fractional Riesz kernels and the Laplace Beltrami operator and following the
approach of [18, 19, 39], [29] has then considered fractional Brownian fields
over manifolds. The previous works [14, 13] on fractional fields indexed by a
fractal set have also followed this approach.

The main goal to this paper is to extend [14, 13] to the stable framework,
that is to define and study

$X_{s}=(-\Delta)^{-s}W_{K,\alpha},$ (1.1)

where $\alpha\in(0,2]$, $s\geq 0$, $\Delta$ is the Neummann or Dirichlet
Laplace operator on the Sierpiński gasket $K$ and $W_{K,\alpha}$ is a
symmetric real-valued $\alpha$-stable random measure with control measure the
Hausdorff measure $\mu$ on $K$. As in [14, 13], the definition (1.1) has to be
interpreted in a distribution sense, that is as

$X_{s,\alpha}(f)=W_{K,\alpha}\left((-\Delta)^{-s}f\right)=\int_{K}(-\Delta)^{-s}f\,dW_{K,\alpha}$

for $f$ in a suitable Sobolev space $H^{-s}(K)$ of functions. Compared to [14,
13], the white noise on $L^{2}(K,\mu)$, that is the Gaussian random measure
with intensity the measure $\mu$, is here replaced by an $\alpha$-stable
measure $W_{K,\alpha}$. As a consequence, $X_{s,\alpha}$ is an $\alpha$-stable
symmetric random field, that is for any functions $f_{1},\ldots,f_{n}$ and any
real numbers $\lambda_{1},\ldots,\lambda_{n}$,
$\sum_{i=1}^{n}\lambda_{i}X_{s,\alpha}(f_{i})$ is a symmetric $\alpha$-stable
random variable. Note that when $\alpha=2$, $W_{K,\alpha}=W_{K,2}$ is nothing
else than a white noise on $L^{2}(K,\mu)$ and hence $X_{s,2}$ is or the
Neumman fractional Gaussian field studied in [14] or the Dirichlet one studied
in [13]. In this paper, we then focus on the case $\alpha<2$, for which the
techniques for studying the sample path regularity differ from the ones used
in the Gaussian case ($\alpha=2$).

As in [14, 13], we are interested in the existence of a random field
$\widetilde{X}_{s,\alpha}$ defined pointwise on $K$ which is a density of
$X_{s,\alpha}$, that is which satisfies

$\forall f\in H^{-s}(K),\quad
X_{s,\alpha}(f)=\int_{K}(-\Delta)^{-s}f(x)\widetilde{X}_{s,\alpha}(x)\,\mu(dx)\quad
a.s.$

Then we establish, see Theorem 3.5 that such density exists if and only if
$s>s_{\alpha}$ which extends Proposition 2.17 of [13] to the $\alpha$-stable
framework. The proof relies on a stochastic Fubini theorem for stable random
fields (see [44]). More or less, up to an additive random variable, this
density field admits as integral representation

$\widetilde{X}_{s,\alpha}(x)=\int_{K}G_{s}(x,y)W_{K,\alpha}(dy),$

with $G_{s}$ the fractional Riesz kernel, and it is itself an $\alpha$-stable
random field. When $\alpha=2$, it corresponds to the Gaussian density field
defined in [14, 13]. Moreover, according to [43], roughly speaking, the sample
paths of $\widetilde{X}_{s,\alpha}$ can not be smoother than the kernel
$G_{s}$: this leads to unboundness of $\widetilde{X}_{s,\alpha}$ for
$s\in(s_{\alpha},s_{0})$, where $s_{0}$ does not depend on the index stability
$\alpha\in(0,2)$ (see Theorems 3.9 and 4.9), whereas in the Gaussian framework
$\alpha=2$, the random field $\widetilde{X}_{s,\alpha}$ has always hölderian
sample paths. In the Euclidean framework, this difference is also observed:
for example, moving average fractional stable motion indexed by
$\mathbb{R}^{d}$ can be unbounded, see e.g.[44, 36, 45, 20], and especially
are always not continuous when $d>2$. Moreover, if $s>s_{0}$, we then
establish that the fractional field $\widetilde{X}_{s,\alpha}$ admits a
continuous modification and obtain an upper bound of its modulus of
continuity. As in [20, 23, 22], our proof relies on its representation as a
conditionally Gaussian LePage random series but slightly differs since an
other key ingredient is the Garsia-Rumsey-Rodemich inequality. This inequality
has first been established in [28] for Euclidean process indexed by
$\mathbb{R}$, extended to random field indexed by a metric space in [3] and
proved for random fields indexed by a fractal in [11]. Compared to [20, 23,
22], it allows us to avoid the construction of a specific dense sequence
$\mathcal{D}\subset K$ on which the increments of $\widetilde{X}_{s,\alpha}$
are controlled.

To conclude, let us remark that fractional stable random fields could be
defined and studied in the framework of the Barlow fractional spaces as in
Section 4 of [14]. Indeed our general analysis of such fields is based on heat
kernel estimates which are available in this general framework. However, for
the sake of presentation we focused on the specific example of the Sierpiński
gasket. In that example further specific properties, like scaling and
symmetries can then be proved, see Section 3.5.

The paper is organized as follows. In Section 2, after some background on the
Sierpiński gasket and Dirichlet form on it, the Neumann Fractional Laplacian
as in [14] but for any parameter $s\geq 0$. The fractional Riesz kernels are
also studied. Section 3 is then devoted to the definition and study of Neumann
fractional stable random fields and Section 4 extends the results to Dirichlet
fractional stable random fields. Appendix A is devoted to the proof of a
technical lemma about the convolution of two Riesz kernels.

##  2 Sierpiński Gasket and Neumann Fractional Riesz kernels

###  2.1 Definition of the gasket

Let us recall the definition of the (standard) Sierpiński gasket, denoted by
$K$ in this paper and of its Hausdorff measure $\mu$. We identify
$\mathbb{R}^{2}\simeq\mathbb{C}$ and consider the three vertices $q_{0}=0$,
$q_{1}=1$ and $q_{2}=e^{\frac{i\pi}{3}}$. The Sierpiński gasket can then be
defined inductively or characterized owing the contraction maps

$F_{i}(z)=\frac{1}{2}(z-q_{i})+q_{i},\quad i\in\\{0,1,2\\}$

as follows. For further details we refer to the book by Kigami [32].

###### Definition 2.1

The Sierpiński gasket is the unique non-empty compact set $K\subset\mathbb{C}$
such that

$K=\bigcup_{i=1}^{3}F_{i}(K).$

Figure 1: Sierpiński gasket.

In this paper, $K$ is equipped with the Euclidean metric denoted by $d$ (so
that its diameter is ${\rm diam}K=1$), and of its Borel $\sigma$-field
$\mathcal{K}$. Then, there exists on $(K,\mathcal{K})$ a unique probability
$\mu$, named normalized Hausdorff measure, such that for every
$i_{1},\cdots,i_{n}\in\\{0,1,2\\}$

$\mu\left(F_{i_{1}}\circ\cdots\circ F_{i_{n}}(K)\right)=3^{-n}.$

As in [14, 13], one key property in our study is the Alhfors property
satisfied by the measure $\mu$. So let us recall that denoting by

$d_{h}=\frac{\ln 3}{\ln 2}$

the Hausdorff dimension of $K$, the continuous probability $\mu$ is
$d_{h}$-Ahlfors regular, that is there exist constants $c,C>0$ such that

$\forall r\in(0,1],\ cr^{d_{h}}\leq\mu(B(x,r))\leq Cr^{d_{h}},$ (2.2)

where $B(x,r)$ is the metric ball of $K$ with center $x$ and radius $r$. The
values of $c$ and $C$ are useless in our study, but for the reader interested
in, [10] has proved that Equation (2.2) holds e.g. with $c=1/3$ and $C=18$.

###  2.2 Canonical Dirichlet form and heat kernel estimates

Following [32], let us introduce in this section the canonical Dirichlet form
the Neumann Laplacian is associated with considering a graph approximation of
the gasket. Let $V_{0}=\\{q_{0},q_{1},q_{2}\\}$, and for any
$\boldsymbol{i}=(i_{1},\ldots,i_{n})\in\\{0,1,2\\}^{n}$,

$W_{\boldsymbol{i}}=F_{i_{1}}\circ\cdots\circ F_{i_{n}}(V_{0})$

and then consider

$V_{n}=\bigcup_{\boldsymbol{i}\in\\{0,1,2\\}^{n}}W_{\boldsymbol{i}}\quad\textrm{and}\quad
V_{*}=\bigcup_{n\geq 0}V_{n}.$

For any $n\in\mathbb{N},$ let us now define the quadratic form
$\mathcal{E}_{n}$ on $\mathbb{R}^{V_{n}}$ by

${\forall
f\in\mathbb{R}^{V_{n}},\quad\mathcal{E}_{n}(f,f)=\frac{1}{2}\left(\frac{5}{3}\right)^{n}\sum_{\boldsymbol{i}\in\\{0,1,2\\}^{n}}\sum_{x,y\in
W_{\boldsymbol{i}}}\left(f(x)-f(y)\right)^{2}.}$

Then, setting

$\mathcal{F}_{*}=\left\\{f\in\mathbb{R}^{V_{*}},\lim_{n\to\infty}\mathcal{E}_{n}\left(f,f\right)<+\infty\right\\}$

for any function $f\in\mathcal{F}_{*}$, let

$\displaystyle\mathcal{E}(f,f)=\lim_{n\to\infty}\mathcal{E}_{n}(f,f).$ (2.3)

In addition, any function $f\in\mathcal{F}_{*}$ can uniquely be extended into
a continuous function defined on the whole $K$. Denoting by $\mathcal{F}$ the
set of such extensions, $(\mathcal{E},\mathcal{F})$ is then a local regular
Dirichlet form on $L^{2}(K,\mu)$ such that for every $f,g\in\mathcal{F}$

$\mathcal{E}(f,g)=\frac{5}{3}\sum_{i=1,2,3}\mathcal{E}(f\circ F_{i},g\circ
F_{i}),$

see the book by Kigami [32]. Moreover, the semigroup $\\{P_{t}\\}_{t\geq 0}$
associated with $\mathcal{E}$ is stochastically complete (i.e. $P_{t}1=1$)
and, from [11], has a jointly continuous heat kernel $p_{t}(x,y)$ with respect
to the reference measure $\mu$. Finally, there exist some constants
$c_{1},c_{2},c_{3},c_{4}\in(0,\infty)$ such that for every $(x,y)\in K\times
K$ and $t\in\bigl{(}0,1)$

$c_{1}t^{-d_{h}/d_{w}}\exp\biggl{(}-c_{2}\Bigl{(}\frac{d(x,y)^{d_{w}}}{t}\Bigr{)}^{\frac{1}{d_{w}-1}}\biggr{)}\leq
p_{t}(x,y)\leq
c_{3}t^{-d_{h}/d_{w}}\exp\biggl{(}-c_{4}\Bigl{(}\frac{d(x,y)^{d_{w}}}{t}\Bigr{)}^{\frac{1}{d_{w}-1}}\biggr{)}$
(2.4)

where we recall that $d_{h}=\frac{\ln 3}{\ln 2}$ is the Hausdorff dimension of
$K$ and where the parameter $d_{w}=\frac{\ln 5}{\ln 2}$ is called the walk
dimension. Since $d_{w}>2$, the heat kernel estimates (2.4) in Equation (2.4)
are said to be sub-Gaussian estimates.

Let $\Delta$ be the generator of the Dirichlet form $\mathcal{E}$. We refer to
$\Delta$ as the Neumann Laplacian on $K$. This terminology comes from the fact
that functions in the domain of $\Delta$ satisfy a Neumann boundary condition,
see Theorem 7.3.9 in [32].

###  2.3 Neumann Fractional Laplacian

In this section, as in [14], we introduce the Neumann fractional Laplacian
$(-\Delta)^{-s}$ on

$L^{2}_{0}(K,\mu)=\left\\{f\in L^{2}(K,\mu),\int_{K}fd\mu=0\right\\}.$

Note that [14] assumes that $s>d_{h}/(2d_{w})$, but following [13] which deals
with the Dirichlet Laplacian, the definition of the Neumann fractional
Laplacian $(-\Delta)^{-s}$ is here easily extended to any $s\geq 0$. To do so,
let us consider
$0<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}\leq\cdots$ the
eigenvalues of $-\Delta$ on $L^{2}_{0}(K,\mu)$ and an orthonormal basis
$\Phi_{j}\in L_{0}^{2}(K,\mu)$, $j\geq 1$ such that

$\Delta\Phi_{j}=-\lambda_{j}\Phi_{h},$

see [32] for details. In addition, from [32], we have
$\Phi_{j}\in\mathrm{dom}(\Delta)\subset\mathcal{F}$, $j\geq 1$, and

$\displaystyle
p_{t}(x,y)=1+\sum_{j=1}^{+\infty}e^{-\lambda_{j}t}\Phi_{j}(x)\Phi_{j}(y)$
(2.5)

where the series converges uniformly on any compact set $(t,x,y)\in[a,b]\times
K\times K$ with $0<a<b$.

Let us now consider $f\in L^{2}_{0}(K,\mu)$, so that in $L^{2}_{0}(K,\mu)$,

$f=\sum_{j=1}^{+\infty}\int_{K}f(x)\Phi_{j}(x)\mu(dx)\,\Phi_{j}.$

Then, for any $s\geq 0$,

$\left|\lambda_{j}^{-s}\int_{K}f(x)\Phi_{j}(x)\mu(dx)\right|^{2}\leq\lambda_{1}^{-2s}\left|\int_{K}f(x)\Phi_{j}(x)\mu(dx)\right|^{2}$

and so the series

$\sum_{j=1}^{+\infty}\lambda_{j}^{-s}\int_{K}f(x)\Phi_{j}(x)\mu(dx)\,\Phi_{j}$

converges in $L^{2}(K,\mu)$ and also in $L_{0}^{2}(K,\mu)$ by continuity of
the scalar product on $L^{2}(K,\mu)$. Then, according to the previous lines,
the Neumann fractional Laplacian of order $s$ is well-defined as follows and
is a bounded operator from $L^{2}_{0}(K,\mu)$ to itself.

######  Definition 2.2 (Neumann Fractional Laplacians)

Let $s\geq 0$. For $f\in L_{0}^{2}(K,\mu)$, the Neumann fractional Laplacian
$(-\Delta)^{-s}$ on $f$ is defined as

$\displaystyle(-\Delta)^{-s}f=\sum_{j=1}^{+\infty}\lambda_{j}^{-s}\int_{K}\Phi_{j}(y)f(y)\mu(dy)\,\Phi_{j}.$
(2.6)

Note that $(-\Delta)^{-s}f\in L_{0}^{2}(K,\mu)$ and
$\left\|(-\Delta)^{-s}f\right\|_{L^{2}(K,\mu)}\leq\lambda_{1}^{-s}\left\|f\right\|_{L^{2}(K,\mu)}$.

###### Remark 2.3

The definition of the Neumann fractional Laplacian $(-\Delta)^{-s}$ on $f$
does not depend on the choice of the orthonormal basis $\Phi_{j}$, $j\geq 1$.
Actually, considering
$0<\lambda_{i_{1}}<\lambda_{i_{2}}<\ldots<\lambda_{i_{n}}<\cdots$ the distinct
eigenvalues of $-\Delta$ on $L_{0}^{2}(K,\mu)$, the Neumann fractional
Laplacian $(-\Delta)^{-s}$ can be rewritten as

$(-\Delta)^{-s}f=\sum_{j=1}^{+\infty}\lambda_{i_{j}}^{-s}\,p_{E_{j}}(f)$

where $p_{E_{j}}$ is the orthogonal projection on the eigenspace $E_{j}\subset
L^{2}_{0}(K,\mu)$ associated with the eigenvalue $\lambda_{i_{j}}$.

###### Remark 2.4

It follows from (2.6) that for $f\in L^{2}_{0}(K,\mu)$ and $s,t\geq 0$,

$\displaystyle(-\Delta)^{-s-t}f=(-\Delta)^{-s}(-\Delta)^{-t}f.$ (2.7)

We then consider the following space of test functions

$\mathcal{S}(K)=\left\\{f\in C_{0}(K),\forall k\geq
0\lim_{n\to+\infty}n^{k}\left|\int_{K}\Phi_{n}(y)f(y)\mu(dy)\right|=0\right\\},$

where

$C_{0}(K)=\left\\{f\in C(K),\int_{K}fd\mu=0\right\\}\subset L_{0}^{2}(K,\mu).$

As in [13], when $f\in\mathcal{S}(K)$, then
$f\in\bigcap_{k\in\mathbb{N}}\mathrm{dom}((-\Delta)^{k})$ and so for every
$k\in\mathbb{N}$, $(-\Delta)^{k}f\in C(K)$ and is Hölder continuous, see [11].
We then endow $\mathcal{S}(K)$ with the topology defined by the family of
norms

$\|f\|_{k}=\|(-\Delta)^{k}f\|_{L^{\infty}(K,\mu)},\quad k\in\mathbb{N}.$

In addition, as in [13], by Lemma 5.1.3 in [32], one can check that

$\sum_{j=1}^{+\infty}\lambda_{j}^{-s}<+\infty\iff s>\frac{d_{h}}{d_{w}}$

and then that the space $\mathcal{S}(K)$ is a nuclear space. Then its dual
space $\mathcal{S}^{\prime}(K)$ for the latter topology is the analogue of the
space of tempered distributions in our setting.

We also note that for $f\in\mathcal{S}(K)$ and $s\geq 0$, it follows from
Lemma 5.1.3 and Theorem 4.5.4 in [32] (see also Lemma 4.8 in [13]) that the
series

$\sum_{j=1}^{+\infty}\lambda_{j}^{-s}\int_{K}\Phi_{j}(y)f(y)\mu(dy)\,\Phi_{j}$

is uniformly convergent on $K$, so that for $f\in\mathcal{S}(K)$,
$(-\Delta)^{-s}f$ is defined pointwise on $K$ by (2.6) and
$(-\Delta)^{-s}f\in\mathcal{S}(K)$.

###  2.4 Neumann Fractional Riesz kernels

Let us recall the definition of a fractional Riesz kernel on $K$ for the
Neumann Laplacian, see [14].

###### Definition 2.5

For a parameter $s>0$, we define the fractional Riesz kernel $G_{s}$ by

$\displaystyle
G_{s}(x,y)=\frac{1}{\Gamma(s)}\int_{0}^{+\infty}t^{s-1}(p_{t}(x,y)-1)dt,\quad
x,y\in K,\,x\neq y,$ (2.8)

with $\Gamma$ the gamma function. Moreover, $G_{s}(x,x)$ is also well-defined
when $s>d_{h}/d_{w}$ (see Lemma 2.4 in [14]).

Note that when $s\in(0,d_{h}/d_{w}]$, the function $y\mapsto G_{s}(x,y)$ is
well-defined $\mu$-almost everywhere, since $\mu\left(\\{x\\}\right)=0$.
Moreover, since the heat kernel $p_{t}$ is symmetric, for any $s>0$ the Riesz
fractional kernel $G_{s}$ is also a symmetric function.

The relationship between the fractional Riesz kernel $G_{s}$ and the
fractional Laplacian defined in the previous section is given by the following
representation formula which can be proved as in [13].

###### Lemma 2.6

Let $s>0$. For $f\in\mathcal{S}(K)$, and $x\in K$

$(-\Delta)^{-s}f(x)=\int_{K}G_{s}(x,y)f(y)\mu(dy).$ (2.9)

The fractional Riesz kernel $G_{s}$ has already been studied in [14]. First,
Proposition 2.6 in [14] gives a control, that allows to study the square
integrability of $G_{s}(x,\cdot)$ needed in the Gaussian framework. In the
stable framework, we are now wondering when $G_{s}(x,\cdot)\in
L^{\alpha}(K,\mu)$, so that its stochastic integral with respect to the random
measure $W_{\alpha}$ is well-defined. This integrability property is the key
point to define pointwise fractional stable fields in next section.
Proposition 2.6 in [14] already allows to state a sufficient condition but
need to be complete in order to characterize completely the integrability of
$G_{s}(x,\cdot)$. In addition Theorem 2.10 in [14] is interested in the Hölder
property in $L^{2}(K,\mu)$ fulfilled by $x\mapsto G_{s}(x,\cdot)$. Especially,
when studying fractional Gaussian random fields, this implies, using classical
entropy methods, a control of their modulus of continuity. However, in the
framework of fractional stable fields, it is not sufficient: in our study, a
pointwise Hölder property for $G_{s}$ is needed. Roughly speaking the stable
random field under study can not be smoother than the fractional Riesz kernel
involved in its definition.

Let us first complete Proposition 2.6 in [14].

###### Proposition 2.7

  1. 1.

If $s\in(0,d_{h}/d_{w})$, then there exist three positive constants
$C_{1},C_{2},C_{3}$ such that for any $x,y\in K$, $x\neq y$,

$C_{1}d(x,y)^{sd_{w}-d_{h}}-C_{2}\leq G_{s}(x,y)\leq
C_{3}d(x,y)^{sd_{w}-d_{h}}.$

  2. 2.

Let $s=d_{h}/d_{w}$. Then there exist three positive constants
$C_{1},C_{2},C_{3}$ such that for any $x,y\in K$ such that $x\neq y$,

$-C_{1}\ln d(x,y)-C_{2}\leq G_{s}(x,y)\leq C_{3}\max(\left|\ln
d(x,y)\right|,1).$

###### Remark 2.8

As a consequence, for $s\leq d_{h}/d_{w}$, $G_{s}$ is unbounded. This will
imply unboundedness of the fractional stable random field associated with
$G_{s}$, see Theorem 3.9 in the sequel.

Proof. The upper bounds have already been established in [14], see Proposition
2.6. So let us now consider $s\in(0,d_{h}/d_{w}]$ and prove the lower bounds.
Let $x,y\in K$, $x\neq y$. As in [14], we have

$G_{s}(x,y)=G_{s}^{(1)}(x,y)+G_{s}^{(2)}(x,y)$

where

$\left\\{\begin{array}[]{rcl}\displaystyle
G_{s}^{(1)}(x,y)&=&\displaystyle\frac{1}{\Gamma(s)}\int_{0}^{1}t^{s-1}(p_{t}(x,y)-1)dt=\frac{1}{\Gamma(s)}\int_{0}^{1}t^{s-1}p_{t}(x,y)dt-\frac{1}{\Gamma(s+1)}\\\\[15.0pt]
\displaystyle
G_{s}^{(2)}(x,y)&=&\displaystyle\frac{1}{\Gamma(s)}\int_{1}^{+\infty}t^{s-1}(p_{t}(x,y)-1)dt.\end{array}\right.$

By the heat kernel lower bound (2.4),

$G_{s}^{(1)}(x,y)\geq\frac{c_{1}}{\Gamma(s)}\int_{0}^{1}t^{s-1-d_{h}/d_{w}}\exp\left(-c_{2}\left(\frac{d(x,y)^{d_{w}}}{t}\right)^{\frac{1}{d_{w}-1}}\right)dt-\frac{1}{\Gamma(s+1)}$

Then using the change of variable $t=ud(x,y)^{d_{w}}$, we have

$G_{s}^{(1)}(x,y)\geq\displaystyle\frac{c_{1}d(x,y)^{sd_{w}-d_{h}}}{\Gamma(s)}\int_{0}^{1/d(x,y)^{d_{w}}}u^{s-1-d_{h}/d_{w}}\exp\left(-c_{2}u^{\frac{1}{1-d_{w}}}\right)du-\frac{1}{\Gamma(s+1)}.$

Then, denoting by $C_{1}$ a positive constant (depending only on
$s,d_{h},d_{w}$) whose value may change at each line, when $s=d_{h}/d_{w}$,
since $d(x,y)\leq 1$,

$\begin{array}[]{rcl}G_{s}^{(1)}(x,y)&\geq&\displaystyle
C_{1}\int_{1}^{1/d(x,y)^{d_{w}}}u^{-1}\exp\left(-c_{2}u^{\frac{1}{1-d_{w}}}\right)du-\frac{1}{\Gamma(s+1)}\\\\[15.0pt]
&\geq&\displaystyle
C_{1}\int_{1}^{1/d(x,y)^{d_{w}}}u^{-1}du-\frac{1}{\Gamma(s+1)}\\\\[15.0pt]
\end{array}$

since $c_{2}>0$ and $d_{w}>1$. Then,

$\begin{array}[]{rcl}G_{s}^{(1)}(x,y)&\geq&\displaystyle-C_{1}\ln
d(x,y)-\frac{1}{\Gamma(s+1)}.\end{array}$

Now when $s<d_{h}/d_{w}$, since $d(x,y)\leq 1$,

$G_{s}^{(1)}(x,y)\geq\displaystyle
C_{1}d(x,y)^{sd_{w}-d_{h}}-\frac{1}{\Gamma(s+1)},$

with

$C_{1}=\frac{c_{1}}{\Gamma(s)}\int_{0}^{1}u^{s-1-d_{h}/d_{w}}\exp\left(-c_{2}{u}^{\frac{1}{1-d_{w}}}\right)du$

a finite positive constant since $c_{2}>0$ and $d_{w}>1$. Moreover, as noticed
in [14], $G_{s}^{(2)}$ is uniformly bounded on $K\times K$, so that there
exists a positive constant $c$ such that for any $x,y\in K$

$G_{s}^{(2)}(x,y)\geq-c$

which concludes the proof choosing $c_{2}=\frac{1}{\Gamma(s+1)}+c$. $\Box$

Let us now study the integrability properties of the fractional Riesz kernels.

###### Lemma 2.9

Let $p\in(0,+\infty)$. Then for any $x\in K$, $y\mapsto G_{s}(x,y)$ is in
$L^{p}(K,\mu)$ iff $s>\max\left(\frac{(p-1)d_{h}}{pd_{w}},0\right)$. Moreover,
for $s>\max\left(\frac{(p-1)d_{h}}{pd_{w}},0\right)$,

$\sup_{x\in K}\int_{K}\left|G_{s}(x,y)\right|^{p}\mu(dy)<+\infty.$

This is a consequence of the following lemma, of Proposition 2.7 and of
Proposition 2.6 in [14].

###### Lemma 2.10

Let $\gamma\in\mathbb{R}$. Then for any $x\in K$,

$\int_{K}\frac{\mu(dy)}{d(x,y)^{\gamma}}<+\infty\iff\gamma<d_{h}.$

Moreover, for any $\gamma<d_{h}$,

$\sup_{x\in K}\int_{K}\frac{\mu(dy)}{d(x,y)^{\gamma}}<+\infty.$

Proof. By the proof of Proposition 2.7 in [14], if $\gamma<d_{h}$, one has

$\forall x\in K,\ \int_{K}\frac{\mu(dy)}{d(x,y)^{\gamma}}<+\infty$

and also

$\sup_{x\in K}\int_{K}\frac{\mu(dy)}{d(x,y)^{\gamma}}<+\infty.$

Let us now consider $\gamma\geq d_{h}$. Then by compactness of $K$, there
exists a positive constant $C$ such that

$\forall x,y\in K,\ d(x,y)^{\gamma}\leq Cd(x,y)^{d_{h}},$

and it is therefore enough to prove that for any $x\in K$,

$\int_{K}\frac{\mu(dy)}{d(x,y)^{d_{h}}}=+\infty.$

We use a dyadic annuli decomposition as follows. Since ${\rm diam}\,K=1$, one
has for every $N\geq 1$ and for any $x\in K$,

$\displaystyle\int_{K}\frac{\mu(dy)}{d(x,y)^{d_{h}}}$
$\displaystyle\geq\sum_{j=0}^{N}\int_{B\left(x,2^{-j}\right)\setminus
B\left(x,2^{-j-1}\right)}\frac{\mu(dy)}{d(x,y)^{d_{h}}}$
$\displaystyle\geq\sum_{j=0}^{N}2^{jd_{h}}\mu\left(B\left(x,2^{-j}\right)\setminus
B\left(x,2^{-j-1}\right)\right)$
$\displaystyle\geq\sum_{j=0}^{N}2^{jd_{h}}\left(\mu\left(B\left(x,2^{-j}\right)\right)-\mu\left(B\left(x,2^{-j-1}\right)\right)\right)$
$\displaystyle\geq\sum_{j=0}^{N}2^{jd_{h}}\mu\left(B\left(x,2^{-j}\right)\right)-\sum_{j=0}^{N}2^{jd_{h}}\mu\left(B\left(x,2^{-j-1}\right)\right)$
$\displaystyle\geq\mu\left(B\left(x,1\right)\right)-2^{Nd_{h}}\mu\left(B\left(x,2^{-N-1}\right)\right)+(1-2^{-d_{h}})\sum_{j=1}^{N}2^{jd_{h}}\mu\left(B\left(x,2^{-j}\right)\right).$

By the Ahlfors regularity (2.2) of the measure $\mu$, for any $x\in K$, when
$N\to+\infty$,

$\sum_{j=1}^{N}2^{jd_{h}}\mu\left(B(x,2^{-j})\right)\to+\infty$

whereas $2^{Nd_{h}}\mu\left(B(x,2^{-N-1}\right))$ remains bounded. Therefore
for any $x\in K$,

$\int_{K}\frac{\mu(dy)}{d(x,y)^{d_{h}}}=+\infty,$

which concludes the proof. $\Box$

We are now ready to prove Lemma 2.9.

Proof. Let us first assume that $s\in(0,d_{h}/d_{w})$. Then, since $\mu$ is a
probability, by Proposition 2.7 and Lemma 2.10,

$\begin{array}[]{rcl}\displaystyle\int_{K}\left|G_{s}(x,y)\right|^{p}\mu(dy)<+\infty&\iff&\displaystyle\int_{K}\frac{\mu(dy)}{d(x,y)^{p\left(d_{h}-sd_{w}\right)}}<+\infty\\\\[15.0pt]
&\iff&\displaystyle p\left(d_{h}-sd_{w}\right)<d_{h}\\\\[15.0pt]
&\iff&s>\frac{(p-1)d_{h}}{pd_{w}}.\end{array}$

Moreover, for $\max\left(0,\frac{(p-1)d_{h}}{pd_{w}}\right)<s<d_{h}/d_{w}$, by
Proposition 2.7 and Lemma 2.10, there exists a positive constant $C$ such that

$\sup_{x\in K}\int_{K}\left|G_{s}(x,y)\right|^{p}\mu(dy)\leq C\sup_{x\in
K}\displaystyle\int_{K}\frac{\mu(dy)}{d(x,y)^{p\left(d_{h}-sd_{w}\right)}}<+\infty.$

Now for $s\geq d_{h}/d_{w}$, using Proposition 2.6 in [14] and compactness of
$K$, for any $\gamma\in(0,d_{h}/p)$, one sees that there exists a positive
constant $C$ (depending on $\gamma)$ such that for any $x,y\in K$, $x\neq y$

$\left|G_{s}(x,y)\right|\leq\frac{C}{d(x,y)^{\gamma}}.$

Then, since $p\gamma<d_{h}$, by Lemma 2.10,

$\sup_{x\in K}\int_{K}\left|G_{s}(x,y)\right|^{p}\mu(dy)\leq C\sup_{x\in
K}\displaystyle\int_{K}\frac{\mu(dy)}{d(x,y)^{p\gamma}}<+\infty,$

which concludes the proof. $\Box$

To conclude this section, next theorem is interested in the smoothness of the
Riesz kernel $G_{s}$, that is in the Hölder property it satisfies. This
property is the key property to study sample paths properties of fractional
stable random fields, which will be defined as an integral of the kernel
$G_{s}$ with respect to a stable random measure (see Theorems 3.5 and 3.9 in
next section). Next theorem is only interested in the case where
$s>d_{h}/d_{w}$ since otherwise, Proposition 2.7 implies that the function
$G_{s}$ is unbounded.

###### Theorem 2.11

  1. 1.

For any $s\in\left(\frac{d_{h}}{d_{w}},1\right)$, there exists a constant
$C>0$ such that for every $x,y,z\in K$,

$|G_{s}(x,z)-G_{s}(y,z)|\leq Cd(x,y)^{sd_{w}-d_{h}}.$

  2. 2.

For $s\geq 1$, there exists a constant $C>0$ such that for every $x,y,z\in K$,

$|G_{s}(x,z)-G_{s}(y,z)|\leq Cd(x,y)^{d_{w}-d_{h}}\max\left(\left|\ln
d(x,y)\right|,1\right).$

where $0^{d_{w}-d_{h}}\max\left(\left|\ln 0\right|,1\right)=0$ by convention.

###### Remark 2.12

In the range $s\in\left(\frac{d_{h}}{d_{w}},1\right)$, it is reasonable to
conjecture that the exponent $sd_{w}-d_{h}$ is optimal. Indeed, by its
definition $x\to G_{s}(x,z)$ is in the domain of the operator $(-\Delta)^{s}$
and, as in Section 9 of [11], one can reasonably conjecture that the only
functions in the domain of the operator $(-\Delta)^{s}$ which are
$sd_{w}-d_{h}+\varepsilon$ Hölder continuous are the constant functions. We
also refer to [15] for further comments on the regularity of functions
associated with the Laplacian.

Proof. Let $x,y,z\in K$. Without loss of restriction, we can assume $x\neq y$.

  1. 1.

Let us first assume that $s\in(d_{h}/d_{w},1)$. Then, for $\delta\in(0,1]$

$G_{s}(x,z)-G_{s}(y,z)=\frac{1}{\Gamma(s)}\int_{0}^{+\infty}t^{s-1}\left(p_{t}(x,z)-p_{t}(y,z)\right)dt=I_{1}(x,y,z)+I_{2}(x,y,z)$

where

$\left\\{\begin{array}[]{rcl}\displaystyle
I_{1}(x,y,z)&=&\displaystyle\frac{1}{\Gamma(s)}\int_{0}^{\delta}t^{s-1}\left(p_{t}(x,z)-p_{t}(y,z)\right)dt\\\\[15.0pt]
\displaystyle
I_{2}(x,y,z)&=&\displaystyle\frac{1}{\Gamma(s)}\int_{\delta}^{+\infty}t^{s-1}\left(p_{t}(x,z)-p_{t}(y,z)\right)dt.\\\\[15.0pt]
\end{array}\right.$

In the following, $C$ is a generic positive constant, independent from
$(\delta,x,y,z)$, which value may change at each line.

First applying the triangle inequality and the upper bound heat kernel (2.4),
for any $t\in(0,1)$,

$\left|p_{t}(x,z)-p_{t}(y,z)\right|\leq Ct^{-d_{h}/d_{w}}$

so that

$\begin{array}[]{rcl}\displaystyle\left|I_{1}(x,y,z)\right|&\leq&C\displaystyle\int_{0}^{\delta}t^{s-1-d_{h}/d_{w}}dt\\\\[15.0pt]
&\leq&C\delta^{s-d_{h}/d_{w}}\end{array}$

since $s-d_{h}/d_{w}>0$.

Moreover, by Lemma 3.4 in [2], for any $t>0$

$\left|p_{t}(x,z)-p_{t}(y,z)\right|\leq\frac{Cd(x,y)^{d_{w}-d_{h}}}{t}$ (2.10)

so that

$\begin{array}[]{rcl}\displaystyle\left|I_{2}(x,y,z)\right|&\leq&\displaystyle
Cd(x,y)^{d_{w}-d_{h}}\int_{\delta}^{+\infty}t^{s-2}dt\\\\[15.0pt]
&\leq&Cd(x,y)^{d_{w}-d_{h}}\delta^{s-1}\end{array}$

since $s<1$.

Hence, by combining the obtained upper bounds for $I_{1}$ and $I_{2}$ and
choosing $\delta=d(x,y)^{d_{w}}\in(0,1]$,

$\begin{array}[]{rcl}\displaystyle\left|G_{s}(x,z)-G_{s}(y,z)\right|&\leq&Cd(x,y)^{sd_{w}-d_{h}}.\end{array}$

  2. 2.

    1. (a)

We first establish Assertion 2. for $s=1$. Then, for $\delta\in(0,1]$

$G_{1}(x,z)-G_{1}(y,z)=\int_{0}^{+\infty}\left(p_{t}(x,z)-p_{t}(y,z)\right)dt=I_{1}(x,y,z)+I_{2}(x,y,z)+I_{3}(x,y,z)$

where

$\left\\{\begin{array}[]{rcl}\displaystyle
I_{1}(x,y,z)&=&\displaystyle\int_{0}^{\delta}\left(p_{t}(x,z)-p_{t}(y,z)\right)dt\\\\[15.0pt]
\displaystyle
I_{2}(x,y,z)&=&\displaystyle\int_{\delta}^{1/\delta}\left(p_{t}(x,z)-p_{t}(y,z)\right)dt\\\\[15.0pt]
\displaystyle
I_{3}(x,y,z)&=&\displaystyle\int_{1/\delta}^{+\infty}\left(p_{t}(x,z)-p_{t}(y,z)\right)dt.\end{array}\right.$

In the following, as previously, $C$ is a generic positive constant,
independent from $(\delta,x,y,z)$, which value may change at each line.

First the same arguments as for $s<1$ lead to

$\begin{array}[]{rcl}\displaystyle\left|I_{1}(x,y,z)\right|&\leq&C\delta^{1-d_{h}/d_{w}}.\end{array}$

Moreover, by applying (2.10),

$\begin{array}[]{rcl}\displaystyle\left|I_{2}(x,y,z)\right|&\leq&\displaystyle
Cd(x,y)^{d_{w}-d_{h}}\int_{\delta}^{1/\delta}\frac{dt}{t}\\\\[15.0pt]
&\leq&-Cd(x,y)^{d_{w}-d_{h}}\ln\delta.\end{array}$

In addition, applying the triangle inequality and Lemma 2.3 in [14], for
$t\geq 1$

$\left|p_{t}(x,z)-p_{t}(y,z)\right|\leq C\exp\left(-\lambda_{1}t\right)$

where $\lambda_{1}>0$ is the lower eigenvalue of $-\Delta$. Hence,

$\begin{array}[]{rcl}\displaystyle\left|I_{3}(x,y,z)\right|&\leq&C\displaystyle\int_{1/\delta}^{+\infty}\exp\left(-\lambda_{1}t\right)dt\\\\[15.0pt]
&\leq&C\exp\left(-\frac{\lambda_{1}}{\delta}\right)\\\\[15.0pt]
&\leq&C\delta^{1-d_{h}/d_{w}}.\end{array}$

Therefore, combining the previous upper bounds and choosing
$\delta=d(x,y)^{d_{w}}\in(0,1]$,

$\begin{array}[]{rcl}\displaystyle\left|G_{1}(x,z)-G_{1}(y,z)\right|&\leq&Cd(x,y)^{d_{w}-d_{h}}\left(1+\left|\ln{d(x,y)}\right|\right)\\\\[15.0pt]
&\leq&Cd(x,y)^{d_{w}-d_{h}}\max\left(\left|\ln{d(x,y)}\right|,1\right).\end{array}$

    2. (b)

Let us now assume that $s>1$. Then, by the Lemma 2.13,

$G_{s}(x,z)-G_{s}(y,z)=\int_{K}\left(G_{1}(x,u)-G_{1}(y,u)\right)G_{s-1}(u,z)\,\mu(du).$

Hence, since $G_{1}$ satisfies Assertion 2.,

$\begin{array}[]{rcl}\displaystyle\left|G_{s}(x,z)-G_{s}(y,z)\right|&\leq&\displaystyle
Cd(x,y)^{d_{w}-d_{h}}\max\left(\left|\ln{d(x,y)}\right|,1\right)\int_{K}\left|G_{s-1}(u,z)\right|\mu(du).\end{array}$

Moreover, since $s-1>0$, by Lemma 2.9 and symmetry of $G_{s-1}$,

$\sup_{z\in K}\int_{K}\left|G_{s-1}(u,z)\right|\mu(du)=\sup_{z\in
K}\int_{K}\left|G_{s-1}(z,u)\right|\mu(du)<+\infty,$

which concludes the proof.

$\Box$

We conclude the section with the following lemma.

###### Lemma 2.13

For any $s>0$ and any $t>0$,

$G_{s+t}(x,y)=\int_{K}G_{s}(x,u)G_{t}(u,y)\mu(du)$

for any $x,y\in K$ such that $x\neq y$. This equation also holds for $x=z$
when $t+s>d_{h}/d_{w}$.

Proof. See Appendix A.

$\Box$

##  3 Neumann Fractional Stable random fields

###  3.1 Background on stable random measures and LePage Series

Given a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ and
$\alpha\in(0,2]$, we consider, on the measurable space $(K,\mathcal{K})$, a
real-valued symmetric stable random measure $W_{K,\alpha}$ with control
measure $\mu$. Let us recall the definition of such a random measure following
[44, Section 3.3]. We first consider

$\mathcal{K}_{0}=\\{A\in\mathcal{K}\,/\,\mu(A)<+\infty\\}.$

Then, $W_{K,\alpha}$ is such that the three following properties hold:

  * •

the random measure $W_{K,\alpha}$ is $\sigma$-additive, i.e. for any sequence
$(A_{n})_{n\in\mathbb{N}}\in\mathcal{K}_{0}^{\mathbb{N}}$ of pairwise disjoint
measurable sets such that $\bigcup_{n\in\mathbb{N}}A_{n}\in\mathcal{K}_{0}$,

$W_{K,\alpha}\left(\bigcup_{n\in\mathbb{N}}A_{n}\right)=\sum_{n\in\mathbb{N}}W_{K,\alpha}\left(A_{n}\right)\
\textrm{almost surely};$

  * •

$W_{K,\alpha}$ is independently scattered, that is for any sequence
$(A_{n})_{n\in\mathbb{N}}\in\mathcal{K}_{0}^{\mathbb{N}}$ of pairwise disjoint
measurable sets, the random variables $W_{K,\alpha}(A_{n})$, $n\in\mathbb{N}$,
are independent;

  * •

for any $A\in\mathcal{K}_{0}$, $W_{K,\alpha}(A)$ is a symmetric
$\alpha$-stable real-valued random variable with scale parameter
$\left(\mu(A)\right)^{1/\alpha}$, that is its characteristic function is given
by

$\mathbb{E}\left(\textup{e}^{iuW_{K,\alpha}(A)}\right)=\textup{e}^{-\left|u\right|^{\alpha}{\mu(A)}},\quad
u\in\mathbb{R}.$

Especially, if $\alpha=2$, then $W_{K,\alpha}$ is a real-valued Gaussian
random measure with intensity $\mu$.

Then for any real-valued $f\in L^{\alpha}(K,\mathcal{K},\mu)$, the stochastic
integral

$W_{K,\alpha}(f)=\int_{K}f(x)\,W_{K,\alpha}(dx)$

is well-defined and is a symmetric $\alpha$-stable real-valued random variable
with scale parameter

$\|f\|_{L^{\alpha}(K,\mu)}=\left(\int_{K}\left|f(x)\right|^{\alpha}\,\mu(dx)\right)^{1/\alpha}$

so that, its characteristic function is then given by

$\mathbb{E}\left(\textup{e}^{iuW_{K,\alpha}(f)}\right)=\textup{e}^{-\left|u\right|^{\alpha}\|f\|_{L^{\alpha}(K,\mu)}^{\alpha}},\quad
u\in\mathbb{R}.$

We refer to [44, Section 3.4] for details on the construction of the
stochastic integral $W_{K,\alpha}(f)$ and on its properties.

In addition, when $\alpha\in(0,2)$, the stochastic integral $W_{K,\alpha}(f)$
can also be represented as a series, and such a series can be chosen to
consider $W_{K,\alpha}(f)$ as a conditionally Gaussian random variable. This
representation will be a key point to study the sample paths smoothness of the
fractional stable fields considered in this paper. Let us then now give this
LePage series representation. To do so, let us assume that the following
assumption holds.

###### Assumption 3.1

$\left(T_{n}\right)_{n\geq 1}$, $\left(g_{n}\right)_{n\geq 1}$ and
$\left(\xi_{n}\right)_{n\geq 1}$ are three independent sequences of random
variables such that

  * •

$T_{n}$ is the $n$th arrival time of a Poisson process with intensity 1;

  * •

$g_{n}$, $n\geq 1$, are i.d.d. real-valued centered Gaussian random variables;

  * •

$\xi_{n}$, $n\geq 1$, are i.d.d. real-valued random variables with common
distribution $\mu$.

Under Assumption 3.1 and for $\alpha\in(0,2)$, according to Equation (3.11.2)
in [44, Section 3.9.1], for any function $f\in L^{\alpha}(K,\mathcal{K},\mu)$,
the series

$S(f)=D_{\alpha}\sum_{n=1}^{+\infty}T_{n}^{-{1/\alpha}}f\left(\xi_{n}\right)g_{n},$
(3.11)

where

$D_{\alpha}=\left(\mathbb{E}\left(\left|g_{1}\right|^{\alpha}\right)\int_{0}^{+\infty}x^{-\alpha}\sin(x)\,dx\right)^{-{1}/{\alpha}},$
(3.12)

converges almost surely and

$W_{K,\alpha}(f)\stackrel{{\scriptstyle(d)}}{{=}}S(f).$ (3.13)

Note also that, conditionally to $(T_{n},\xi_{n})_{n\geq 1}$, $S(f)$ is a
centered real-valued Gaussian random variable with variance

$\sigma^{2}(f)=D_{\alpha}^{2}\,\mathbb{E}\left(g_{1}^{2}\right)\,\sum_{n=1}^{+\infty}T_{n}^{-{2/\alpha}}\left|f\left(\xi_{n}\right)\right|^{2}.$

###  3.2 Definition of the Neumann Fractional Stable Field

We first introduce the Neumman fractional stable field as a random variable
which takes its values in the space $\mathcal{S}^{\prime}(K)$ of tempered
distributions associated with the space $\mathcal{S}(K)$ of test functions.
Next theorem, which states its existence, extends Theorem 2.12 of [13]
considering the Neumann Laplacian and stable random variables.

###### Theorem 3.2

Let $s\geq 0$ and $\alpha\in(0,2]$. There exists a probability measure
$\nu_{s,\alpha}$ on $\mathcal{S}^{\prime}(K)$ such that for every
$f\in\mathcal{S}(K)$

$\int_{\mathcal{S}^{\prime}(K)}e^{iX(f)}\nu_{s,\alpha}(dX)=\textup{e}^{-\|(-\Delta)^{-s}f\|_{L^{\alpha}(K,\mu)}^{\alpha}}.$

Proof. By the Bochner-Minlos theorem in nuclear spaces (see [46], Theorem A)
it is enough to prove that the functional

$\varphi:f\to\exp\left(-\int_{K}|(-\Delta)^{-s}f|^{\alpha}d\mu\right),$

which is defined on $\mathcal{S}(K)$ since
$(-\Delta)^{-s}f\in\mathcal{S}(K)\subset\mathcal{C}(K)$ for
$f\in\mathcal{S}(K)$, is continuous at $0$ and positive definite. Let us first
prove it is positive definite. Let $a_{1},\cdots,a_{n}\in\mathbb{C}$ and
$f_{1},\cdots,f_{n}\in\mathcal{S}(K)$. We have

$\displaystyle\sum_{i,j=1}^{n}a_{i}\overline{a_{j}}\varphi(f_{i}-f_{j})=$
$\displaystyle\sum_{i,j=1}^{n}a_{i}\overline{a_{j}}\exp\left(-\int_{K}|(-\Delta)^{-s}f_{i}-(-\Delta)^{-s}f_{j}|^{\alpha}d\mu\right)$
$\displaystyle=$
$\displaystyle\sum_{i,j=1}^{n}a_{i}\overline{a_{j}}\mathbb{E}\left(\exp\left(iW_{K,\alpha}\left[(-\Delta)^{-s}f_{i}-(-\Delta)^{-s}f_{j})\right]\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i,j=1}^{n}a_{i}\overline{a_{j}}\mathbb{E}\left(\exp\left(iW_{K,\alpha}\left[(-\Delta)^{-s}f_{i}\right]\right)\exp\left(-iW_{K,\alpha}\left[(-\Delta)^{-s}f_{j}\right]\right)\right)$
$\displaystyle=$
$\displaystyle\left|\sum_{i=1}^{n}a_{i}\mathbb{E}\left(\exp\left(iW_{K,\alpha}\left[(-\Delta)^{-s}f_{i}\right]\right)\right)\
\right|^{2}\geq 0.$

Next, we prove that $\varphi$ is continuous at 0. Since $\mu$ is a probability
measure and $0<\alpha\leq 2$, for every $f\in\mathcal{S}(K)\subset
L^{2}(K,\mu)$,

$\left(\int_{K}|(-\Delta)^{-s}f|^{\alpha}d\mu\right)^{2/\alpha}\leq\int_{K}|(-\Delta)^{-s}f|^{2}d\mu\leq\lambda_{1}^{-2s}\int_{K}f^{2}d\mu$

where the last inequality follows from Definition 2.2. Since the convergence
in $\mathcal{S}(K)$ implies the convergence in $L^{2}(K,\mu)$, we conclude
that $\varphi$ is indeed continuous at $0$. $\Box$

The previous result yields for $s\geq 0$ and $\alpha\in(0,2]$ the existence of
a random variable $X_{s,\alpha}$ on $\mathcal{S}^{\prime}(K)$ such that for
every $f\in\mathcal{S}(K)$

$\mathbb{E}\left(e^{iX_{s,\alpha}(f)}\right)=\textup{e}^{-\|(-\Delta)^{-s}f\|_{L^{\alpha}(K,\mu)}^{\alpha}}.$

On the other hand, since $(-\Delta)^{-s}:\mathcal{S}(K)\to\mathcal{S}(K)$, one
can define the operator
$(-\Delta)^{-s}:\mathcal{S}^{\prime}(K)\to\mathcal{S}^{\prime}(K)$ thanks to
the duality formula

$((-\Delta)^{-s}X)(f)=X((-\Delta)^{-s}f),\quad
X\in\mathcal{S}^{\prime}(K),f\in\mathcal{S}(K)$

In distribution, we have then

$(-\Delta)^{-s^{\prime}}X_{s,\alpha}=X_{s+s^{\prime},\alpha}.$

Despite the previous nice interpretations as random variables in
$\mathcal{S}^{\prime}(K)$, in this paper we will work with the following
concrete and more constructive realization of the fractional stable random
field as a stochastic integral; we a priori lose the a.s. continuity in the
$\mathcal{S}(K)$ topology but this does not play any role in the sequel.

Similarly to [13] we consider the Sobolev space $H^{-s}(K)$ which is defined
as the completion of $\mathcal{S}(K)$ with respect to the norm
$\|f\|_{H^{-s}(K)}=\|(-\Delta)^{-s}f\|_{L^{2}(K,\mu)}$. This Sobolev space
does not depend on the choice of the basis since it is also the completion of
$L^{2}_{0}$ with respect to that norm.

###### Definition 3.3

Let $s\geq 0$ and $\alpha\in(0,2]$. The Neumann fractional $\alpha$-stable
field $\left\\{X_{s}(f),f\in H^{-s}(K)\right\\}$ is defined as

$X_{s,\alpha}(f)=W_{K,\alpha}((-\Delta)^{-s}f)=\int_{K}(-\Delta)^{-s}f(z)\,W_{K,\alpha}(dz).$
(3.14)

For $s=0$, by convention, $(-\Delta)^{-s}f=f$. Note that the field
$X_{s,\alpha}$ is well-defined since for $f\in H^{-s}(K)$, $(-\Delta)^{-s}f\in
L^{2}\subset L^{\alpha}$.

###### Remark 3.4

It follows from Definition 3.3 that for every $f\in H^{-s}(K)$,

$\mathbb{E}\left(\textup{e}^{iX_{s,\alpha}(f)}\right)=\textup{e}^{-\|(-\Delta)^{-s}f\|_{L^{\alpha}(K,\mu)}^{\alpha}}.$

As in [13] which deals with the case $\alpha=2$, this property could be used
as a definition of a Neumann fractional $\alpha$-stable field. However, it
will be more convenient here to work with the concrete representation (3.14).

###  3.3 Density field

As in [14, 13], let us now study the existence of a density with respect to
the Hausdorff measure $\mu$ of the Neumann fractional stable field
$X_{s,\alpha}$. We only consider the case $\alpha<2$ since the case $\alpha=2$
has already been considered in [14], even though next result is more precise
(see Remark 3.6 below).

###### Theorem 3.5

Let $\alpha\in(0,2)$.

  1. 1.

If $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$, then for every
$f\in\mathcal{S}(K)$, we have a.s.

$\int_{K}f(x)\widetilde{X}_{s,\alpha}(x)\mu(dx)=X_{s,\alpha}(f)$

with $\widetilde{X}_{s,\alpha}=\left(\widetilde{X}_{s,\alpha}(x)\right)_{x\in
K}$ the fractional $\alpha$-stable field well-defined by

$\widetilde{X}_{s,\alpha}(x)=\int_{K}G_{s}(x,z)W_{K,\alpha}(dz),\ x\in K.$
(3.15)

  2. 2.

Moreover, if there exists a random field $(Y(x))_{x\in K}$ such that a.s

$\int_{K}|Y(x)|\mu(dx)<+\infty$

and such that for every $f\in\mathcal{S}(K)$ we have a.s.

$\int_{K}f(x)Y(x)\mu(dx)=X_{s}(f),$

then we have $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$ and
for a.e. $x\in K$ with probability one
$Y(x)=\widetilde{X}_{s,\alpha}(x)+\int_{K}Y(z)d\mu(z)$.

###### Remark 3.6

  1. 1.

When $\alpha=2$, the random field $\widetilde{X}_{s,\alpha}$ is still well-
defined by (3.15) when $s>\frac{d_{h}}{2d_{w}}$ and is the fractional Brownian
motion of index $H=sd_{\omega}-\frac{d_{h}}{2}$ introduced in [14]. Moreover
according Theorem 1.1 in [14], it is also the density field of $X_{s,2}$ when
$s<1-\frac{d_{h}}{2d_{w}}$, and this can still be proved for any
$s>\frac{d_{h}}{2d_{w}}$ as done in [13] for the Dirichlet Laplacian.

  2. 2.

In the Euclidean framework, that is when we considered the Laplacian on
$\mathbb{R}^{d}$, the heat kernel is given by

$p_{t}(x,y)=\frac{1}{(4\pi
t)^{d/2}}\exp\left(-\frac{\left\|x-y\right\|^{2}}{4t}\right),\quad
t>0,\,x\in\mathbb{R}^{d},\,y\in\mathbb{R}^{d}.$

In this case, due to the behavior of the heat kernel $p_{t}$, the Riesz kernel
$G_{s}$ cannot be defined by Equation (2.8) for any real $s$. Following e.g.
[29] which deals with fractional Brownian fields (so $\alpha=2$) indexed by
manifolds, we can define it by

$G_{s}(x,y)=\frac{1}{\Gamma(s)}\int_{0}^{+\infty}t^{s-1}\left(p_{t}(x,y)-p_{t}(0,y)\right)\,dt,\quad
x\neq y,$

as soon as $s\in(-\infty,d/2+1)$. In addition, for $x\neq y$ and $y\neq 0$,

$G_{s}(x,y)=\left\\{\begin{array}[]{ll}b_{s}\left(\|x-y\|^{2s-d}-\|y\|^{2s-d}\right)&\textrm{
when $s\neq\frac{d}{2}$}\\\\[10.0pt]
b_{s}\left(\log\|y\|-\log\|x-y\|\right)&\textrm{ when
$s=\frac{d}{2}$}\end{array}\right.$

with

$b_{s}=\left\\{\begin{array}[]{ll}\displaystyle\frac{\Gamma\left(\frac{d}{2}+1-s\right)}{4^{s}\pi^{\frac{d}{2}}\left(\frac{d}{2}-s\right)\Gamma(s)}&\textrm{
when $s\neq\frac{d}{2}$}\\\\[15.0pt]
\displaystyle\frac{2}{\left(4\pi\right)^{\frac{d}{2}}\Gamma(s)}&\textrm{ when
$s=\frac{d}{2}$}.\end{array}\right.$

Then replacing in Equation (3.15), $K$ by $\mathbb{R}^{d}$ and $W_{K,\alpha}$
by $W_{\alpha}$ a real symmetric $\alpha$-stable random measure with control
measure the Lebesgue measure $\mu$, the random field
$\widetilde{X}_{s,\alpha}$ is well-defined when
$s\in\left(\frac{d}{2}-\frac{d}{2\alpha},\frac{d}{2}-\frac{d}{2\alpha}+\frac{1}{2}\right)$.
Note that in this case, $\widetilde{X}_{s,\alpha}(0)=0$. Moreover, when
$\alpha=2$, the random field $\widetilde{X}_{s,\alpha}$ is the fractional
Brownian field of Hurst index $H=2s-d/2$; when $\alpha<2$, it is a fractional
moving average stable field of index
$H=2s-d+\frac{d}{\alpha}\neq\frac{d}{\alpha}$ when $s\neq d/2$ and a log-
fractional stable field when $s=d/2$, see e.g. [44].

Before turning to the proof we start with a preliminary lemma.

###### Lemma 3.7

Let $\alpha\in(0,2)$ and $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha
d_{w}},0\right)$. Then $\widetilde{X}_{s,\alpha}$ is well-defined by (3.15)
and a.s.

$\int_{K}\left|\widetilde{X}_{s,\alpha}(x)\right|\mu(dx)<+\infty$

and

$\int_{K}\widetilde{X}_{s,\alpha}(x)\mu(dx)=0.$

Proof. Let $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$. Then
by Lemma 2.9, the Riesz kernel $z\mapsto G_{s}(x,z)$ is in
$L^{\alpha}(K,\mathcal{K},\mu)$ and so $\widetilde{X}_{s}$ is well-defined by
(3.15).

Let us now assume that $\alpha>1$. By Lemma 2.9 applied with $p=\alpha$,

$C_{1}=\sup_{x\in K}\int_{K}\left|G_{s}(x,y)\right|^{\alpha}\mu(dy)<+\infty$

and then

$\int_{K}\left(\int_{K}\left|G_{s}(x,y)\right|^{\alpha}\mu(dy)\right)^{1/\alpha}\\!\\!\mu(dx)<C_{1}^{1/\alpha}\mu(K)=C_{1}^{1/\alpha}<+\infty.$

Hence, by Theorem 11.3.2 in [44], since $\mu$ is a $\sigma$-finite measure on
$K$ and since $\alpha>1$, a.s.

$\int_{K}\left|\widetilde{X}_{s,\alpha}(x)\right|\mu(dx)<+\infty.$

Let us now assume that $\alpha\in(0,1]$. Then
$\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)=0$ and so $s>0$. We
can choose $\varepsilon>0$ (depending on $s$) small enough such that
$s>\frac{\varepsilon d_{h}}{(1+\varepsilon)d_{w}}>0$. Then applying Lemma 2.9
with $p=1+\varepsilon$,

$C_{2}=\sup_{x\in
K}\int_{K}\left|G_{s}(x,y)\right|^{1+\varepsilon}\mu(dy)<+\infty.$

Moreover by symmetry of $G_{s}$,

$C_{2}=\sup_{y\in K}\int_{K}\left|G_{s}(x,y)\right|^{1+\varepsilon}\mu(dx)$

and then

$\int_{K}\left(\int_{K}\left|G_{s}(x,y)\right|^{1+\varepsilon}\mu(dx)\right)^{\alpha/(1+\varepsilon)}\mu(dy)\leq
C_{2}^{\alpha/(1+\varepsilon)}\mu(K)=C_{2}^{\alpha/(1+\varepsilon)}<+\infty.$

Hence by Theorem 11.3.2 in [44], since $\mu$ is a $\sigma$-finite measure on
$K$ and since $1+\varepsilon>\alpha$, a.s.

$\int_{K}\left|\widetilde{X}_{s,\alpha}(x)\right|^{1+\varepsilon}\mu(dx)<+\infty.$

This implies that a.s.

$\int_{K}\left|\widetilde{X}_{s,\alpha}(x)\right|\mu(dx)<+\infty$

since $\mu$ is a probability. Then, applying Theorem 11.4.1 in [44], that is a
stochastic Fubini theorem for stable random fields, it follows that a.s.

$\int_{K}\widetilde{X}_{s,\alpha}(x)\mu(dx)=\int_{K}\left(\int_{K}G_{s}(x,z)\mu(dx)\right)W_{K,\alpha}(dz)=0$

since for every $z\in K$

$\int_{K}G_{s}(x,z)\mu(dx)=0.$

$\Box$

Proof. Let $f\in\mathcal{S}(K)$. Then $f$ is continuous on $K$ and bounded on
the compact $K$. Hence

$\int_{K}\left|f(x)\widetilde{X}_{s,\alpha}(x)\right|\mu(dx)\leq
C\int_{K}\left|\widetilde{X}_{s,\alpha}(x)\right|\mu(dx)<+\infty\quad\textrm{a.s.}$

by Lemma 3.7. Therefore applying Theorem 11.4.1 in [44], that is a stochastic
Fubini theorem, and Lemma 2.6, almost surely,

$\displaystyle\int_{K}f(x)\widetilde{X}_{s,\alpha}(x)\mu(dx)=$
$\displaystyle\int_{K}f(x)\left(\int_{K}G_{s}(x,y)W_{K,\alpha}(dy)\right)\mu(dx)$
$\displaystyle=$
$\displaystyle\int_{K}\left(\int_{K}G_{s}(x,y)f(x)d\mu(x)\right)W_{K,\alpha}(dy)$
$\displaystyle=$ $\displaystyle\int_{K}(-\Delta)^{-s}f(y)W_{K,\alpha}(dy).$

Next, assume that there exists a random field $(Y(x))_{x\in K}$ such that a.s

$\int_{K}|Y(x)|\mu(dx)<+\infty$

and such that for every $f\in\mathcal{S}(K)$ we have a.s.

$\int_{K}f(x)Y(x)\mu(dx)=X_{s}(f).$

We can assume $\int_{K}Y(x)\mu(dx)=0$, otherwise we could consider
$Y(x)-\int_{K}Y(x)\mu(dx)$. Then, we have

$\mathbb{E}\left(e^{iu\int_{K}f(x)Y(x)\mu(dx)}\right)=\textup{e}^{-\left|u\right|^{\alpha}\|(-\Delta)^{-s}f\|_{\alpha}^{\alpha}}.$

We use this equality with $f=p_{t}(y,\cdot)-1\in\mathcal{S}(K)$. This yields

$\displaystyle\mathbb{E}\left(e^{iu\int_{K}p_{t}(y,x)Y(x)\mu(dx)}\right)=\textup{e}^{-\left|u\right|^{\alpha}\|(-\Delta)^{-s}(p_{t}(y,\cdot)-1)\|_{\alpha}^{\alpha}}$
(3.16)

If $s\leq\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$, then
$\|(-\Delta)^{-s}(p_{t}(y,\cdot)-1)\|_{\alpha}\to+\infty$ when $t\to 0$.
Indeed, applying Lemma 2.6, we first note that

$(-\Delta)^{-s}(p_{t}(y,\cdot)-1)(x)=\int_{K}G_{s}(x,z)(p_{t}(y,z)-1)\mu(dz)=\int_{K}G_{s}(x,z)p_{t}(y,z)\mu(dz).$

Then, we have the estimates for $x,y\in K$, $x\neq y$,

$G_{s}(x,y)\geq C_{1}d(x,y)^{sd_{w}-d_{h}}-C_{2},$

by Proposition 2.7, and by (2.4),

$p_{t}(y,z)\geq
c_{1}t^{-d_{h}/d_{w}}\exp\biggl{(}-c_{2}\Bigl{(}\frac{d(y,z)^{d_{w}}}{t}\Bigr{)}^{\frac{1}{d_{w}-1}}\biggr{)}$

which imply

$\displaystyle\int_{K}G_{s}(x,z)p_{t}(y,z)d\mu(z)$ $\displaystyle\geq
c_{3}t^{-d_{h}/d_{w}}\int_{K}\exp\biggl{(}-c_{2}\Bigl{(}\frac{d(y,z)^{d_{w}}}{t}\Bigr{)}^{\frac{1}{d_{w}-1}}\biggr{)}d(x,z)^{sd_{w}-d_{h}}\mu(dz)-c_{4}$
$\displaystyle\geq
c_{5}t^{-d_{h}/d_{w}}\int_{B(y,t^{1/d_{w}})}d(x,z)^{sd_{w}-d_{h}}\mu(dz)-c_{4}$
$\displaystyle\geq c_{6}(d(x,y)+t^{1/d_{w}})^{sd_{w}-d_{h}}-c_{4},$

where the last inequality follows from the Ahlfors property (2.2) and the
triangle inequality since $sd_{w}-d_{h}<0$. We then see that for
$s\leq\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$

$\lim_{t\to
0}\int_{K}(d(x,y)+t^{1/d_{w}})^{\alpha(sd_{w}-d_{h})}\mu(dx)=+\infty.$

Therefore, if $s\leq\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$,
then $\|(-\Delta)^{-s}(p_{t}(y,\cdot)-1)\|_{\alpha}\to+\infty$ when $t\to 0$.
From (3.16) we deduce that for $s\leq\max\left(\frac{(\alpha-1)d_{h}}{\alpha
d_{w}},0\right)$ we have for every $y\in K$

$\lim_{t\to 0}\mathbb{E}\left(e^{iu\int_{K}p_{t}(y,x)Y(x)\mu(dx)}\right)=0.$

This is not possible because a.s, when $t\to 0$,
$y\to\int_{K}p_{t}(y,x)Y(x)\mu(dx)$ converges in $L^{1}(K,\mu)$ to $Y$, see
[12, Page 114], since we have a.s.

$\int_{K}|Y(x)|\mu(dx)<+\infty.$

We conclude that $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$.
In that case, from the first part of the proof we deduce then that for every
$f\in\mathcal{S}(K)$, we have a.s.

$\int_{K}f(x)Y(x)\mu(dx)=X_{s,\alpha}(f)=\int_{K}f(x)\widetilde{X}_{s,\alpha}(x)\mu(dx).$

This implies that we have a.s. for a.e. $x\in K$

$Y(x)=\widetilde{X}_{s,\alpha}(x)$

since as before we can use $f=p_{t}(x,\cdot)-1$ and let $t\to 0.$ $\Box$

When $\alpha<2$, the study of the sample path properties of the pointwise
Neumann $fractional\alpha$-stable random field $\widetilde{X}_{s,\alpha}$ is
based on its representation as a conditionally Gaussian LePage series. It
follows from the series representation (3.11) of any stochastic integral
$W_{K,\alpha}(f)$ and Equation (3.13).

###### Proposition 3.8

Let $\alpha\in(0,2)$ and $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha
d_{w}},0\right)$. Assume that Assumption 3.1 is fulfilled and let us consider
the Neumann $\alpha$-stable fractional field $\widetilde{X}_{s,\alpha}$
defined by (3.15). Then for any $x\in K$, the series

$Y_{s,\alpha}(x)=S\left(G_{s}\left(x,\cdot\right)\right)=D_{\alpha}\sum_{n=1}^{+\infty}T_{n}^{-{1/\alpha}}G_{s}\left(x,\xi_{n}\right)g_{n},$
(3.17)

with $D_{\alpha}$ given by (3.12), converges almost surely and

$\left(\widetilde{X}_{s,\alpha}(x)\right)_{x\in
K}\stackrel{{\scriptstyle(fdd)}}{{=}}\left(Y_{s,\alpha}(x)\right)_{x\in K}.$

Then conditionally to $(T_{n},\xi_{n})_{n\geq 1}$, $Y_{s,\alpha}$ is a
centered real-valued Gaussian random field and the increment
$Y_{s,\alpha}(x)-Y_{s,\alpha}(y)$ is a centered real-valued Gaussian variable
with variance

$s_{\alpha}(x,y)^{2}=D_{\alpha}^{2}\,\mathbb{E}\left(g_{1}^{2}\right)\sum_{n=1}^{+\infty}T_{n}^{-2/\alpha}\left|G_{s}(x,\xi_{n})-G_{s}(y,\xi_{n})\right|^{2}.$
(3.18)

###  3.4 Sample paths regularity

In this section, we are now interested in the sample path smoothness of the
Neumann fractional $\alpha$-stable random field $X_{s,\alpha}$ on $K$. This
extend the sample path study done in the Gaussian framework in [14],
considering the whole range of parameter $s$. The proof is not based on the
control of the variance of the increments, since the stable field
$X_{s,\alpha}$ is not square integrable but as in several studies about stable
random fields (see e.g. [36, 21, 23, 22]), one main ingredient is the LePage
series representation stated in Proposition 3.8.

###### Theorem 3.9

Let $\alpha\in(0,2)$, $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha
d_{w}},0\right)$ and let $\widetilde{X}_{s,\alpha}$ be the Neumann fractional
$\alpha$-stable field. Let us set $\eta_{s}=\min(s,1)d_{w}-d_{h}$ and

$\beta_{s}=\left\\{\begin{array}[]{ll}0&\textrm{si $s<1$}\\\ 1&\textrm{si
$s\geq 1$.}\end{array}\right.$

  1. 1.

If $s>\frac{d_{h}}{d_{w}}$, then there exists a modification
$\widetilde{X}_{s,\alpha}^{*}$ of $\widetilde{X}_{s,\alpha}$ such that

$\lim_{\delta\to 0}\underset{\underset{x,y\in K}{\scriptsize
0<d(x,y)}\leq\delta}{\sup}\
\frac{\left|\widetilde{X}_{s,\alpha}^{*}(x)-\widetilde{X}_{s,\alpha}^{*}(y)\right|}{d(x,y)^{\eta_{s}}|\ln
d(x,y)|^{\beta_{s}+\frac{1}{2}}}<\infty.$ (3.19)

  2. 2.

If $s\leq\frac{d_{h}}{d_{w}}$, a.s., the sample paths of
$\widetilde{X}_{s,\alpha}$ are unbounded on the compact $K$.

###### Remark 3.10

  1. 1.

For $\alpha=2$, i.e. for the Neumann fractional Gaussian field
$\widetilde{X}_{s,2}$, Equation (3.19) has been stated in Theorem 3.8 of [14]
when $\frac{d_{h}}{2d_{w}}<s<1-\frac{d_{h}}{2d_{w}}$ and can be extended to
any $s>\frac{d_{h}}{2d_{w}}$ following [13] which deals with the Dirichlet
fractional Laplacian.

  2. 2.

Theorem 3.8 extends the known results for moving average and log-fractional
stable fields indexed by the Euclidean space $\mathbb{R}^{d}$. Actually, an
analogous of the upper bound (3.19) has been established in [36] when $d=1$,
$1<\alpha<2$ and $1/\alpha<H<1$ (which corresponds to
$\frac{1}{2}<s<1-\frac{1}{2\alpha}$). In all the other cases, moving average
fractional stable random fields have almost surely unbounded sample paths on
any compact subset of $\mathbb{R}^{d}$ and so have log-fractional stable
random fields, see [44, 20].

  3. 3.

As in the Euclidean framework, contrary to Neumann fractional Gaussian random
fields ($\alpha=2$), when $\alpha<2$, the Neumann fractional stable random
field $\widetilde{X}_{s,\alpha}$ does not always have continuous sample paths.
Note also that the restriction of any Euclidean moving average or log-
fractional stable field is unbounded almost surely on the Sierpiński gasket
$K$ whereas the Neumann fractional stable field $\widetilde{X}_{s,\alpha}$ has
Hölderian sample paths on $K$ as soon as $s>\frac{d_{h}}{d_{w}}$.

We divide the proof in several steps. We first assume that $s>d_{h}/d_{w}$ and
first study the sample paths regularity of the series $Y_{s,\alpha}$, which
has the same finite distributions as $\widetilde{X}_{s,\alpha}$. To do so, we
first consider the partial sum $Y_{s,\alpha,N}$ defined by

$Y_{s,\alpha,N}(x)=D_{\alpha}\sum_{n=1}^{N}T_{n}^{-1/\alpha}G_{s}(x,\xi_{n})g_{n},\quad
x\in K.$

Note that since $s>d_{h}/d_{w}$, by Theorem 2.11, each $G_{s}(\cdot,\xi_{n})$
is continuous on $K$ and then $Y_{s,\alpha,N}$ has continuous sample paths on
$K$. Moreover, conditionally to $(T_{n},\xi_{n})_{n\geq 1}$, for any integer
$N\geq 1$, $Y_{s,\alpha,N}$ is a centered real-valued Gaussian random field
and the increment $Y_{s,\alpha,N}(x)-Y_{s,\alpha,N}(y)$ is a centered real-
valued Gaussian variable with variance

$s_{\alpha,N}(x,y)^{2}=D_{\alpha}^{2}\,\mathbb{E}\left(g_{1}^{2}\right)\sum_{n=1}^{N}T_{n}^{-2/\alpha}\left|G_{s}(x,\xi_{n})-G_{s}(y,\xi_{n})\right|^{2}\leq
s_{\alpha}(x,y)^{2},$

where we recall that $s_{\alpha}(x,y)^{2}$ is given by (3.18).

###### Proposition 3.11

Let $\alpha\in(0,2)$ and $s>\frac{d_{h}}{d_{w}}$. Let $\eta_{s}$ and
$\beta_{s}$ be defined as in Theorem 3.9.

  1. 1.

There exist $h_{s}>0$ and a finite positive random variable $C$ such that a.s
for any $x,y\in K$ such that $d(x,y)\leq h_{s}$,

$\sup_{N\geq 1}\left|Y_{s,\alpha,N}(x)-Y_{s,\alpha,N}(y)\right|\leq
Cd(x,y)^{\eta_{s}}(-\ln d(x,y))^{\beta_{s}+\frac{1}{2}}.$ (3.20)

  2. 2.

Moreover, almost surely, the series $(Y_{s,\alpha,N})_{N\geq 1}$ converges
uniformly on $K$ to $Y_{s,\alpha}$, and $Y_{s}$ also satisfies (3.20), that is
a.s for any $x,y\in K$ such that $d(x,y)\leq h_{s}$,

$\sup_{N\geq 1}\left|Y_{s,\alpha}(x)-Y_{s,\alpha}(y)\right|\leq
Cd(x,y)^{\eta_{s}}(-\ln d(x,y))^{\beta_{s}+\frac{1}{2}}.$ (3.21)

The proof relies on two main ingredients. The following lemma which gives an
upper bound for the conditional variance $s_{N}^{2}(x,y)$, uniformly in $N$,
and the Garsia-Rodemich-Rumsey inequality.

###### Lemma 3.12

Let $\alpha\in(0,2)$ and $s>\frac{d_{h}}{d_{w}}$. Then there exists a finite
positive random variable $A$ such that a.s for any $x,y\in K$,

$\sup_{N\geq 1}s_{\alpha,N}(x,y)=s_{\alpha}(x,y)\leq
Ad(x,y)^{\eta_{s}}\max\left(\left|\ln d(x,y)\right|^{\beta_{s}},1\right)$

with $\eta_{s}$ and $\beta_{s}$ given in Theorem 3.9.

Proof. As a direct consequence of Theorem 2.11, by definition of $\eta_{s}$
and $\beta_{s}$,

$\sup_{N\geq 1}s_{\alpha,N}(x,y)^{2}=s_{\alpha}(x,y)^{2}\leq
A^{2}d(x,y)^{2\eta_{s}}\max\left(\left|\ln
d(x,y)\right|^{\beta_{s}},1\right)^{2}$

where $A^{2}=c\sum_{n=1}^{+\infty}T_{n}^{-2/\alpha}$, with $c$ a positive
constant, converges almost surely since $2/\alpha>1$ (see e.g. Theorem 1.4.5
in [44]). $\Box$

Let us now recall the Garsia-Rodemich-Rumsey inequality for fractals,
established by Barlow and Perkins in lemma 6.1 of [11].

###### Lemma 3.13

Let $p$ be an increasing continuous function on $[0,\infty)$ such that
$p(0)=0,$ and $\psi:\mathbb{R}\rightarrow\mathbb{R}_{+}$ be a non-negative
symmetric continuous convex function such that
$\lim_{u\to\infty}\psi(u)=\infty$. Let $f:K\rightarrow\mathbb{R}$ be a
measurable function such that

$\Gamma=\int_{K\times
K}\psi\left(\frac{\left|f(x)-f(y)\right|}{p(d(x,y))}\right)\mu(dx)\mu(dy)<\infty.$

Then there exists a constant $c_{K}$ depending only on $d_{h}$ such that

$\left|f(x)-f(y)\right|\leq
8\int_{0}^{d(x,y)}\psi^{-1}\left(\frac{c_{K}\Gamma}{u^{2d_{h}}}\right)\,p({\rm
d}u)$ (3.22)

for $\mu\times\mu$-almost all $(x,y)\in K\times K$. Moreover if $f$ is
continuous on $K$, then (3.22) holds for any $(x,y)\in K\times K$.

In the sequel, we choose

$\psi(u)=\exp\left(\frac{u^{2}}{4}\right)-1.$ (3.23)

The choice of $p=p_{s}(d(x,y))$ is a slight modification of the upper bound
stated in Lemma 3.12 in order to obtain an increasing function. Let us first
note that since $\eta_{s}>0$, there exists $h_{s}\in(0,1/\textup{e}]$ small
enough such that the function $p_{s}$ defined by

$p_{s}(h)=\left\\{\begin{array}[]{ll}0&\textrm{if $h=0$}\\\
h^{\eta_{s}}\max\left(\left|\ln
h\right|^{\beta_{s}},1\right)=h^{\eta_{s}}\left(-\ln
h\right)^{\beta_{s}}&\textrm{if $h\in(0,h_{s}]$}\\\ \end{array}\right.$

is a continuous and increasing on $(0,h_{s}]$. As a consequence, there exist
some positive constants $d_{1},d_{2}$ such that for any $h\in(0,1],$

$d_{1}p_{s}(h)\leq h^{\eta_{s}}\max\left(\left|\ln
h\right|^{\beta_{s}},1\right)\leq d_{2}p_{s}(h).$ (3.24)

Then we will apply Garsia-Rodemich-Rumsey inequality choosing
$f=Y_{s,N}/(Ad_{2})$ with $A$ the positive random variable introduced in Lemma
3.12. This leads to study

$\Gamma_{N}=\int_{K\times
K}\psi\left(\frac{\left|Y_{s,\alpha,N}(x)-Y_{s,\alpha,N}(y)\right|}{Ad_{2}\,p_{s}(d(x,y))}\right)\mu(dx)\mu(dy).$

###### Lemma 3.14

Let $\alpha\in(0,2)$ and $s>\frac{d_{h}}{d_{w}}$. Then, with the previous
notations, $\mathbb{E}\left(\sup_{N\geq 1}\Gamma_{N}\right)<+\infty.$

Proof. For any integer $M\geq 1$,

$\sup_{1\leq N\leq M}\Gamma_{N}\leq\int_{K\times K}\psi\left(\sup_{1\leq N\leq
M}\frac{\left|Y_{s,\alpha,N}(x)-Y_{s,\alpha,N}(y)\right|}{{A}d_{2}p_{s}(d(x,y))}\right)\mu(dx)\mu(dy)$

since the function $\psi$ given by (3.23) a non-decreasing function. Then, by
Fubini-Tonelli theorem,

$\mathbb{E}\left(\sup_{1\leq N\leq M}\Gamma_{N}\right)\leq\int_{K\times
K}I_{M}(x,y)\mu(dx)\mu(dy)$

where for $x\neq y$,

$I_{M}(x,y)=\mathbb{E}\left(\psi\left(\sup_{1\leq N\leq
M}\frac{\left|Y_{s,\alpha,N}(x)-Y_{s,\alpha,N}(y)\right|}{{A}d_{2}\,p_{s}(d(x,y))}\right)\right).$

Moreover, since $\psi$ is non-negative,

$I_{M}(x,y)=\displaystyle\int_{0}^{+\infty}\mathbb{P}\left(\psi\left(\sup_{1\leq
N\leq
M}\frac{\left|Y_{s,\alpha,N}(x)-Y_{s,\alpha,N}(y)\right|}{{A}d_{2}p_{s}(d(x,y))}\right)>t\right)dt$

so that, by definition (3.23) of $\psi$ and a simple change of variables,

$I_{M}(x,y)=\int_{1}^{+\infty}\mathbb{P}\left(\sup_{1\leq N\leq
M}\left|Y_{s,\alpha,N}(x)-Y_{s,\alpha,N}(y)\right|>2{A}d_{2}p_{s}(d(x,y))\sqrt{\ln
t}\right)dt.$

Now by Lemma 3.12 and Equation (3.24), since $d(x,y)\leq 1$,

${A}d_{2}p_{s}(d(x,y))\geq s_{M}(x,y)$

and then

$I_{M}(x,y)=\int_{1}^{+\infty}\mathbb{P}\left(\sup_{1\leq N\leq
M}\left|Y_{s,\alpha,N}(x)-Y_{s,\alpha,N}(y)\right|>2s_{M}(x,y)\sqrt{\ln
t}\right)dt.$

Then, conditioning to $(T_{n},\xi_{n})_{n\geq 1}$ and applying Proposition 2.3
of Ledoux-Talagrand lead to

$I_{M}(x,y)\leq\int_{1}^{+\infty}\mathbb{E}\left(\mathbb{P}\left(\left.\left|Y_{s,\alpha,M}(x)-Y_{s,\alpha,M}(y)\right|>2s_{M}(x,y)\sqrt{\ln
t}\,\right|(T_{n},\xi_{n})_{n\geq 1}\right)\right)dt$

since the random variables $g_{n}$ are i.i.d. and symmetric. Therefore, since,
conditionally to $(T_{n},\xi_{n})_{n\geq 1}$,
$Y_{s,\alpha,M}(x)-Y_{s,\alpha,M}(y)$ is a centered real-valued Gaussian
variable with variance $s_{M}(x,y)^{2}$ and so

$I_{M}(x,y)\leq\int_{1}^{+\infty}\mathbb{P}\left(|Z|>2\sqrt{\ln t}\right)dt$

with $Z$ a standard normal random variable. Moreover,

$\forall\lambda>0,\
\mathbb{P}\left(|Z|>\lambda\right)\leq\sqrt{\frac{2}{\pi}}\frac{\textup{e}^{-\frac{\lambda^{2}}{2}}}{\lambda}$

and so

$I_{M}(x,y)\leq\frac{1}{\sqrt{2\pi}}\int_{1}^{+\infty}\frac{1}{t^{2}\sqrt{\ln
t}}\,dt:=c<+\infty.$

Therefore,

$\mathbb{E}\left(\sup_{1\leq N\leq M}\Gamma_{N}\right)\leq c\mu(K)^{2}=c$

and then by the monotone convergence theorem, $\mathbb{E}\left(\sup_{N\geq
1}\Gamma_{N}\right)\leq c<+\infty.$ $\Box$

We are now ready to prove Proposition 3.11.

Proof.

  1. 1.

Let $\tilde{\Omega}=\left\\{\sup_{N\geq 1}\Gamma_{N}<+\infty\right\\}$. By
Lemma 3.14, $\mathbb{P}\left(\tilde{\Omega}\right)=1$ and then it is then
sufficient to establish (3.20) for $\omega\in\tilde{\Omega}$. So let from now
$\omega\in\tilde{\Omega}$. Then by Garsia-Rodemich-Rumsey inequality (see
Lemma 3.13), for each integer $N\geq 1$,

$\forall x,y\in K,\
\left|Y_{s,\alpha,N}(x,\omega)-Y_{s,\alpha,N}(y,\omega)\right|\leq
8Ad_{2}\int_{0}^{d(x,y)}\psi^{-1}\left(\frac{c_{K}\Gamma_{N}(\omega)}{u^{2d_{h}}}\right)p_{s}(du)$

since each $f_{N}=Y_{s,\alpha,N}/(Ad_{2})$ is continuous on $K$. Then since
$\psi$ is defined by Equation (3.23),

$\forall x,y\in K,\ \sup_{N\geq
1}\left|Y_{s,\alpha,N}(x,\omega)-Y_{s,\alpha,N}(y,\omega)\right|\leq
16Ad_{2}\int_{0}^{d(x,y)}\sqrt{\ln\left(1+\frac{c_{K}\Gamma_{\infty}(\omega)}{u^{2d_{h}}}\right)}p_{s}(du)$

where $\Gamma_{\infty}(\omega)=\sup_{N\geq 1}\Gamma_{N}(\omega)<+\infty$. Note
that for any ${u\in(0,1/\textup{e}]}$

$0\leq
16Ad_{2}\sqrt{\ln\left(1+\frac{c_{K}\Gamma_{\infty}}{u^{2d_{h}}}\right)}\leq
C\sqrt{|\ln u|}$

with $C$ a finite positive random variable which does not depend on $u$.
Hence, for any $x,y$ such that $d(x,y)\leq 1/\textup{e}$,

$\sup_{N\geq 1}\left|Y_{s,N}(x,\omega)-Y_{s,N}(y,\omega)\right|\leq
C(\omega)\int_{0}^{d(x,y)}\sqrt{|\ln u|}\,p_{s}(du).$

Moreover, denoting by $c$ a constant which value may change at each line and
applying two integrations by parts, for any
$r\in(0,h_{s}]\subset(0,1/\textup{e}]$, we have:

$\begin{array}[]{rcl}\displaystyle\int_{0}^{r}\sqrt{|\ln
u|}\,p_{s}(du)&\leq&\displaystyle cr^{\eta_{s}}\left(-\ln
r\right)^{\beta_{s}+\frac{1}{2}}+cr^{\eta_{s}}\left(-\ln
r\right)^{\beta_{s}-\frac{1}{2}}+c\int_{0}^{r}u^{\eta_{s}-1}\left(-\ln
u\right)^{\beta_{s}-\frac{3}{2}}du\\\\[10.0pt] &\leq&\displaystyle
cr^{\eta_{s}}\left(-\ln
r\right)^{\beta_{s}+\frac{1}{2}}+c\int_{0}^{r}u^{\eta_{s}-1}du\\\\[10.0pt]
&\leq&\displaystyle cr^{\eta_{s}}\left(-\ln
r\right)^{\beta_{s}+\frac{1}{2}}+cr^{\eta_{s}}\\\\[10.0pt] &\leq&\displaystyle
cr^{\eta_{s}}\left(-\ln r\right)^{\beta_{s}+\frac{1}{2}}\end{array}$

since $\eta_{s}>0$ and $\beta_{s}-3/2<0$. Hence for $x,y\in K$ such that
$d(x,y)\leq h_{s}$, up to change the value of $C$,

$\displaystyle\sup_{N\geq
1}\left|Y_{s,N}(x,\omega)-Y_{s,N}(y,\omega)\right|\leq\displaystyle
C(\omega)d(x,y)^{\eta_{s}}(-\ln d(x,y))^{\beta_{s}+\frac{1}{2}},$

which concludes the proof of Assertion 1.

  2. 2.

Since $K$ is a separable set, there exists a dense countable set
$\mathcal{D}\subset K$. We now follow the fourth step of the proof of Theorem
3.1 in [22]. Let

$\Omega^{\prime}=\tilde{\Omega}\cap\bigcap_{x\in\mathcal{D}}\left\\{\lim_{N\to+\infty}Y_{s,\alpha,N}=Y_{s,\alpha}\right\\}.$

By Proposition 3.8 and since $\mathcal{D}$ is countable,
$\mathbb{P}\left(\Omega^{\prime}\right)=\mathbb{P}\left(\tilde{\Omega}\right)=1.$
Moreover, for each $\omega\in\Omega^{\prime}$, according to the proof of
Assertion 1, the real-valued sequence $(Y_{s,\alpha,N}(\cdot,\omega))_{N\geq
1}$ satisfies (3.20) and then is uniformly equicontinuous on $K$ since
$\eta_{s}>0$. Then, by Theorem I.26 and by generalizing Theorem I.27 of [42]
to the compact set $K$, for each $\omega\in\Omega^{\prime}$,
$(Y_{s,\alpha,N}(\cdot,\omega))_{N\geq 1}$ converges uniformly on $K$ toward
its limits $Y_{s,\alpha}(\cdot,\omega)$. Moreover, Equation (3.20) is
fulfilled for any $\omega\in\Omega^{\prime}$, which leads to Equation (3.21)
letting $N\to+\infty$ and concludes the proof.

$\Box$

Let us now prove Theorem 3.9.

Proof.

  1. 1.

Let us assume that $s>d_{h}/d_{w}$. We follow the same lines as the end of the
proof of Proposition 5.1 in [22]. Let us consider a dense countable set
$\mathcal{D}\subset K$ and $\Omega^{\prime\prime}:=\left\\{C<\infty\right\\}$
with

$\widetilde{C}:=\sup_{\underset{0<d(x,y)\leq h_{s}}{x,y\in
D}}\frac{\left|\widetilde{X}_{s,\alpha}(x)-\widetilde{X}_{s,\alpha}(y)\right|}{d(x,y)^{\eta_{s}}\left(-\ln
d(x,y)\right)^{\beta_{s}+\frac{1}{2}}}.$

Then, since $\widetilde{X}_{s,\alpha}$ and $Y_{s,\alpha}$ have the same finite
distributions, and since $Y_{s,\alpha}$ satisfies (3.21),
$\mathbb{P}\left(\Omega^{\prime\prime}\right)=1$ and
$\widetilde{X}_{s,\alpha}$ is stochastically continuous.

Let us now define a modification $\widetilde{X}_{s,\alpha}^{*}$ of
$\widetilde{X}_{s,\alpha}$ with Hölder sample paths. First, for
$\omega\notin\Omega^{\prime\prime}$, let
$\widetilde{X}_{s,\alpha}^{*}(u,\omega)=0$ for any $u\in K$. Then let us
consider $\omega\in\Omega^{\prime\prime}$. We set

$\forall u\in\mathcal{D},\
\widetilde{X}_{s,\alpha}^{*}(u,\omega)=\widetilde{X}_{s,\alpha}(u,\omega).$

Then, by definition of $\widetilde{C}$, for any $x,y\in\mathcal{D}$ such that
$d(x,y)\leq h_{s}$,

$\left|\widetilde{X}^{*}_{s,\alpha}(x,\omega)-\widetilde{X}^{*}_{s,\alpha}(y,\omega)\right|\leq\widetilde{C}(\omega)d(x,y)^{\eta_{s}}\left(-\ln
d(x,y)\right)^{\beta_{s}+\frac{1}{2}}.$ (3.25)

Let us now consider $u\in K$ and $(u_{n})_{n\in\mathbb{N}}$ be a sequence of
$\mathcal{D}$ such that $\lim_{n\to+\infty}u_{n}=u$.

Since $\widetilde{C}(\omega)<+\infty,$ Equation (3.25) implies that the
sequence
$\left(\widetilde{X}_{s,\alpha}^{*}(u_{n},\omega)\right)_{n\in\mathbb{N}}$ is
a real-valued Cauchy sequence and then converges. Note also that the limits
does not depend on choice of $(u_{n})_{n}$, so that we can define
$\widetilde{X}^{*}_{s,\alpha}(u,\omega)$ by

$\widetilde{X}^{*}_{s,\alpha}(u,\omega)=\lim_{n\to+\infty}\widetilde{X}_{s,\alpha}\left(u_{n},\omega\right).$

Then, by Equation (3.25), by definition of $\widetilde{X}_{s,\alpha}^{*}$ and
continuity of the distance $d$,

$\left|\widetilde{X}^{*}_{s,\alpha}(x,\omega)-\widetilde{X}^{*}_{s,\alpha}(y,\omega)\right|\leq\widetilde{C}(\omega)d(x,y)^{\eta_{s}}\left(-\ln
d(x,y)\right)^{\beta_{s}+\frac{1}{2}}$

for any $x,y\in K$ such that $d(x,y)\leq h_{s}$. This inequality also holds
for $\omega\notin\Omega^{\prime\prime}$ up to replace $\widetilde{C}(\omega)$
by $0$, so that $\widetilde{X}_{s,\alpha}^{*}$ satisfies Equation (3.19).
Moreover, by stochastic continuity of $\widetilde{X}_{s,\alpha}$ and since
$\mathbb{P}(\Omega^{\prime\prime})=1$, the random field
$\widetilde{X}_{s,\alpha}^{*}$ is a modification of
$\widetilde{X}_{s,\alpha}$, which concludes the proof of Assertion 1.

  2. 2.

Let us now assume that $s\leq d_{h}/d_{w}$ and consider $\mathcal{D}\subset K$
a countable dense subset of $K$. Then, by the lower bounds in Lemma 2.7, for
any $y\in K\backslash\mathcal{D}$

$\sup_{x\in\mathcal{D}}\left|G_{s}(x,y)\right|=+\infty$

so that, since $\mu\left(K\backslash\mathcal{D}\right)>0$,

$\int_{K}\sup_{x\in\mathcal{D}}\left|G_{s}(x,y)\right|^{\alpha}\mu(dy)=+\infty.$

Therefore, according to Theorem 10.2.3 in [44], a.s. the sample paths of
$\widetilde{X}_{s}$ are unbounded on $K$.

$\Box$

Finally, when $\alpha<1$, the upper bound (3.19) of the modulus of continuity
stated in Theorem 3.9 for $s>\frac{d_{h}}{d_{w}}$ can be improved: the
$\frac{1}{2}$ in the logarithm term can be removed. Actually, since
$\alpha<1$, the series (3.17) converges absolutely almost surely and then the
increments $Y_{s,\alpha}(x)-Y_{s,\alpha}(y)$ can be directly dominated,
without controlling the conditional variance $s^{2}(x,y)$. It is then
unnecessary to apply the Garsia-Rodemich-Rumsey inequality, which gives the
$(-\ln d(x,y))^{\frac{1}{2}}$ term in the upper bound (3.19).

###### Theorem 3.15

Let $\alpha\in(0,1)$ and $s>\frac{d_{h}}{d_{w}}$. Let
$\widetilde{X}_{s,\alpha}$ be the Neumann fractional $\alpha$-stable field,
and $\eta_{s}$ and $\beta_{s}$ be defined as in Theorem 3.9. Then there exists
a the modification $\widetilde{X}^{*}_{s,\alpha}$ of
$\widetilde{X}_{s,\alpha}$ such that

$\lim_{\delta\to 0}\underset{\underset{x,y\in K}{\scriptsize
0<d(x,y)}\leq\delta}{\sup}\
\frac{\left|\widetilde{X}_{s,\alpha}^{*}(x)-\widetilde{X}_{s,\alpha}^{*}(y)\right|}{d(x,y)^{\eta_{s}}|\ln
d(x,y)|^{\beta_{s}}}<\infty.$

Proof. For any $x,y\in K$,

$\left|Y_{s,\alpha}(x)-Y_{s,\alpha}(y)\right|\leq
D_{\alpha}\sum_{n=1}^{+\infty}T_{n}^{-1/\alpha}\left|G_{s}(x,\xi_{n})-G_{s}(y,\xi_{n})\right|\left|g_{n}\right|$

and then by Theorem 2.11, there exists a constant $c>0$ such that for any
$x,y\in K$

$\left|Y_{s,\alpha}(x)-Y_{s,\alpha}(y)\right|\leq
cd(x,y)^{\eta_{s}}\max\left(\left|\ln
d(x,y)\right|^{\beta_{s}},1\right)\sum_{n=1}^{+\infty}T_{n}^{-1/\alpha}\left|g_{n}\right|.$

In addition, since $0<\alpha<1$ and since Assumption 3.1 is fulfilled, by
Theorem 1.4.5 in [44], the series

$A=c\sum_{n=1}^{+\infty}T_{n}^{-1/\alpha}\left|g_{n}\right|$

converges almost surely to an $\alpha$-stable random variable, and in
particular $A<+\infty$ almost surely. Then the random field $Y_{s,\alpha}$
satisfies (3.21) with $C=A$. Hence Theorem 3.15 follows from the proof of
Theorem 3.9 (replacing $\beta_{s}+\frac{1}{2}$ by $\beta_{s}$). $\Box$

###  3.5 Symmetries and self-similarity properties

####  3.5.1 Invariance by reflections

Let $\sigma_{1},\sigma_{2},\sigma_{3}$ be the reflections about the lines
dividing the triangle with vertices $q_{0},q_{1},q_{2}$ into two equal parts.
The Sierpiński gasket is invariant by this reflections, and so is its
Hausdorff measure $\mu$. This leads to invariance of the random measure
$W_{K,\alpha}$ and then of the fractional stable random fields ${X}_{s}$ and
$\widetilde{X}_{s,\alpha}$. Such properties have already been established in
[14], studying the covariance function when $\alpha=2$.

###### Proposition 3.16

  1. 1.

For any $f\in L^{\alpha}(K,\mathcal{K},\mu)$, for every $i=1,2,3$,
$f\circ\sigma_{i}\in L^{\alpha}(K,\mathcal{K},\mu)$ and

$W_{K,\alpha}(f\circ\sigma_{i})\stackrel{{\scriptstyle(d)}}{{=}}W_{K,\alpha}(f).$

  2. 2.

As a consequence, for every $i=1,2,3$, and every $s>0$

$\left({X}_{s,\alpha}\left(f\circ\sigma_{i}(x)\right)\right)_{f\in
H^{-s}(K)}\stackrel{{\scriptstyle(fdd)}}{{=}}\left({X}_{s,\alpha}\left(f\right)\right)_{f\in
H^{-s}(K)}.$

  3. 3.

Moreover, if $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$, then
for every $i=1,2,3$,

$\left(\widetilde{X}_{s,\alpha}\left(\sigma_{i}(x)\right)\right)_{x\in
K}\stackrel{{\scriptstyle(fdd)}}{{=}}\left(\widetilde{X}_{s,\alpha}\left(x\right)\right)_{x\in
K}.$

Proof.

  1. 1.

Let $f\in L^{\alpha}(K,\mu)$ and $i\in\left\\{1,2,3\right\\}$. Note that by
the change of variables $y=\sigma_{i}(x)$,

$\int_{K}\left|f\circ\sigma_{i}(x)\right|^{\alpha}\,\mu(dx)=\int_{K}\left|f(y)\right|^{\alpha}\,\mu(dy)<+\infty$

since $K$ and $\mu$ are invariant by $\sigma_{i}$. In particular,
$f\circ\sigma_{i}\in L^{\alpha}(K,\mu)$. Moreover,

$\left\|W_{K,\alpha}(f\circ\sigma_{i})\right\|_{L^{\alpha}(K,\mu)}^{\alpha}=\int_{K}\left|f\circ\sigma_{i}(x)\right|^{\alpha}\,\mu(dx)=\int_{K}\left|f(y)\right|^{\alpha}\,\mu(dy)=\left\|W_{K,\alpha}(f)\right\|_{L^{\alpha}(K,\mu)}^{\alpha}.$

Hence, the two $S\alpha S$ random variables $W_{K,\alpha}(f\circ\sigma_{i})$
and $W_{K,\alpha}(f)$ have the same distribution, which concludes the proof of
the first assertion.

  2. 2.

Let $i\in\\{1,2,3\\}.$ Since $\Phi_{j}\in\mathcal{F},$ $j\geq 1$ is an
orthonormal basis of $L_{0}^{2}(K,\mu)$, using the invariance of $\mu$ by the
continuous bijective function $\sigma_{i}$, one can check that
$\Phi_{j}\circ\sigma_{i}\in\mathcal{F},$ $j\geq 1$, is also an orthonormal
basis of $L_{0}^{2}(K,\mu)$. Moreover, since the Dirichlet form $\mathcal{E}$
is invariant by $\sigma_{i}$, that is since

$\mathcal{E}\left(f\circ\sigma_{i},f\circ\sigma_{i}\right)=\mathcal{E}\left(f,f\right),$

we have:

$\Delta\left(f\circ\sigma_{i}\right)=\left(\Delta f\right)\circ\sigma_{i}.$

and then

$\Delta\left(\Phi_{j}\circ\sigma_{i}\right)=\left(\Delta\Phi_{j}\right)\circ\sigma_{i}=-\lambda_{j}\Phi_{j}\circ\sigma_{i}$

that is, as the function $\Phi_{j}$, $\Phi_{j}\circ\sigma_{i}$ is an
eigenfunction of $-\Delta$ associated to the eigenvalue $-\lambda_{j}$. By
unicity of the fractional Laplacian $(-\Delta)^{-s}$, for any $g\in
L_{0}^{2}(K,\mu)$, in the basis $\Phi_{j}\circ\sigma_{i}\in\mathcal{F},$
$j\geq 1$,

$(-\Delta)^{-s}g=\sum_{j=1}^{+\infty}\frac{1}{\lambda_{j}^{s}}\left(\int_{K}\Phi_{j}(\sigma_{i}(y))g(y)\mu(dy)\right)\Phi_{j}\circ\sigma_{i}.$

Hence, since for any $f\in L_{0}^{2}(K,\mu)$, $g=f\circ\sigma_{i}\in
L_{0}^{2}(K,\mu)$ and we have:

$(-\Delta)^{-s}(f\circ\sigma_{i})=\sum_{j=1}^{+\infty}\frac{1}{\lambda_{j}^{s}}\left(\int_{K}\Phi_{j}(\sigma_{i}(y))\,(f\circ\sigma_{i})(y)\mu(dy)\right)\Phi_{j}\circ\sigma_{i}.$

Then, applying the change of variable $x=\sigma_{i}(y)$ in the integrals and
since $\mu$ is invariant by $\sigma_{i}$, we get: for any $f\in
L_{0}^{2}(K,\mu)$,

$(-\Delta)^{-s}(f\circ\sigma_{i})=\sum_{j=1}^{+\infty}\frac{1}{\lambda_{j}^{s}}\left(\int_{K}\Phi_{j}(x)f(y)\mu(dy)\right)\Phi_{j}\circ\sigma_{i}=\left((-\Delta)^{-s}f\right)\circ\sigma_{i}$
(3.26)

Moreover from the definition of the Sobolev space $H^{-s}(K)$, it follows that
for $f\in H^{-s}(K)$, $f\circ\sigma_{i}\in H^{-s}(K)$ and
$\|f\|_{H^{-s}(K)}=\|f\circ\sigma_{i}\|_{H^{-s}(K)}$ and that Equation (3.26)
still holds (by density) for $f\in H^{-s}(K)$. Hence, for any $f\in
H^{-s}(K)$, the two random variables $X_{s,\alpha}(f)$ and
$X_{s,\alpha}(f\circ\sigma_{i})$ are well-defined and we have:

$X_{s}\left(f\circ\sigma_{i}\right)=W_{K,\alpha}\left((-\Delta)^{-s}\left(f\circ\sigma_{i}\right)\right)=W_{K,\alpha}\left(\left((-\Delta)^{-s}f\right)\circ\sigma_{i}\right)$

and then, by Assertion 1.,

$X_{s,\alpha}\left(f\circ\sigma_{i}\right)\stackrel{{\scriptstyle(d)}}{{=}}W_{K,\alpha}\left((-\Delta)^{-s}f\right),$

that is

$X_{s,\alpha}\left(f\circ\sigma_{i}\right)\stackrel{{\scriptstyle(d)}}{{=}}X_{s,\alpha}\left(f\right),$

which implies Assertion 2. by linearity of $X_{s,\alpha}$ on $H^{-s}(K)$.

  3. 3.

Let $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$ and let
$i\in\\{1,2,3\\}$. Then, for any $x\in K$,

$\widetilde{X}_{s,\alpha}\left(\sigma_{i}(x)\right)=\int_{K}G_{s}\left(\sigma_{i}(x),y\right)W_{K,\alpha}(dy).$

Hence, by linearity of the stochastic integrals with respect to $W_{K,\alpha}$
and by Assertion 1,

$\left(\widetilde{X}_{s,\alpha}\left(\sigma_{i}(x)\right)\right)_{x\in
K}\stackrel{{\scriptstyle(fdd)}}{{=}}\left(\int_{K}G_{s}\left(\sigma_{i}(x),\sigma_{i}(z)\right)W_{K,\alpha}(dz)\right)_{x\in
K}.$

Moreover, according to the proof of Proposition 3.10 in [14], for any $x,z\in
K$ such that $x\neq z$,

$G_{s}\left(\sigma_{i}(x),\sigma_{i}(z)\right)=G_{s}(x,y).$

Therefore,

$\left(\widetilde{X}_{s,\alpha}\left(\sigma_{i}(x)\right)\right)_{x\in
K}\stackrel{{\scriptstyle(fdd)}}{{=}}\left(\int_{K}G_{s}\left(x,z\right)W_{K,\alpha}(dz)\right)_{x\in
K},$

which concludes the proof.

$\Box$

####  3.5.2 Invariance by scaling

Let $w=(i_{1},\cdots,i_{n})\in\\{1,2,3\\}^{n}$, and denote

$F_{w}=F_{i_{1}}\circ\cdots\circ F_{i_{n}}$

where we recall that

$F_{i}(z)=\frac{1}{2}(z-q_{i})+q_{i}.$

The compact set $K_{w}:=F_{w}(K)\subset K$ is itself a Sierpiński gasket. We
equip $K_{w}$ with the restriction $\mu_{|_{K_{w}}}$ of the Hausdorff measure
$\mu$. Let us from now indicate with a superscript or subscript $w$ the
objects related to the Sierpiński gasket $K_{w}$ (Dirichlet form
$\mathcal{E}^{w}$, Laplacian $\Delta_{w}$, heat kernel $p^{w}_{t}(x,y)$,
etc…).

We now introduce the Neumann fractional $\alpha$-stable random field on the
smaller Sierpiński $K_{w}$ as previously done on $K$. Let us first note that
the function $F_{w}:K\rightarrow K_{w}$ is continuous and bijective and that
according to the proof of Proposition 3.10 in [14],

$\mathcal{F}^{w}=\left\\{f\circ F_{w}^{-1},\,f\in\mathcal{F}\right\\}$

and for $f\in\mathcal{F}$,

$\Delta_{w}\left(f\circ F_{w}^{-1}\right)=5^{n}\left(\Delta f\right)\circ
F_{w}^{-1}.$ (3.27)

Let us now consider the functions

$\Phi_{j}^{w}=3^{\frac{n}{2}}\,\Phi_{j}\circ F_{w}^{-1}\in\mathcal{F}^{w},\
j\geq 1.$

Since each $\Phi_{j}$, $j\geq 1$ is an orthonormal basis of
$L_{0}^{2}(K,\mu)$, from the self-similarity of the Hausdorff measure $\mu,$
it follows that $\Phi_{j}^{w}\in\mathcal{F}^{w}$, $j\geq 1$, is an orthonormal
basis of $L_{0}^{2}\left(K_{w},\mu\right)$. Moreover, since
$\Delta\Phi_{j}=-\lambda_{j}\Phi_{j}$, by Equation (3.27),

$-\Delta_{w}\Phi_{j}^{w}=\lambda_{j}^{w}\Phi_{j}^{w}$

with $\lambda_{j,w}=5^{n}\lambda_{j}$. Then, the heat kernel admits the
spectral expansion:

$p^{w}_{t}(x,y)=\frac{1}{\mu(K_{w})}+\sum_{j=1}^{+\infty}\textup{e}^{-\lambda_{j,w}}\Phi_{j}^{w}(x)\Phi_{j}^{w}(y)=3^{n}+\sum_{j=1}^{+\infty}\textup{e}^{-\lambda_{j,w}}\,\Phi_{j}^{w}(x)\Phi_{j}^{w}(y).$

We next consider the following space of test functions

$\mathcal{S}\left(K_{w}\right)=\left\\{f\in C_{0}\left(K_{w}\right),\forall
k\geq
1\lim_{n\to+\infty}n^{k}\left|\int_{K_{w}}\Phi_{n}^{w}(y)f(y)\mu(dy)\right|=0\right\\},$

where

$C_{0}\left(K_{w}\right)=\left\\{f\in
C\left(K_{w}\right),\int_{K_{w}}fd\mu=0\right\\}\subset
L_{0}^{2}\left(K_{w},\mu\right).$

We define then the Sobolev space $H^{-s}\left(K_{w}\right)$ as the completion
of $\mathcal{S}\left(K_{w}\right)$ with respect to the norm
$\|f\|_{H^{-s}(K_{w})}=\|(-\Delta)^{-s}f\|_{L^{2}(K_{w},\mu)}$. For $s\geq 0$,
we then consider $X_{s}^{w}$ the fractional stable field defined on
$H^{-s}\left(K_{w}\right)$ by

$X_{s,\alpha}(f)=\int_{K_{w}}\left(-\Delta_{w}\right)^{-s}f(z)\,W_{K_{w},\alpha}(dz),\quad
f\in H^{-s}\left(K_{w}\right)$

where $W_{K_{w},\alpha}$ is a real-valued symmetric $\alpha$-stable random
measure with control measure $\mu_{|_{K_{w}}}$. Note that the volume of
$K_{w}$ is $\mu(K_{w})=3^{n}$, which leads to define the Riesz kernel by

$G_{s}^{w}(x,y)=\int_{0}^{+\infty}t^{s-1}\left(p_{t}^{w}(x,y)-\frac{1}{\mu(K_{w}))}\right)dt=\int_{0}^{+\infty}t^{s-1}\left(p_{t}^{w}(x,y)-3^{n}\right)dt,\quad
x,y\in K_{w},\,x\neq y.$

Let us mention that there is a typo in the proof of Proposition 3.10 in[14],
since the constant $3^{n}$ is missing in the definition of $G_{s}^{w}$.

Finally, for $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$, we
also consider

$\widetilde{X}_{s,\alpha}^{w}(x)=\int_{K_{w}}G_{s}^{w}(x,y)\,W_{K_{w},\alpha}(dy),\quad
x\in K_{w}$

and Theorem 3.5 still holds: that is $\widetilde{X}_{s}^{w}$ is the density of
$X_{s,\alpha}^{w}$ with respect to the measure $\mu_{|_{K_{w}}}$. Next
Theorem, which generalized the self-similarity property established in the
Gaussian framework in [14], states that up to a normalization and a scaling,
we recover the Neumann fractional $\alpha$-stable random field defined on the
standard Sierpiński gasket $K$.

###### Proposition 3.17

Let $s>0$ and $H=sd_{w}-\frac{(\alpha-1)d_{h}}{\alpha}.$

  1. 1.

For any $f\in L^{\alpha}\left(K_{w},\mathcal{K}_{w},\mu_{|_{K_{w}}}\right)$,
then $f\circ F_{w}\in L^{\alpha}(K,\mathcal{K},\mu)$ and

$3^{n/\alpha}\,W_{K_{w},\alpha}(f)\stackrel{{\scriptstyle(d)}}{{=}}W_{K,\alpha}(f\circ
F_{w}).$

  2. 2.

As a consequence, $\displaystyle
2^{nH}3^{n}\left({X}^{w}_{s,\alpha}\left(f\right)\right)_{f\in
H^{-s}\left(K_{w}\right)}\stackrel{{\scriptstyle(fdd)}}{{=}}\left({X}_{s,\alpha}\left(f\circ
F_{w}\right)\right)_{f\in H^{-s}\left(K_{w}\right)}.$

  3. 3.

Moreover, for $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$,
$(2^{nH}\widetilde{X}_{s,\alpha}^{w}(F_{w}(x)))_{x\in K}$ is the Neumann
fractional $\alpha$-stable field defined on $K$ by (3.15), that is

$2^{nH}(\widetilde{X}_{s,\alpha}^{w}(F_{w}(x)))_{x\in
K}\stackrel{{\scriptstyle(fdd)}}{{=}}(\widetilde{X}_{s,\alpha}(x))_{x\in K}.$

###### Remark 3.18

By analogy with the Fractional Brownian fields of index $H$, in view of the
last proposition, the random fields defined in Definition 3.3 as a
distribution or pointwise by (3.15) are named Neumann fractional
$\alpha$-stable random fields on $K$ of index
$H=sd_{w}-\frac{(\alpha-1)d_{h}}{\alpha}$.

Proof.

  1. 1.

Let $f\in L^{\alpha}\left(K_{w},\mathcal{K}_{w},\mu_{|_{K_{w}}}\right)$. Then,

$\left\|W_{K_{w},\alpha}(f)\right\|_{L^{\alpha}(K_{w},\mu)}^{\alpha}=\int_{K_{w}}\left|f(y)\right|^{\alpha}\mu(dy)$

so that, by self-similarity of the Hausdorff measure $\mu$,

$\left\|W_{K_{w},\alpha}(f)\right\|_{L^{\alpha}(K_{w},\mu)}^{\alpha}=\frac{1}{3^{n}}\int_{K}\left|f(F_{w}(z))\right|^{\alpha}\mu(dz).$

In particular, $f\circ F_{w}\in L^{\alpha}(K,\mathcal{K},\mu)$, so that the
stable random variable $W_{K,\alpha}(f\circ F_{w})$ is well-defined. Moreover,

$\left\|W_{K,\alpha}(f\circ
F_{\omega})\right\|_{L^{\alpha}(K_{w},\mu)}^{\alpha}=\int_{K}\left|f(F_{w}(z))\right|^{\alpha}\mu(dz)=\left\|3^{n/\alpha}\,W_{K_{w},\alpha}(f)\right\|_{L^{\alpha}(K,\mu)}^{\alpha}$

so that the two $S\alpha S$ random variables $W_{K,\alpha}(f\circ F_{\omega})$
and $3^{n/\alpha}\,W_{K_{w},\alpha}(f)$ have the same distribution, which
establishes Assertion 1.

  2. 2.

Let $f\in L^{2}_{0}(K_{w},\mu)$. Then, $f\circ F_{w}\in L^{2}_{0}(K,\mu)$ and

$\begin{array}[]{rcl}\displaystyle\left(-\Delta\right)^{-s}\left(f\circ
F_{w}\right)&=&\displaystyle\sum_{j=1}^{+\infty}\frac{1}{\lambda_{j}^{s}}\left(\int_{K}\Phi_{j}(y)f\left(F_{w}(y)\right)\mu(dy)\right)\Phi_{j}\\\\[15.0pt]
&=&\displaystyle
5^{ns}3^{-n}\sum_{j=1}^{+\infty}\frac{1}{\lambda_{j,w}^{s}}\left(\int_{K}\Phi_{j}^{w}\left(F_{w}(y)\right)f\left(F_{w}(y)\right)\mu(dy)\right)\Phi_{j}^{w}\circ
F_{w}\end{array}$

Hence, applying the change of variable $x=F_{w}(y)$ in the integrals and using
the self-similarity of $\mu$, we get:

$\begin{array}[]{rcl}\displaystyle\left(-\Delta\right)^{-s}\left(f\circ
F_{w}\right)&=&\displaystyle
5^{ns}\sum_{j=1}^{+\infty}\frac{1}{\lambda_{j,w}^{s}}\left(\int_{K_{w}}\Phi_{j}^{w}\left(x\right)f\left(x\right)\mu(dy)\right)\Phi_{j}^{w}\circ
F_{w}\\\\[20.0pt] &=&\displaystyle
5^{ns}\left(\left(-\Delta_{w}\right)^{-s}f\right)\circ F_{w}.\end{array}$

By definition of $H^{-s}(K)$, one deduces that if $f\in H^{-s}(K_{w})$, then
$f\circ F_{w}\in H^{-s}(K)$ and

$\left(-\Delta\right)^{-s}\left(f\circ
F_{w}\right)=5^{ns}\left(\left(-\Delta_{w}\right)^{-s}f\right)\circ F_{w}.$

Hence, for $f\in H^{-s}(K_{w})$,

$X_{s,\alpha}\left(f\circ
F_{w}\right)=W_{K,\alpha}\left(\left(-\Delta\right)^{-s}\left(f\circ
F_{w}\right)\right)=5^{ns}W_{K,\alpha}\left(\left(\left(-\Delta_{w}\right)^{-s}f\right)\circ
F_{w}\right)$

and then by Assertion 1.,

$X_{s,\alpha}\left(f\circ
F_{w}\right)\stackrel{{\scriptstyle(d)}}{{=}}5^{ns}3^{n/\alpha}W_{K_{w},\alpha}\left(\left(-\Delta_{w}\right)^{-s}f\right)$

that is

$X_{s,\alpha}\left(f\circ
F_{w}\right)\stackrel{{\scriptstyle(d)}}{{=}}5^{ns}3^{n/\alpha}X_{s,\alpha}^{w}\left(f\right).$

Moreover, since $s=\frac{H}{d_{w}}+\frac{(\alpha-1)d_{h}}{\alpha
d_{w}}=\frac{H\ln 2}{\ln 5}+\left(1-\frac{1}{\alpha}\right)\frac{\ln 3}{\ln
5},$

$5^{ns}=2^{nH}3^{n-n/\alpha}$

and so

$X_{s,\alpha}\left(f\circ
F_{w}\right)\stackrel{{\scriptstyle(d)}}{{=}}2^{nH}3^{n}X_{s,\alpha}^{w}\left(f\right).$

Assertion 2. then follows by linearity of $X_{s,\alpha}$ and of
$X_{s,\alpha}^{w}$.

  3. 3.

For any $x\in K$,

$\widetilde{X}_{s,\alpha}^{w}\left(F_{w}(x)\right)=\int_{K_{w}}G_{s}^{w}\left(F_{w}(x),y\right)W_{K_{w},\alpha}(dy).$

Hence, by linearity of the stochastic integrals with respect to
$W_{K_{w},\alpha}$ and $W_{K,\alpha}$ and by Assertion 1,

$3^{n/\alpha}\left(\widetilde{X}_{s,\alpha}^{w}\left(F_{w}(x)\right)\right)_{x\in
K}\stackrel{{\scriptstyle(fdd)}}{{=}}\left(\int_{K}G_{s}^{w}\left(F_{w}(x),F_{w}(z)\right)W_{K,\alpha}(dz)\right)_{x\in
K}.$

Moreover, according to the proof of Proposition 3.10 in [14], for any $x,y\in
K$ such that $x\neq y$,

$G_{s}^{w}(F_{w}(x),F_{w}(y))=\frac{3^{n}}{5^{ns}}G_{s}(x,y).$

Therefore,

$\frac{5^{ns}}{3^{n-n/\alpha}}\left(\widetilde{X}_{s,\alpha}^{w}\left(F_{w}(x)\right)\right)_{x\in
K}\stackrel{{\scriptstyle(fdd)}}{{=}}\left(\int_{K}G_{s}\left(x,z\right)W_{K,\alpha}(dz)\right)_{x\in
K},$

that is

$2^{nH}\left(\widetilde{X}_{s,\alpha}^{w}\left(F_{w}(x)\right)\right)_{x\in
K}\stackrel{{\scriptstyle(fdd)}}{{=}}\left(\widetilde{X}_{s,\alpha}(x)\right)_{x\in
K},$

which concludes the proof.

$\Box$

##  4 Dirichlet Fractional Stable random fields

The previous sections have focused on the construction of the fractional
stable random fields which were constructed using the Dirichlet form (2.3) and
its associated Laplacian. In this section we show how to perform a similar
construction with Dirichlet boundary conditions at the 3 vertices of the
Sierpiński triangle. For fractional Gaussian fields this construction was done
in [13].

###  4.1 Dirichlet fractional Riesz kernels

Let us first recall the construction of the Laplacian with Dirichlet boundary
conditions; we refer to [13] for further details. Let

$\mathcal{F}_{D}=\\{f\in
C(K):\lim_{m\to+\infty}\mathcal{E}_{m}(f,f)<\infty,f=0\text{ on }V_{0}\\}.$

By Theorem 4.1 and Lemma 4.1 in [27], $(\mathcal{E},\mathcal{F}_{D})$ is a
local regular Dirichlet form on $L^{2}(K,\mu)$. The generator of this
Dirichlet form $\Delta_{D}$ is referred to as the Dirichlet Laplacian on $K$.
This Laplacian generates the Dirichlet heat semigroup $\\{P_{t}^{D}\\}_{t\geq
0}$ on $L^{2}(K,\mu)$ and the associated Dirichlet heat kernel denoted by
$p_{t}^{D}(x,y)$, for $t>0$ and $x,y\in K$, admits a uniformly convergent
spectral expansion:

$\displaystyle
p_{t}^{D}(x,y)=\sum_{j=1}^{+\infty}e^{-\lambda^{D}_{j}t}\Psi_{j}(x)\Psi_{j}(y)$
(4.28)

where
$0<\lambda^{D}_{1}\leq\lambda^{D}_{2}\leq\cdots\leq\lambda^{D}_{j}\leq\cdots$
are the eigenvalues of $-\Delta_{D}$ and $(\Psi_{j})_{j\geq
1}\subset\mathcal{D}(\Delta_{D})\subset\mathcal{F}_{D}$ is an orthonormal
basis of $L^{2}(K,\mu)$ such that

$\Delta_{D}\Psi_{j}=-\lambda^{D}_{j}\Psi_{j}.$

From [38], this heat kernel satisfies for some $c_{1},c_{2}\in(0,\infty)$,

$p_{t}^{D}(x,y)\leq
c_{1}t^{-\frac{d_{h}}{d_{w}}}\\!\exp\biggl{(}\\!-c_{2}\Bigl{(}\frac{d(x,y)^{d_{w}}}{t}\Bigr{)}^{\frac{1}{d_{w}-1}}\\!\biggr{)}$
(4.29)

for every $(x,y)\in K\times K$ and $t\in\bigl{(}0,+\infty)$ and also satisfies
the sub-Gaussian lower bound (see [38]):

$p_{t}^{D}(x,y)\geq
c_{1}\Psi_{1}(x)\Psi_{1}(y)t^{-\frac{d_{h}}{d_{w}}}\\!\exp\biggl{(}\\!-c_{2}\Bigl{(}\frac{d(x,y)^{d_{w}}}{t}\Bigr{)}^{\frac{1}{d_{w}-1}}\\!\biggr{)}$
(4.30)

for every $(x,y)\in K\times K$ and $t\in\bigl{(}0,1)$ where $\Psi_{1}$ is the
first eigenfunction of $-\Delta_{D}$.

The main differences induced by the Dirichlet boundary conditions are the
following:

  * •

$\Psi_{j}(q)=0$ whenever $q\in V_{0}$;

  * •

for all $x,y\in K$, $p_{t}^{D}(x,q)=p_{t}^{D}(q,y)=0$ whenever $q\in V_{0}$;

  * •

uniformly on $K\times K$, $p_{t}^{D}(x,y)\to 0$ when $t\to+\infty$.

###### Definition 4.1

Let $s\geq 0$. For $f\in L^{2}(K,\mu)$, the Dirichlet fractional Laplacian
$(-\Delta_{D})^{-s}$ on $f$ is defined as

$(-\Delta_{D})^{-s}f=\sum_{j=1}^{+\infty}\frac{1}{(\lambda^{D}_{j})^{s}}\Psi_{j}\int_{K}\Psi_{j}(y)f(y)\mu(dy).$

###### Definition 4.2

For a parameter $s\geq 0$, we define the Dirichlet fractional Riesz kernel
$G^{D}_{s}$ by

$\displaystyle
G^{D}_{s}(x,y)=\frac{1}{\Gamma(s)}\int_{0}^{+\infty}t^{s-1}p^{D}_{t}(x,y)dt,\quad
x,y\in K,\,x\neq y.$ (4.31)

For the Dirichlet boundary case, the analogue of the Schwartz space is given
by

$\mathcal{S}_{D}(K)=\left\\{f\in C(K),f=0\text{ on }V_{0},\forall k\geq
0\lim_{n\to+\infty}n^{k}\left|\int_{K}\Psi_{n}(y)f(y)\mu(dy)\right|=0\right\\}.$

We note that from [13], for $s>0$, $f\in\mathcal{S}_{D}(K)$, and $x\in K$

$(-\Delta_{D})^{-s}f(x)=\int_{K}G^{D}_{s}(x,y)f(y)\mu(dy).$ (4.32)

The Sobolev space $\mathcal{H}^{-s}_{D}(K)$ is defined as the completion of
$\mathcal{S}_{D}(K)$ with respect to the inner product

$\left\langle
f,g\right\rangle_{\mathcal{H}^{-s}_{D}(K)}=\int_{K}(-\Delta_{D})^{-s}f(y)(-\Delta_{D})^{-s}g(y)\mu(dy).$

The following result can be proved as in the case of the Neumann boundary
condition and the details are let to the reader.

###### Proposition 4.3

  1. 1.

If $s\in[0,d_{h}/d_{w})$, there exist constants $c,C>0$ such that for every
$x,y\in K$, $x\neq y$,

$c\frac{\Psi_{1}(x)\Psi_{1}(y)}{d(x,y)^{d_{h}-sd_{w}}}\leq
G_{s}^{D}(x,y)\leq\frac{C}{d(x,y)^{d_{h}-sd_{w}}}.$

  2. 2.

If $s=d_{h}/d_{w}$, there exists a constant $C>0$ such that for every $x,y\in
K$, $x\neq y$

$c\Psi_{1}(x)\Psi_{1}(y)\max(1,|\ln d(x,y)|)\leq G_{s}^{D}(x,y)\leq
C\max(1,|\ln d(x,y)|).$

###### Lemma 4.4

Let $p\in(0,+\infty)$. Then for any $x\in K\setminus V_{0}$, $y\mapsto
G_{s}(x,y)$ is in $L^{p}(K,\mu)$ iff
$s>\max\left(\frac{(p-1)d_{h}}{pd_{w}},0\right)$. Moreover, for
$s>\max\left(\frac{(p-1)d_{h}}{pd_{w}},0\right)$,

$\sup_{x\in K}\int_{K}\left|G^{D}_{s}(x,y)\right|^{p}\mu(dy)<+\infty.$

Proof. The proof is similar to that of Lemma 2.9. We however point out a
slight difference which comes from the lower bounds in Proposition 4.3
involving $\Psi_{1}$. Assume $0<s<d_{h}/d_{w}$ and that $x\in K\setminus
V_{0}$ is such that $y\to G_{s}(x,y)\in L^{p}(K,\mu)$. Then, from Proposition
4.3 we have
$\int_{K}\frac{\Psi_{1}(y)}{d(x,y)^{p(d_{h}-sd_{w})}}\mu(dy)<+\infty$. Let now
$\varepsilon>0$ be small enough such that the closed ball
$B(x,\varepsilon)\subset K\setminus V_{0}$, since $\psi_{1}$ is positive and
continuous on $B(x,\varepsilon)$ we deduce that is bounded below by a positive
constant on $B(x,\varepsilon)$. Therefore
$\int_{B(x,\varepsilon)}\frac{1}{d(x,y)^{p(d_{h}-sd_{w})}}\mu(dy)<+\infty$
which yields $p(d_{h}-sd_{w})<d_{h}$ using a proof similar to that of Lemma
2.10. $\Box$

###### Theorem 4.5

  1. 1.

For any $s\in\left(\frac{d_{h}}{d_{w}},1\right)$, there exists a constant
$C>0$ such that for every $x,y,z\in K$,

$|G^{D}_{s}(x,z)-G^{D}_{s}(y,z)|\leq Cd(x,y)^{sd_{w}-d_{h}}.$

  2. 2.

For $s\geq 1$, there exists a constant $C>0$ such that for every $x,y,z\in K$,

$|G^{D}_{s}(x,z)-G^{D}_{s}(y,z)|\leq Cd(x,y)^{d_{w}-d_{h}}\max\left(\left|\ln
d(x,y)\right|,1\right).$

###  4.2 Dirichlet fractional stable fields

###### Definition 4.6

Let $s\geq 0$ and $\alpha\in(0,2]$. The Dirichlet fractional $\alpha$-stable
field $\left\\{X_{s,\alpha}^{D}(f),f\in{\mathcal{H}^{-s}_{D}(K)}\right\\}$ is
defined as

$X_{s,\alpha}^{D}(f)=W_{K,\alpha}((-\Delta_{D})^{-s}f)=\int_{K}(-\Delta_{D})^{-s}f(z)\,W_{K,\alpha}(dz).$

###### Definition 4.7

Let $\alpha\in(0,2]$ and $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha
d_{w}},0\right)$. Then for any $x\in K$, by Lemma 2.9, the Dirichlet Riesz
kernel $z\mapsto G^{D}_{s}(x,z)$ is in $L^{\alpha}(K,\mathcal{K},\mu)$ and so
a fractional $\alpha$-stable field
$\widetilde{X}^{D}_{s,\alpha}=\left(\widetilde{X}_{s,\alpha}^{D}(x)\right)_{x\in
K}$ is well-defined by

$\widetilde{X}_{s,\alpha}^{D}(x)=\int_{K}G^{D}_{s}(x,z)W_{K,\alpha}(dz),\ x\in
K.$

This field, defined pointwise, is called the Dirichlet fractional
$\alpha$-stable of order $H=sd_{\omega}-\frac{(\alpha-1)d_{h}}{\alpha}$.

Note that since $G_{s}(q,z)=0$ whenever $q\in V_{0}$, as expected we get that
the Dirichlet fractional $\alpha$-stable field vanishes at the 3 vertices of
the Sierpiński triangle.

The following theorem can be proved as Theorem 3.5.

###### Theorem 4.8

Let $\alpha\in(0,2)$.

  * •

If $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$, then for every
$f\in\mathcal{S}_{D}(K)$, we have a.s.

$\int_{K}f(x)\widetilde{X}_{s,\alpha}^{D}(x)\mu(dx)=X_{s,\alpha}^{D}(f)$

with
$\widetilde{X}_{s,\alpha}^{D}=\left(\widetilde{X}_{s,\alpha}^{D}(x)\right)_{x\in
K}$ the fractional $\alpha$-stable field well-defined by

$\widetilde{X}_{s,\alpha}(x)=\int_{K}G^{D}_{s}(x,z)W_{K,\alpha}(dz),\ x\in K.$
(4.33)

  * •

Moreover, if there exists a random field $(Y(x))_{x\in K}$ such that a.s

$\int_{K}|Y(x)|\mu(dx)<+\infty$

and such that for every $f\in\mathcal{S}(K)$ we have a.s.

$\int_{K}f(x)Y(x)\mu(dx)=X_{s,\alpha}^{D}(f),$

then we have $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha d_{w}},0\right)$ and
for a.e. $x\in K$ with probability one $Y(x)=\widetilde{X}_{s,\alpha}^{D}(x)$.

The following result can then be proved as Theorem 3.9 and Theorem 3.15.

###### Theorem 4.9

Let $\alpha\in(0,2)$, $s>\max\left(\frac{(\alpha-1)d_{h}}{\alpha
d_{w}},0\right)$ and let $\widetilde{X}_{s,\alpha}^{D}$ be the Dirichlet
fractional $\alpha$-stable field. Let us set $\eta_{s}=\min(s,1)d_{w}-d_{h}$
and

$\beta_{s}=\left\\{\begin{array}[]{ll}0&\textrm{if $s<1$}\\\ 1&\textrm{if
$s\geq 1$}\end{array}\right.$

  1. 1.

If $s>\frac{d_{h}}{d_{w}}$ and if $\alpha\geq 1$, then there exists a
modification $\widetilde{X}_{s,\alpha}^{D,*}$ of
$\widetilde{X}_{s,\alpha}^{D}$ such that

$\lim_{\delta\to 0}\underset{\underset{x,y\in K}{\scriptsize
0<d(x,y)}\leq\delta}{\sup}\
\frac{\left|\widetilde{X}_{s,\alpha}^{D,*}(x)-\widetilde{X}_{s,\alpha}^{D,*}(y)\right|}{d(x,y)^{\eta_{s}}|\ln
d(x,y)|^{\beta_{s}+\frac{1}{2}}}<\infty.$ (4.34)

  2. 2.

If $s>\frac{d_{h}}{d_{w}}$ and if $\alpha<1$, then there exists a modification
$\widetilde{X}_{s,\alpha}^{D,*}$ of $\widetilde{X}_{s,\alpha}^{D}$ such that

$\lim_{\delta\to 0}\underset{\underset{x,y\in K}{\scriptsize
0<d(x,y)}\leq\delta}{\sup}\
\frac{\left|\widetilde{X}_{s,\alpha}^{D,*}(x)-\widetilde{X}_{s,\alpha}^{D,*}(y)\right|}{d(x,y)^{\eta_{s}}|\ln
d(x,y)|^{\beta_{s}}}<\infty.$ (4.35)

  3. 3.

If $s\leq\frac{d_{h}}{d_{w}}$, a.s., the sample paths of
$\widetilde{X}_{s,\alpha}$ are unbounded on the compact $K$.

Finally, we conclude by pointing out that the Dirichlet fractional
$\alpha$-stable field on the Sierpinski gasket satisfies the same invariance
properties as for the Neumann field which are stated in Section 3.5.

##  Appendix A Proof of Lemma 2.13

We first observe that according to Proposition 2.7 and Lemma 2.10 one has for
every $x\in K$

$\displaystyle\int_{K}\int_{K}\left|G_{s}(x,u)G_{t}(u,y)\right|\mu(du)\mu(dy)<+\infty.$
(A.36)

Now, on one hand, from Lemma 2.6, we have for every $f\in\mathcal{S}(K)$

$(-\Delta)^{-s-t}f(x)=\int_{K}G_{s+t}(x,y)f(y)\mu(dy).$

On the other hand, from (2.7), Lemma 2.6 and Fubini theorem which is justified
thanks to (A.36), we have

$\displaystyle(-\Delta)^{-s-t}f(x)$
$\displaystyle=\int_{K}G_{s}(x,y)(-\Delta)^{-t}f(y)\mu(dy)$
$\displaystyle=\int_{K}G_{s}(x,y)\left(\int_{K}G_{t}(y,u)f(u)\mu(du)\right)\mu(dy)$
$\displaystyle=\int_{K}\left(\int_{K}G_{s}(x,u)G_{t}(u,y)\mu(du)\right)f(y)\mu(dy)$

This gives that for every $f\in\mathcal{S}(K)$,

$\int_{K}\left(\int_{K}G_{s}(x,u)G_{t}(u,y)\mu(du)\right)f(y)\mu(dy)=\int_{K}G_{s+t}(x,y)f(y)\mu(dy).$

We use this equality with $f=p_{t}(z,\cdot)\in\mathcal{S}(K)$ and let $t\to
0$. Since for every function $g\in L^{1}(K,\mu)$,
$z\to\int_{K}p_{t}(z,y)g(y)\mu(dy)$ converges in $L^{1}$ to $g$ when $t\to 0$
(see [12, Page 114]), this gives that for $x\in K$ and $\mu$ a.e. $z\in K$

$G_{s+t}(x,z)=\int_{K}G_{s}(x,u)G_{t}(u,z)\mu(du).$

The conclusion will then follow from the following lemma.

###### Lemma A.1

  1. 1.

Let $s>0$. The map $(x,y)\to G_{s}(x,y)$ is continuous on the set $\\{(x,y)\in
K^{2},x\neq y\\}$. If moreover $s>d_{h}/d_{w}$, it is continuous on $K\times
K$.

  2. 2.

Let $s,t>0$ and $x\in K$. The map $z\to\int_{K}G_{s}(x,u)G_{t}(u,z)\mu(du)$ is
continuous on the set $\\{z\in K,z\neq x\\}$. If moreover $s+t>d_{h}/d_{w}$,
then it is continuous on $K$.

Proof. We recall that

$\displaystyle
G_{s}(x,y)=\frac{1}{\Gamma(s)}\int_{0}^{+\infty}t^{s-1}(p_{t}(x,y)-1)dt,\quad
x,y\in K,\,x\neq y$ (A.37)

and that the map $(t,x,y)\to p_{t}(x,y)$ is continuous on $(0,+\infty)\times
K\times K$.

It follows from Lemma 2.3 in [14] that there exists a constant $C>0$ such that
for $t\geq 1$ and $x,y\in K$, $|p_{t}(x,y)-1|\leq Ce^{-\lambda_{1}t}$. On the
other hand, if $\varepsilon>0$, $0<t<1$, $d(x,y)\geq\varepsilon$, then from
(2.4) one has

$|p_{t}(x,y)-1|\leq
c_{3}t^{-d_{h}/d_{w}}\exp\biggl{(}-c_{4}\Bigl{(}\frac{\varepsilon^{d_{w}}}{t}\Bigr{)}^{\frac{1}{d_{w}-1}}\biggr{)}+1.$

It then easily follows from the representation (A.37) that $(x,y)\to
G_{s}(x,y)$ is continuous on the set $\\{(x,y)\in K^{2},d(x,y)>\varepsilon\\}$
and the conclusion of Assertion 1 follows since $\varepsilon$ is arbitrary. To
prove Assertion 1, for $s>d_{h}/d_{w}$, one can simply use the upper bound
$p_{t}(x,y)\leq c_{3}t^{-d_{h}/d_{w}}$ which is valid for all $t\in(0,1)$,
$x,y\in K.$ The proof of Assertion 2 relies on Assertion 1 and the upper
bounds in Proposition 2.7. The details are left to the reader.

$\Box$

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Fabrice Baudoin<EMAIL_ADDRESS>
Department of Mathematics, Aarhus University, Denmark

Céline Lacaux<EMAIL_ADDRESS>
LMA Université d’Avignon, EA 2151, 84029, Avignon, France