# Parabolic Fractal Geometry
of Stable Lévy Processes with Drift

Peter Kern  Peter Kern, Mathematical Institute, Faculty of Mathematics and
Natural Sciences, Heinrich Heine University Düsseldorf, Universitätsstr. 1,
D-40225 Düsseldorf, Germany<EMAIL_ADDRESS>and  Leonard Pleschberger  Leonard
Pleschberger, Mathematical Institute, Faculty of Mathematics and Natural
Sciences, Heinrich Heine University Düsseldorf, Universitätsstr. 1, D-40225
Düsseldorf, Germany<EMAIL_ADDRESS>
###### Abstract.

We explicitly calculate the Hausdorff dimension of the graph and range of an
isotropic stable Lévy process $X$ plus deterministic drift function $f$. For
that purpose we use a restricted version of the genuine Hausdorff dimension
which is called the parabolic Hausdorff dimension. It turns out that covers by
parabolic cylinders are optimal for treating self-similar processes, since
their distinct non-linear scaling between time and space geometrically matches
the self-similarity of the processes. We provide explicit formulas for the
Hausdorff dimension of the graph and the range of $X+f$. In sum the parabolic
Hausdorff dimension of the drift term $f$ alone contributes to the Hausdorff
dimension of $X+f$. Further, we derive some formulas and bounds for the
parabolic Hausdorff dimension.

###### Key words and phrases:

Stable Lévy process, drift function, fractal path behavior, parabolic
Hausdorff dimension, self-similarity

###### 2020 Mathematics Subject Classification:

Primary 60G51; Secondary 28A78, 28A80, 60G17, 60G18, 60G52.

##  1\. Introduction

Let $X=(X_{t})_{t\geq 0}$ be a Lévy process in ${\mathbb{R}^{d}}$ which is a
stochastic process on some probability space $(\Omega,\pazocal{A},\mathbb{P})$
with the following properties:

  1. (i)

The process $\mathbb{P}$-almost surely starts in $0\in{\mathbb{R}^{d}}$.

  2. (ii)

$X$ possesses independent increments, i.e. for any $0\leq t_{0}<\dots<t_{n}$,
the random variables
$X_{t_{0}},X_{t_{1}}-X_{t_{0}},\dots,X_{t_{n}}-X_{t_{n-1}}$ are independent.

  3. (iii)

$X$ has stationary increments, i.e. the distribution of
$X_{t+h}-X_{t}\overset{\mathrm{d}}{=}X_{h}$ does not depend on $t$, where the
symbol $\overset{\mathrm{d}}{=}$ denotes equality in distribution.

  4. (iv)

$X$ is stochastically continuous, i.e.
$\mathbb{P}(|X_{t+h}-X_{t}|>\varepsilon)\rightarrow 0$ as $h\rightarrow 0$ for
any $t\geq 0$ and $\varepsilon>0$.

Additionally assuming self-similarity, the Lévy process is called stable. In
this paper we only deal with the special case of an isotropic $\alpha$-stable
Lévy process in which case the self-similarity is given by

(1.1) $(X_{\,c\cdot t})_{t\geq 0}\ \overset{\mathrm{fd}}{=}\
(c^{1/\alpha}\cdot X_{t})_{t\geq 0}\quad\text{ for all }c>0,$

where $\overset{\mathrm{fd}}{=}$ denotes equality of all finite-dimensional
distributions which characterise the stochastic processes in law. In this case
the Hurst index $H=1/\alpha$ is restricted to $H\geq\frac{1}{2}$, i.e.
$\alpha\in(0,2]$ and the isotropic $\alpha$-stable Lévy process is also
uniquely determined by the characteristic function
${\mathbb{E}}\big{[}{e^{i\langle\xi,X_{t}\rangle}}\big{]}=e^{-t\,C\cdot\|\xi\|^{\alpha}}$
with Lévy exponent $\Psi(\xi)=C\cdot\|\xi\|^{\alpha}$ for some constant $C>0$.
In case of $\alpha=2$ we obtain Brownian motion. For details on stable Lévy
processes we refer to the monograph [16].

The integrability of $\exp(-t\,C\cdot||\xi||^{\alpha})$ ensures the
applicability of the Fourier inversion formula. Therefore, for any $t>0$ the
random variable $X_{t}$ possesses the continuous density function

$x\mapsto
p(t,x):=(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{-i\langle\xi,x\rangle}e^{-t\Psi(\xi)}\,\textnormal{d}\xi=(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{-i\langle\xi,x\rangle}e^{-t\,C\cdot\|\xi\|^{\alpha}}\,\textnormal{d}\xi$

which for $\alpha\in(0,2)$ cannot be expressed in simple terms but belongs to
$C^{\infty}({\mathbb{R}^{d}})$ with all derivatives in
$L^{1}({\mathbb{R}^{d}})\cap C_{0}({\mathbb{R}^{d}})$; see [16]. Further, from
the self-similarity property (1.1) it easily follows that

(1.2) $p(t,x)=t^{-d/\alpha}\cdot
p\bigg{(}1,\frac{x}{t^{1/\alpha}}\bigg{)}\quad\text{ for all $t>0$ and
$x\in{\mathbb{R}^{d}}$.}$

Thus we define $p(x):=p(1,x)$ as the density at time $t=1$ and by Theorem 2.1
in [3] we have the tail estimate

(1.3) $p(x)\in\pazocal{O}\big{(}\|x\|^{-d-\alpha}\big{)}\quad\text{ as
}\|x\|\rightarrow\infty.$

This density is bounded and rotationally symmetric, i.e. writing $x=ry$ with
$r=\|x\|>0$ and $y=x/\|x\|\in S^{d-1}$ the density $p(x)=p(ry)$ does not
depend on $y$ and due to unimodality, see [16], $r\mapsto p(ry)$ is non-
increasing.

Our aim is to analyse isotropic stable Lévy processes plus (arbitrary) Borel
measurable drift functions by methods from fractal geometry. In particular, we
determine formulas for the Hausdorff dimension of the graph and the range of
an isotropic $\alpha$-stable Lévy process plus drift. Starting with Brownian
motion, in the past decades much effort has been made to explicitly calculate
the Hausdorff dimension of the range and the graph of stable Lévy processes
with an even more general self-similarity relation than (1.1), e.g. see [4, 5,
15, 7, 10, 1, 11, 18] in chronological order or the excellent review article
[20]. Only recently, Peres and Sousi started to deal with Hausdorff dimension
results of self-similar processes with an additional drift function by
considering Brownian motion [13] and fractional Brownian motion [14]. We will
follow the method in [14] to prove corresponding results for isotropic stable
Lévy processes. The restriction to isotropic stable Lévy processes is due to
rotational symmetry which is needed in the proof method. Compared to the
method in [14] we have to overcome with some additional issues:

  1. (1)

An isotropic $\alpha$-stable Lévy process for $\alpha\in(0,2)$ is a pure jump
process. Hence we cannot use Hölder continuity of the sample paths to derive
upper bounds for the Hausdorff dimension as in case of fractional Brownian
motion in [14].

  2. (2)

The Hurst index $H=1/\alpha$ of an isotropic $\alpha$-stable Lévy process is
restricted to $H\geq 1/2$, whereas $H\in(0,1)$ for fractional Brownian motion.
It will turn out that the case $H\geq 1$ needs different arguments than the
blueprint given for $H\in(0,1)$ in [14].

  3. (3)

By (1.3) the tail of the probability density of an isotropic $\alpha$-stable
Lévy process falls off as a power function, whereas the normal density of
fractional Brownian motion decreases exponentially fast.

We introduce a generalised version of the genuine Hausdorff dimension which is
called the $\alpha$-parabolic Hausdorff dimension in Section 2. We also give a
priori upper and lower bounds for the $\alpha$-parabolic Hausdorff dimension
in terms of the genuine Hausdorff dimension. It turns out that covers by
$\alpha$-parabolic cylinders are optimal for treating self-similar processes,
since their distinct non-linear scaling between time and space geometrically
matches the self-similarity of the processes. We provide explicit formulas for
the Hausdorff dimension of the graph and the range of an isotropic
$\alpha$-stable Lévy process plus Borel measurable drift function in Section 3
and defer the proofs to Sections 4–6. In sum the $\alpha$-parabolic Hausdorff
dimension of the drift term $f$ alone contributes to the Hausdorff dimension
of $X+f$. We derive new formulas and estimates for the $\alpha$-parabolic
Hausdorff dimension of constant functions and Hölder continuous functions in
Section 7.

##  2\. On Parabolic Fractal Geometry

We start with the definition of the $\alpha$-parabolic Hausdorff dimension
which inheres a distinct non-linear scaling between time and space. Throughout
this paper the symbol $|\cdot|$ denotes the diameter of a set in real
Euclidean space. For real functions $f,g$ the symbol $f\lesssim g$ denotes the
existence of a constant $C\in(0,\infty)$ not depending on the variables such
that $f\leq C\cdot g$ and $f\asymp g$ is short for $f\lesssim g$ and
$g\lesssim f$.

######  Definition 2.1.

Let $A\subseteq\mathbb{R}^{1+d}$ be an any set and
$\alpha,\beta\in(0,\infty)$. The $\alpha$-_parabolic $\beta$-Hausdorff (outer)
measure_ of $A$ is defined as

$\displaystyle\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta}(A):=\lim_{\delta\downarrow
0}\
\inf\Bigg{\\{}\sum_{k=1}^{\infty}|\mathsf{P}_{k}|^{\beta}:A\subseteq\bigcup_{n=1}^{\infty}\mathsf{P}_{k},\
\mathsf{P}_{k}\in\mathscr{P}^{\alpha},\ |\mathsf{P}_{k}|\leq\delta\Bigg{\\}},$

where the _$\alpha$ -parabolic cylinders_
$(\mathsf{P}_{n})_{n\in{\mathbb{N}}}$ are contained in

(2.1) $\mathscr{P}^{\alpha}:=\Bigg{\\{}[t,t+c]\times\prod_{j=1}^{d}\
[x_{j},x_{j}+c^{1/\alpha}]:t,\,x_{j}\in\mathbb{R},\ c\in(0,1]\Bigg{\\}}.$

We define the _$\alpha$ -parabolic Hausdorff dimension_ of $A$ as

$\mathscr{P}^{\alpha}\text{-}\dim
A:=\inf\big{\\{}\beta>0:\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta}(A)=0\big{\\}}=\sup\,\big{\\{}\beta>0:\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta}(A)=\infty\big{\\}}.$

The case $\alpha=1$ equals the genuine Hausdorff dimension which is simply
denoted by the symbol $\dim$.

Let us compare the $\alpha$-parabolic Hausdorff dimension to other parabolic
Hausdorff dimensions appearing in the literature. For the Gaussian case,
Taylor and Watson [17] introduced the _parabolic Hausdorff dimension_
$\mathscr{P}\text{-}\dim$ in the same way we did for $\alpha=2$ in order to
determine polar sets for the heat equation. On the contrary, for $H\in(0,1]$
Peres and Sousi [14] defined the $H$_-parabolic $\beta$-Hausdorff content_

$\Psi_{H}^{\beta}(A):=\inf\bigg{\\{}\sum_{k=1}^{\infty}c_{k}^{\beta}:A\subseteq\bigcup_{k=1}^{\infty}\mathsf{P}_{k},\
\mathsf{P}_{k}\in\mathscr{P}^{1/H}\bigg{\\}}$

with $1/H$-parabolic cylinders $\mathscr{P}^{1/H}$ from Equation (2.1). This
results in what they call the $H$-_parabolic Hausdorff dimension_

$\dim_{\Psi,H}A:=\inf\big{\\{}\beta>0:\Psi_{H}^{\beta}(A)=0\big{\\}}=\sup\big{\\{}\beta>0:\Psi_{H}^{\beta}(A)>0\big{\\}}.$

######  Proposition 2.2.

Let $A\subseteq\mathbb{R}^{1+d}$ be an arbitrary set,
$\alpha=1/H\in[1,\infty)$ and $\beta\in(0,\infty)$. Then one has

(2.2) $\mathscr{P}^{\alpha}\text{-}\dim A=(\dim_{\Psi,H}A)/H,$

i.e. Peres and Sousi’s $H$-parabolic Hausdorff dimension differs from our
$\alpha$-parabolic Hausdorff dimension by a constant factor.

###### Proof.

First we introduce the auxiliary _$\alpha$ -parabolic $\beta$-Hausdorff
content_

$\Phi^{\beta}_{\alpha}(A):=\inf\Bigg{\\{}\sum_{k=1}^{\infty}|\mathsf{P}_{k}|^{\beta}:A\subseteq\bigcup_{k=1}^{\infty}\mathsf{P}_{k},\
\mathsf{P}_{k}\in\mathscr{P}^{\alpha}\Bigg{\\}}.$

Since for $\alpha\geq 1$ and
$\mathsf{P}_{k}=[t_{k},t_{k}+c_{k}]\times\prod_{j=1}^{d}\
[x_{j,k},x_{j,k}+c_{k}^{1/\alpha}]\in\mathscr{P}^{\alpha}$ we have
$|\mathsf{P}_{k}|^{\beta}\asymp\ c_{k}^{\beta/\alpha}=\ c_{k}^{H\cdot\beta}$
one has

(2.3) $\Phi^{\beta}_{\alpha}(A)=\Psi_{H}^{\beta/H}(A).$

Following the arguments in Proposition 4.9 of [12] one gets
$\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta}(A)=0$ if and only if
$\Phi_{\alpha}^{\beta}(A)=0$, which together with Equation (2.3) shows (2.2).
∎

The $\alpha$-parabolic Hausdorff dimension fulfils the following countable
stability property which easily follows from monotonicity and
$\sigma$-subadditivity of the $\alpha$-parabolic Hausdorff measure as argued
for the genuine Hausdorff dimension on page 29 in [6].

######  Proposition 2.3.

For every $\alpha\in(0,\infty)$ and
$(A_{k})_{k\in{\mathbb{N}}}\subseteq\mathbb{R}^{1+d}$ the $\alpha$-parabolic
Hausdorff dimension fulfils the _countable stability property_

$\mathscr{P}^{\alpha}\text{-}\dim\bigcup_{k=1}^{\infty}A_{k}=\sup_{k\in{\mathbb{N}}}\
\mathscr{P}^{\alpha}\text{-}\dim A_{k}.$

Moreover, we derive the following a priori estimates for the Hausdorff
dimension in terms of the parabolic Hausdorff dimension.

######  Theorem 2.4.

Let $A\subseteq\mathbb{R}^{1+d}$ be any set. Let
$\phi_{\alpha}=\mathscr{P}^{\alpha}\text{-}\dim A$. Then one has

$\dim A\leq\begin{cases}\phi_{\alpha}\ \wedge\
\alpha\cdot\phi_{\alpha}+1-\alpha,&\alpha\in(0,1],\\\ \phi_{\alpha}\ \wedge\
\frac{1}{\alpha}\cdot\phi_{\alpha}+\big{(}1-\frac{1}{\alpha}\big{)}\cdot
d,&\alpha\in[1,\infty)\end{cases}$

and

$\dim A\geq\begin{cases}\phi_{\alpha}+\big{(}1-\frac{1}{\alpha}\big{)}\cdot
d,&\alpha\in(0,1],\\\ \phi_{\alpha}+1-\alpha,&\alpha\in[1,\infty).\end{cases}$

###### Proof.

_(i)_ Let $\alpha\in(0,\infty)$. By the definition of the $\alpha$-parabolic
$\beta$-Hausdorff measure there are only coverings by
$\mathscr{P}^{\alpha}$-sets permitted. So besides $\mathscr{P}^{\alpha}$ there
could exist more efficient covers of $A$ with respect to their shape.
Therefore
$\pazocal{H}^{\beta}(A)\leq\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta}(A)$
which implies $\dim A\leq\phi_{\alpha}$.

_(ii)_ Let $\alpha\in(0,1]$ and $\varepsilon>0$ be arbitrary. If
$\beta>\alpha\cdot\phi_{\alpha}+1-\alpha$, then

$\frac{\beta}{\alpha}+\bigg{(}1-\frac{1}{\alpha}\bigg{)}>\phi_{\alpha}.$

We can cover $A$ by the $\alpha$-parabolic cylinders

$\big{(}\mathsf{P}_{c_{k}}\big{)}_{k\in{\mathbb{N}}}=\Bigg{(}\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\prod_{j=1}^{d}\Big{[}x_{j,k},\,x_{j,k}+c_{k}^{1/\alpha}\Big{]}\Bigg{)}_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{\alpha}$

with $\big{|}\mathsf{P}_{c_{k}}\big{|}\asymp c_{k}$ for every
$k\in{\mathbb{N}}$ such that
$\sum_{k=1}^{\infty}\big{|}\mathsf{P}_{c_{k}}\big{|}^{\beta/\alpha+1-1/\alpha}\leq\varepsilon$.
Each $\mathsf{P}_{c_{k}}$ can be covered by
$\Big{\lceil}c_{k}^{1-1/\alpha}\Big{\rceil}$ hypercubes
$\square_{c_{k}^{1/\alpha}}$ with sidelength $c_{k}^{1/\alpha}$. Hence

$\displaystyle\pazocal{H}^{\beta}(A)$ $\displaystyle\leq\
\sum_{k=1}^{\infty}\Big{\lceil}c_{k}^{1-1/\alpha}\Big{\rceil}\cdot\big{|}\square_{c_{k}^{1/\alpha}}\big{|}^{\beta}\lesssim\
\sum_{k=1}^{\infty}c_{k}^{\beta/\alpha+1-1/\alpha}\lesssim\sum_{k=1}^{\infty}|\mathsf{P}_{c_{k}}|^{\beta/\alpha+1-1/\alpha}\leq\
\varepsilon.$

Since $\beta>\alpha\cdot\phi_{\alpha}+1-\alpha$ is arbitrary we have $\dim
A\leq\alpha\cdot\phi_{\alpha}+1-\alpha$.

_(iii)_ Let $\alpha\in[1,\infty)$ and $\varepsilon>0$ be arbitrary. If
$\beta>1/\alpha\cdot\phi_{\alpha}+(1-1/\alpha)\cdot d$, then

$\alpha\beta+(1-\alpha)\cdot d>\phi_{\alpha}.$

We can cover $A$ by the $\alpha$-parabolic cylinders

$\big{(}\mathsf{P}_{c_{k}^{1/\alpha}}\big{)}_{k\in{\mathbb{N}}}=\Bigg{(}\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\prod_{j=1}^{d}\Big{[}x_{j,k},\,x_{j,k}+c_{k}^{1/\alpha}\Big{]}\Bigg{)}_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{\alpha}$

with $|\mathsf{P}_{c_{k}^{1/\alpha}}|\asymp c_{k}^{1/\alpha}$ such that
$\sum_{k=1}^{\infty}\big{|}\mathsf{P}_{c_{k}^{1/\alpha}}\big{|}^{\alpha\beta+(1-\alpha)\cdot
d}\leq\varepsilon$. Each $\mathsf{P}_{c_{k}^{1/\alpha}}$ can be covered by
$\Big{\lceil}c_{k}^{1/\alpha-1}\Big{\rceil}^{d}$ hypercubes $\square_{c_{k}}$
with sidelength $c_{k}$. Then

$\displaystyle\pazocal{H}^{\beta}(A)$ $\displaystyle\leq\
\sum_{k=1}^{\infty}\Big{\lceil}c_{k}^{1/\alpha-1}\Big{\rceil}^{d}\cdot\big{|}\square_{c_{k}}\big{|}^{\beta}\lesssim\
\sum_{k=1}^{\infty}\big{(}c_{k}^{1/\alpha}\big{)}^{\alpha\beta+(1-\alpha)\cdot
d}\lesssim\sum_{k=1}^{\infty}|\mathsf{P}_{c_{k}^{1/\alpha}}|^{\alpha\beta+(1-\alpha)\cdot
d}\leq\ \varepsilon.$

Since $\beta>1/\alpha\cdot\phi_{\alpha}+(1-1/\alpha)\cdot d$ is arbitrary we
have $\dim
A\leq\frac{1}{\alpha}\cdot\phi_{\alpha}+\big{(}1-\frac{1}{\alpha}\big{)}\cdot
d$.

_(iv)_ Let $\alpha\in(0,1]$. Further, let $\beta>\dim A$ and $\varepsilon>0$
be arbitrary. Then we can cover $A$ with hypercubes

$\big{(}\square_{c_{k}}\big{)}_{k\in{\mathbb{N}}}=\Bigg{(}\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\prod_{j=1}^{d}\big{[}x_{j,k},\,x_{j,k}+c_{k}\big{]}\Bigg{)}_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{1}$

of sidelength $c_{k}\leq 1$ such that
$\sum_{k=1}^{\infty}\big{|}\square_{c_{k}}\big{|}^{\beta}\leq\varepsilon$.
Each $\square_{c_{k}}$ can be covered by
$\Big{\lceil}c_{k}^{1-1/\alpha}\Big{\rceil}^{d}$ $\alpha$-parabolic cylinders

$\big{(}\mathsf{P}_{c_{k}}\big{)}_{k\in{\mathbb{N}}}=\Bigg{(}\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\prod_{j=1}^{d}\Big{[}x_{j,k},\,x_{j,k}+c_{k}^{1/\alpha}\Big{]}\Bigg{)}_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{\alpha}$

with $|P_{c_{k}}|\asymp c_{k}$. By choosing $\gamma=\beta+(1/\alpha-1)\cdot d$
one has

$\displaystyle\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\gamma}(A)\leq\sum_{k=1}^{\infty}\Big{\lceil}c_{k}^{1-1/\alpha}\Big{\rceil}^{d}\cdot\big{|}\mathsf{P}_{c_{k}}\big{|}^{\gamma}\lesssim\sum_{k=1}^{\infty}c_{k}^{(1-1/\alpha)d+\gamma}=\sum_{k=1}^{\infty}c_{k}^{\beta}\lesssim\sum_{k=1}^{\infty}|\square_{c_{k}}|^{\beta}\leq\varepsilon.$

Since $\beta>\dim A$ is arbitrary, one has $\mathscr{P}^{\alpha}\text{-}\dim
A\leq\dim A+\big{(}\frac{1}{\alpha}-1\big{)}\cdot d$.

_(v)_ Let $\alpha\in[1,\infty)$. Further, let $\beta>\dim A$ and
$\varepsilon>0$ be arbitrary. Then we can cover $A$ with hypercubes

$\big{(}\square_{c_{k}}\big{)}_{k\in{\mathbb{N}}}=\Bigg{(}\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\prod_{j=1}^{d}\big{[}x_{j,k},\,x_{j,k}+c_{k}\big{]}\Bigg{)}_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{1}$

of sidelength $c_{k}\leq 1$ such that
$\sum_{k=1}^{\infty}\big{|}\square_{c_{k}}\big{|}^{\beta}\leq\varepsilon$.
Each $\square_{c_{k}}$ can be covered by
$\Big{\lceil}c_{k}^{1-\alpha}\Big{\rceil}$ $\alpha$-parabolic cylinders

$\big{(}\mathsf{P}_{c_{k}}\big{)}_{k\in{\mathbb{N}}}=\Bigg{(}\Big{[}t_{k},\,t_{k}+c_{k}^{\alpha}\Big{]}\times\prod_{j=1}^{d}\big{[}x_{j,k},\,x_{j,k}+c_{k}\big{]}\Bigg{)}\subseteq\mathscr{P}^{\alpha}$

with $|P_{c_{k}}|\asymp c_{k}$. By choosing $\gamma=\beta+\alpha-1$ one has

$\displaystyle\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\gamma}(A)\leq\sum_{k=1}^{\infty}\Big{\lceil}c_{k}^{1-\alpha}\Big{\rceil}\cdot\big{|}\mathsf{P}_{c_{k}}\big{|}^{\gamma}\lesssim\sum_{k=1}^{\infty}c_{k}^{1-\alpha+\gamma}=\sum_{k=1}^{\infty}c_{k}^{\beta}\lesssim\sum_{k=1}^{\infty}\big{|}\square_{c_{k}}\big{|}^{\beta}\leq\varepsilon.$

Since $\beta>\dim A$ is arbitrary, one has $\mathscr{P}^{\alpha}\text{-}\dim
A\leq\dim A+(\alpha-1)$ and the theorem is proven. ∎

##  3\. Main Results

So far our considerations regarding the parabolic Hausdorff dimension were
purely of geometric nature. Now we will apply it to stochastic processes. We
unite the cogitations of the following sections and begin with the Hausdorff
dimension of the graph $\pazocal{G}_{T}(X+f)=\\{(t,X_{t}+f(t)):t\in T\\}$ of
an isotropic stable Lévy process $X$ plus Borel measurable drift function $f$.

######  Theorem 3.1.

Let $T\subseteq\mathbb{R}_{+}$ be a Borel set and $\alpha\in(0,2]$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable Lévy process in
$\mathbb{R}^{d}$. Further, let $f:T\rightarrow\mathbb{R}^{d}$ be a Borel
measurable function. Let
$\varphi_{\alpha}=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$. Then
one $\mathbb{P}$-almost surely has

$\dim\pazocal{G}_{T}(X+f)=\begin{cases}\varphi_{1},&\alpha\in(0,1],\\\
\varphi_{\alpha}\ \wedge\
\frac{1}{\alpha}\cdot\varphi_{\alpha}+\big{(}1-\frac{1}{\alpha}\big{)}\cdot
d,&\alpha\in[1,2].\end{cases}$

###### Proof.

The Gaussian case where $\alpha=2$ follows from Theorem 1.2 in [14] together
with Proposition 2.2. The other cases will follow by the combination of
Corollary 4.2 with Theorem 5.7 for drift functions $f$ with $||f(t)-f(s)||\leq
1$ for all $s,t\in T$. For the treatment of arbitrary drift functions $f$ we
write $T=\bigcup_{z\in({\mathbb{Z}}/2)^{d}}T_{z}$ where $T_{z}:=\\{t\in
T:\|f(t)-z\|\leq\tfrac{1}{2}\\}$. The claim now follows easily by using the
countable stability property from Proposition 2.3. ∎

The formula for the Hausdorff dimension of the range
$\pazocal{R}_{T}(X+f)=\\{X_{t}+f(t):t\in T\\}$ of an isotropic stable Lévy
process $X$ plus Borel measurable drift function $f$ reads as follows.

######  Theorem 3.2.

Let $T\subseteq\mathbb{R}_{+}$ be a Borel set and $\alpha\in(0,2]$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable Lévy process in
$\mathbb{R}^{d}$ and $f:T\rightarrow{\mathbb{R}^{d}}$ be a Borel measurable
function. Let
$\varphi_{\alpha}=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$. Then
one $\mathbb{P}$-almost surely has

$\dim\pazocal{R}_{T}(X+f)=\begin{cases}\alpha\cdot\varphi_{\alpha}\ \wedge\
d,&\alpha\in(0,1],\\\ \varphi_{\alpha}\ \wedge\ d,&\alpha\in[1,2].\end{cases}$

###### Proof.

The Gaussian case where $\alpha=2$ follows from Theorem 1.2 in [14]. The rest
will follow by Theorem 6.1 and Theorem 6.4 analogously to the proof of Theorem
3.1. ∎

Our main theorems imply an improvement of Theorem 2.4 for graph sets.

######  Corollary 3.3.

Let $\varphi_{\alpha}=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$. In
case of $\alpha\in(0,1]$ one has

$\varphi_{1}\geq\alpha\cdot\varphi_{\alpha}\ \vee\
\varphi_{\alpha}+\bigg{(}1-\frac{1}{\alpha}\bigg{)}\cdot d.$

###### Proof.

For $\alpha\in(0,1]$ the combination of Theorem 3.1 and Theorem 3.2 directly
yields

$\varphi_{1}=\dim\pazocal{G}_{T}(f)\geq\dim\pazocal{G}_{T}(X+f)\geq\dim\pazocal{R}_{T}(X+f)=\alpha\cdot\varphi_{\alpha}\
\wedge\ d.$

Further we have

$\alpha\cdot\varphi_{\alpha}\geq\varphi_{\alpha}+\bigg{(}1-\frac{1}{\alpha}\bigg{)}\cdot
d\quad\text{ if and only if }\quad\alpha\cdot\varphi_{\alpha}\leq d$

and

$d\geq\varphi_{\alpha}+\bigg{(}1-\frac{1}{\alpha}\bigg{)}\cdot d\quad\text{ if
and only if }\quad\alpha\cdot\varphi_{\alpha}\leq d$

which proves the claim. ∎

##  4\. Graph: Upper Bound via Geometric Measure Theory

We calculate an upper bound for the Hausdorff dimension of the graph of an
isotropic stable Lévy process $X$ plus drift function by means of an efficient
covering.

######  Theorem 4.1.

Let $T\subseteq\mathbb{R}_{+}$ be any set and $\alpha\in(0,2]$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable Lévy process in
$\mathbb{R}^{d}$ and $f:T\rightarrow{\mathbb{R}^{d}}$ be any function. Further
let $\varphi_{\alpha}=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$ be
the $\alpha$-parabolic Hausdorff dimension of the graph of $f$ over $T$. Then
for $\alpha\in(0,1]$ one $\mathbb{P}$-almost surely has

$\displaystyle\dim\pazocal{G}_{T}(X+f)\leq\dim\pazocal{G}_{T}(f)=\varphi_{1},$

and for $\alpha\in[1,2]$ one $\mathbb{P}$-almost surely has

$\displaystyle\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(X+f)\leq\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)=\varphi_{\alpha}.$

###### Proof.

(i) Let $\alpha\in(0,1]$, $\beta=\varphi_{1}$ and let $\delta,\,\varepsilon>0$
be arbitrary. Then $\pazocal{G}_{T}(f)$ can be covered by hypercubes

$\big{(}\square_{c_{k}}\big{)}_{k\in{\mathbb{N}}}=\left(\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\prod_{i=1}^{d}\Big{[}x_{i,k},\,x_{i,k}+c_{k}\Big{]}\right)_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{1}$

such that
$\sum_{k=1}^{\infty}\big{|}\square_{c_{k}}\big{|}^{\beta+\delta}\lesssim\sum_{k=1}^{\infty}c_{k}^{\,\beta+\delta}\leq\varepsilon$.
Let $M_{k}(\omega)$ be the random number of a fixed $2^{d}$-nested collection
of hypercubes with sidelength $c_{k}^{1/\alpha}$ that the path $t\mapsto
X_{t}(\omega)$ hits at some time $t\in[t_{k},\,t_{k}+c_{k}]$. Let
$\bigcup_{k\in{\mathbb{N}}}\
\mathsf{P}_{c_{k}}(\omega)\supseteq\pazocal{G}_{T}(X(\omega))$ with

$\big{(}\mathsf{P}_{c_{k}}(\omega)\big{)}_{k\in{\mathbb{N}}}=\Bigg{(}\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\bigcup_{j=1}^{M_{k}(\omega)}\prod_{i=1}^{d}\Big{[}\xi_{i,j,k}(\omega),\,\xi_{i,j,k}(\omega)+c_{k}^{1/\alpha}\Big{]}\Bigg{)}_{k\in{\mathbb{N}}}$

being a corresponding random parabolic cover of the graph of this path. Then
for all $t\in[t_{k},\,t_{k}+c_{k}]$ there exists
$j\in\\{1,\dots,M_{k}(\omega)\\}$ such that for the $i$-th component of $X+f$
we have

$\xi_{i,j,k}(\omega)+x_{i,k}\leq
X_{t}^{(i)}(\omega)+f_{i}(t)\leq\xi_{i,j,k}(\omega)+x_{i,k}+c_{k}^{1/\alpha}+c_{k}\leq\xi_{i,j,k}(\omega)+x_{i,k}+2c_{k}.$

Hence we obtain a random cover $\bigcup_{k\in{\mathbb{N}}}\
\widetilde{\square}_{c_{k}}(\omega)\supseteq\pazocal{G}_{T}(X(\omega)+f)$
where

$\displaystyle\widetilde{\square}_{c_{k}}(\omega)=\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\bigcup_{j=1}^{M_{k}(\omega)}\prod_{i=1}^{d}$
$\displaystyle\left(\Big{[}\xi_{i,j,k}(\omega)+x_{i,k},\,\xi_{i,j,k}(\omega)+x_{i,k}+c_{k}\Big{]}\right.$
$\displaystyle\cup\ $
$\displaystyle\,\left.\Big{[}\xi_{i,j,k}(\omega)+x_{i,k}+c_{k},\,\xi_{i,j,k}(\omega)+x_{i,k}+2c_{k}\Big{]}\right)$

This is a union of $M_{k}(\omega)\cdot 2^{d}$ sets with diameter
$\sqrt{d+1}\cdot c_{k}$. An application of Pruitt and Taylor’s covering Lemma
6.1 in [15] and Lemma 3.4 in [11] shows that for all $\delta^{\prime}>0$ one
has

${\mathbb{E}}[M_{k}]\lesssim\frac{c_{k}}{{\mathbb{E}}\Big{[}T\big{(}c_{k}^{1/\alpha}/3,c_{k}\big{)}\Big{]}}\lesssim
c_{k}^{-\delta^{\prime}/\alpha},$

where $T\big{(}c_{k}^{1/\alpha}/3,c_{k}\big{)}$ is the sojourn time of the
process $(X_{t})_{t\in[0,c_{k}]}$ in a ball of radius $c_{k}^{1/\alpha}/3$
centred at the origin. Hence we get for
$\varepsilon^{\prime}=\delta+\delta^{\prime}/\alpha>0$

$\displaystyle{\mathbb{E}}\Big{[}\pazocal{H}^{\beta+\varepsilon^{\prime}}(\pazocal{G}_{T}(X+f))\Big{]}\leq{\mathbb{E}}\Bigg{[}\sum_{k=1}^{\infty}|\widetilde{\square}_{c_{k}}|^{\beta+\varepsilon^{\prime}}\Bigg{]}$
$\displaystyle\quad\lesssim\sum_{k=1}^{\infty}{\mathbb{E}}[M_{k}]\cdot
c_{k}^{\beta+\varepsilon^{\prime}}\lesssim\sum_{k=1}^{\infty}c_{k}^{\beta+\delta}\leq\varepsilon.$

Since $\varepsilon,\varepsilon^{\prime}>0$ are arbitrary, we get for all
$\alpha\in(0,1]$ and $\beta^{\prime}>\beta$

${\mathbb{E}}\Big{[}\pazocal{H}^{\beta^{\prime}}(\pazocal{G}_{T}(X+f))\Big{]}=0$

which implies

$\pazocal{H}^{\beta^{\prime}}(\pazocal{G}_{T}(X+f))=0\quad\mathbb{P}\text{-almost
surely}.$

Since $\beta^{\prime}>\beta$ is arbitrary we finally get
$\dim\pazocal{G}_{T}(X+f)\leq\beta=\varphi_{1}\,\,\mathbb{P}\text{-almost
surely}$.

(ii) Let $\alpha\in[1,2]$,
$\beta=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$ and let
$\varepsilon,\delta>0$ be arbitrary. Then $\pazocal{G}_{T}(f)$ can be covered
by $\alpha$-parabolic cylinders

$\big{(}\mathsf{P}_{c_{k}^{1/\alpha}}\big{)}_{k\in{\mathbb{N}}}=\left(\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\prod_{i=1}^{d}\Big{[}x_{i,k},\,x_{i,k}+c_{k}^{1/\alpha}\Big{]}\right)_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{\alpha}$

such that
$\sum_{k=1}^{\infty}\big{|}\mathsf{P}_{c_{k}^{1/\alpha}}\big{|}^{\beta+\delta}\lesssim\sum_{k=1}^{\infty}c_{k}^{(\beta+\delta)/\alpha}\leq\varepsilon$.
Let $M_{k}(\omega)$ be the random number of a fixed $2^{d}$-nested collection
of hypercubes with sidelength $c_{k}^{1/\alpha}$ that the path $t\mapsto
X_{t}(\omega)$ hits at some time $t\in[t_{k},\,t_{k}+c_{k}]$. As in _(i)_ , we
obtain a random parabolic cover $\bigcup_{k\in{\mathbb{N}}}\
\widetilde{\mathsf{P}}_{c_{k}^{1/\alpha}}(\omega)\supseteq\pazocal{G}_{T}(X(\omega)+f)$
where

$\displaystyle\widetilde{\mathsf{P}}_{c_{k}^{1/\alpha}}(\omega)=\big{[}t_{k},\,t_{k}+c_{k}\big{]}\times\bigcup_{j=1}^{M_{k}(\omega)}\prod_{i=1}^{d}$
$\displaystyle\left(\Big{[}\xi_{i,j,k}(\omega)+x_{i,k},\,\xi_{i,j,k}(\omega)+x_{i,k}+c_{k}^{1/\alpha}\Big{]}\right.$
$\displaystyle\cup\ $
$\displaystyle\,\left.\Big{[}\xi_{i,j,k}(\omega)+x_{i,k}+c_{k}^{1/\alpha},\,\xi_{i,j,k}(\omega)+x_{i,k}+2c_{k}^{1/\alpha}\Big{]}\right).$

This is a union of $M_{k}(\omega)\cdot 2^{d}$ sets with diameter
$|\widetilde{\mathsf{P}}_{c_{k}^{1/\alpha}}(\omega)|\lesssim
c_{k}^{1/\alpha}$. An application of Pruitt and Taylor’s covering Lemma 6.1 in
[15] and Lemma 3.4 in [11] show that

${\mathbb{E}}[M_{k}]\leq\frac{c_{k}}{{\mathbb{E}}\Big{[}T\big{(}c_{k}^{1/\alpha}/3,c_{k}\big{)}\Big{]}}\lesssim
c_{k}^{-\delta^{\prime}/\alpha}.$

where $T\big{(}c_{k}^{1/\alpha}/3,c_{k}\big{)}$ denotes the sojourn time of
the process $(X_{t})_{t\in[0,c_{k}]}$ in a ball of radius $c_{k}^{1/\alpha}/3$
centred at the origin. Hence we get for
$\varepsilon^{\prime}=\delta+\delta^{\prime}>0$ with the same calculations as
above

$\displaystyle{\mathbb{E}}\Big{[}\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta+\varepsilon^{\prime}}(\pazocal{G}_{T}(X+f))\Big{]}\lesssim\sum_{k=1}^{\infty}{\mathbb{E}}[M_{k}]\cdot
c_{k}^{(\beta+\varepsilon^{\prime})/\alpha}$
$\displaystyle\quad\lesssim\sum_{k=1}^{\infty}c_{k}^{(\beta+\varepsilon^{\prime}-\delta^{\prime})/\alpha}=\sum_{k=1}^{\infty}c_{k}^{(\beta+\delta)/\alpha}\leq\varepsilon.$

Since $\varepsilon,\varepsilon^{\prime}>0$ are arbitrary, as in (i) we finally
get

$\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(X+f)\leq\beta=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)\quad\mathbb{P}\text{-almost
surely},$

as claimed. ∎

######  Corollary 4.2.

Let $T\subseteq\mathbb{R}_{+}$ be any set and $\alpha\in(0,2]$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable Lévy process in
$\mathbb{R}^{d}$ and $f:T\rightarrow{\mathbb{R}^{d}}$ be any function. Further
let $\varphi_{\alpha}=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$.
Then one $\mathbb{P}$-almost surely has

$\dim\pazocal{G}_{T}(X+f)\leq\begin{cases}\varphi_{1},&\alpha\in(0,1],\\\
\varphi_{\alpha}\ \wedge\
\frac{1}{\alpha}\cdot\varphi_{\alpha}+\big{(}1-\frac{1}{\alpha}\big{)}\cdot
d,&\alpha\in[1,2].\end{cases}$

###### Proof.

The Gaussian case $\alpha=2$ follows from Corollary 2.3 in [14] and
Proposition 2.2. The rest follows directly from Theorem 2.4 and Theorem 4.1. ∎

##  5\. Graph: Lower Bound via Potential Theory

Next we want to calculate a lower bound for the Hausdorff dimension of
isotropic stable Lévy processes with drift. This will be accomplished by the
energy method, see Section 4.3 in [12]. This method makes use of the Lebesgue
integral. Hence for the first time we have to impose restrictions on the
domain $T\subseteq\mathbb{R}_{+}$ and the drift function
$f:T\rightarrow{\mathbb{R}^{d}}$ with regard to their measurability. We
introduce some notions from potential theory.

######  Definition 5.1.

Let $A\subseteq\mathbb{R}^{1+d}$ be a Suslin set and $\mu$ be a probability
measure supported on $A$, i.e. $\mu\in\pazocal{M}^{1}(A)$. Further, let
$K:\mathbb{R}^{1+d}\rightarrow[0,\infty]$ be a Lebesgue measurable function
which is called the _difference kernel_. The _K-energy of a probability
measure $\mu$_ is defined to be

$\pazocal{E}_{K}(\mu):=\int_{A}\int_{A}K(t-s,x-y)\,\textnormal{d}\mu(t,x)\,\textnormal{d}\mu(s,y)$

and the _equilibrium value of A_ is defined as
$\pazocal{E}^{*}_{K}:=\inf_{\mu\in\pazocal{M}^{1}(A)}\pazocal{E}_{K}(\mu)$. We
define the _K-capacity of $A$_ as

$\operatorname{Cap}_{K}(A):=\frac{1}{\pazocal{E}^{*}_{K}}.$

Whenever the kernel has the form $K(t,x)=||(t,x)||^{-\beta}$, we write
$\pazocal{E}_{\beta}(\mu)$ for $\pazocal{E}_{K}(\mu)$ and
$\operatorname{Cap}_{\beta}(A)$ for $\operatorname{Cap}_{K}(A)$ and we refer
to them as the _$\beta$ -energy of a probability measure $\mu$_ and the _Riesz
$\beta$-capacity of $A$_, respectively. Next we state Frostman’s theorem.

######  Theorem 5.2.

Let $\alpha>0$. For any Suslin set $A\subseteq\mathbb{R}^{1+d}$ one has

$\mathscr{P}^{\alpha}\text{-}\dim A\geq\dim
A=\sup\\{\beta:\,\operatorname{Cap}_{\beta}(A)>0\\}.$

###### Proof.

This follows from Theorem 2.4 together with Appendix B in [2]. ∎

The next lemma shows that we can work with an energy integral where the stable
process $X$ is transformed into the kernel.

######  Lemma 5.3.

Let $T\subseteq\mathbb{R}_{+}$ be a Borel set and $\alpha\in(0,2]$. Let
$X=(X_{t})_{t\geq 0}$ be a stochastic process in $\mathbb{R}^{d}$ with
stationary increments and $f:\mathbb{R}_{+}\rightarrow{\mathbb{R}^{d}}$ be a
Borel measurable function. Define the difference kernel

$K^{\beta}(t,x):={\mathbb{E}}\big{[}||(t,\mathrm{sign}(t)\cdot
X_{|t|}+x)||^{-\beta}\big{]}.$

Then from $\operatorname{Cap}_{K^{\beta}}(\pazocal{G}_{T}(f))>0$ it follows
that $\mathbb{P}$-almost surely
$\operatorname{Cap}_{\beta}(\pazocal{G}_{T}(X+f))>0$ holds. Hence
$\pazocal{E}_{K^{\beta}}(\mu)<\infty$ for some probability measure
$\mu\in\pazocal{M}^{1}(\pazocal{G}_{T}(f))$ implies

$\dim\pazocal{G}_{T}(X+f)\geq\beta\quad\mathbb{P}\text{-almost surely}.$

###### Proof.

For every $\omega\in\Omega$, the pathwise bijection
$(t,f(t))\in\pazocal{G}_{T}(f)$ if and only if
$(t,X_{t}(\omega)+f(t))\in\pazocal{G}_{T}(X_{t}(\omega)+f)$ yields the
existence of some random probability measure
$\nu_{\omega}\in\pazocal{M}^{1}(\pazocal{G}_{T}(X(\omega)+f))$ with
$\nu_{\omega}(\widetilde{A}_{\omega})=\mu(A)$ for all Borel sets
$A\subseteq\pazocal{G}_{T}(f)$ where
$\widetilde{A}_{\omega}:=\\{(t,x+X_{t}(\omega)):\ (t,x)\in A\\}$. Therefore,
Tonelli’s theorem and the stationarity of the increments of $X$ yield

$\displaystyle{\mathbb{E}}\big{[}\pazocal{E}_{\beta}(\nu_{\omega})\big{]}=\
{\mathbb{E}}\bigg{[}\int_{\pazocal{G}_{T}(X(\omega)+f)}\int_{\pazocal{G}_{T}(X(\omega)+f)}||(t-s,x-y)||^{-\beta}\,\textnormal{d}\nu_{\omega}(t,x)\,\textnormal{d}\nu_{\omega}(s,y)\bigg{]}$
$\displaystyle=\
{\mathbb{E}}\bigg{[}\int_{\pazocal{G}_{T}(f)}\int_{\pazocal{G}_{T}(f)}||(t-s,x+X_{t}(\omega)-(y+X_{s}(\omega)))||^{-\beta}\,\textnormal{d}\mu(t,x)\,\textnormal{d}\mu(s,y)\bigg{]}$
$\displaystyle=\
\int_{\pazocal{G}_{T}(f)}\int_{\pazocal{G}_{T}(f)}{\mathbb{E}}\big{[}||(t-s,X_{t}(\omega)-X_{s}(\omega)+x-y)||^{-\beta}\big{]}\,\textnormal{d}\mu(t,x)\,\textnormal{d}\mu(s,y)$
$\displaystyle=\
\int_{\pazocal{G}_{T}(f)}\int_{\pazocal{G}_{T}(f)}{\mathbb{E}}\big{[}||(t-s,\mathrm{sign}(t-s)\cdot
X_{|t-s|}(\omega)+x-y)||^{-\beta}\big{]}\,\textnormal{d}\mu(t,x)\,\textnormal{d}\mu(s,y)$
$\displaystyle=\ \pazocal{E}_{K^{\beta}}(\mu)$

By assumption, there exists $\mu\in\pazocal{M}^{1}(\pazocal{G}_{T}(f))$ such
that $\pazocal{E}_{K^{\beta}}(\mu)<\infty$, therefore
$\pazocal{E}_{\beta}(\nu_{\omega})<\infty\,\,\mathbb{P}\text{-almost surely}.$
The rest of the claim follows by Frostman’s theorem 5.2. ∎

Frostman’s lemma provides the suitable candidate for the probability measure
$\mu$. We give a parabolic version of it.

######  Theorem 5.4.

Let $A\subseteq\mathbb{R}^{1+d}$ be a Borel set. If
$\mathscr{P}^{\alpha}\text{-}\dim A>\beta,$ then there exists
$\mu\in\pazocal{M}^{1}(A)$ such that we have

$\mu\Bigg{(}\big{[}t,\,t+c\big{]}\times\prod_{i=1}^{d}\Big{[}x_{i},\,x_{i}+c^{1/\alpha}\Big{]}\Bigg{)}\lesssim\begin{cases}c^{\,\beta},&\alpha\in(0,1],\\\
c^{\,\beta/\alpha},&\alpha\in[1,\infty)\end{cases}$

for every $c\in(0,1]$ and $t,x_{1},\dots,x_{d}\in\mathbb{R}$.

###### Proof.

The parabolic case can easily be proven along the lines of the classical case,
see e.g. section 4.4 in [12]. ∎

The following lemma is a refinement of (2.7) in [14].

######  Lemma 5.5.

Let $t\in\mathbb{R}$ be fixed and $h:{\mathbb{R}^{d}}\rightarrow\mathbb{R},\
h(x)=\|(t,x)\|^{-\beta}=(t^{2}+||x||^{2})^{-\beta/2}$. Then $h$ is
rotationally symmetric and the mapping $r\mapsto h(r\cdot y)$ is non-
increasing for $r=||x||$ and does not not depend on $y=x/||x||\in S^{d-1}$.
Further, let $p:{\mathbb{R}^{d}}\rightarrow\mathbb{R}$ be a rotationally
symmetric function such that also $r\mapsto p(r\cdot y)$ is non-increasing for
$r=||x||$ and $y=x/||x||\in S^{d-1}$. Then for all $u\in{\mathbb{R}^{d}}$ we
have

$\int_{\mathbb{R}^{d}}h(x+u)\cdot
p(x)\,\textnormal{d}x\lesssim\int_{\mathbb{R}^{d}}h(x)\cdot
p(x)\,\textnormal{d}x,$

provided that the integrals exist.

###### Proof.

The first part is obvious. Further, by monotonicity we have

$\displaystyle\int_{\mathbb{R}^{d}}h(x+u)\cdot p(x)\,\textnormal{d}x$
$\displaystyle=\ \int_{\\{||x||<||x+u||\\}}\underbrace{h(x+u)}_{\leq
h(x)}\cdot\
p(x)\,\textnormal{d}x+\int_{\\{||x||\geq||x+u||\\}}h(x+u)\cdot\underbrace{p(x)}_{\leq
p(x+u)}\,\textnormal{d}x$ $\displaystyle\leq\ 2\int_{\mathbb{R}^{d}}h(x)\cdot
p(x)\,\textnormal{d}x,$

as claimed. ∎

Inspired by Lemma 2.5 in [14], we give a priori estimates for the difference
kernel $K^{\beta}={\mathbb{E}}\big{[}||(t,\mathrm{sign}(t)\cdot
X_{|t|}(\omega)+x)||^{-\beta}\big{]}$ from Lemma 5.3.

######  Lemma 5.6.

Let $\alpha\in(0,2)$ and $X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable
Lévy process in ${\mathbb{R}^{d}}$. Let $\beta\geq 0$ and $\tau\in\mathbb{R}$,
$\delta\in{\mathbb{R}^{d}}$ be such that $|\tau|\in(0,1]$,
$||\delta||\in[0,1]$. Then appropriate choices for estimating the difference
kernel from Lemma 5.3
$K^{\beta}(\tau,\delta):={\mathbb{E}}\big{[}||(\tau,\mathrm{sign}(\tau)\cdot
X_{|\tau|}+\delta)||^{-\beta}\big{]}$ are

(5.1) $\displaystyle
K^{\beta}(\tau,\delta)\lesssim\begin{cases}|\tau|^{-\beta},\\\
|\tau|^{-\beta/\alpha},&\text{ for }\beta<d,\\\
|\tau|^{(1-1/\alpha)d-\beta},&\text{ for }\beta>d.\end{cases}$

and one has

(5.2) $\displaystyle
K^{\beta}(\tau,\delta)\lesssim\begin{cases}||\delta||^{-\beta},&\text{ for
}\alpha\in(0,1],|\tau|\leq||\delta||,\\\ ||\delta||^{-\beta},&\text{ for
}\alpha\in[1,2),\beta<d,|\tau|\leq||\delta||^{\alpha},\\\
||\delta||^{(\alpha-1)d-\alpha\beta},&\text{ for
}\alpha\in[1,2),\,\beta>d,|\tau|\leq||\delta||^{\alpha}.\end{cases}$

###### Proof.

Let $p(x)$ denote the density function of
$X_{1}\overset{\mathrm{d}}{=}|\tau|^{-1/\alpha}X_{|\tau|}$. We define rescaled
increments $\widetilde{\tau}:=\tau/|\tau|^{1/\alpha}$ and
$\widetilde{\delta}:=\delta/|\tau|^{1/\alpha}$. Trivial estimation always
yields

${\mathbb{E}}\big{[}||(\tau,\mathrm{sign}(\tau)\cdot
X_{|\tau|}+\delta)||^{-\beta}\big{]}\leq|\tau|^{-\beta}.$

The self-similarity of the stable Lévy process and Lemma 5.5 yield

(5.3) $\begin{split}&{\mathbb{E}}\big{[}||(\tau,\mathrm{sign}(\tau)\cdot
X_{|\tau|}+\delta)||^{-\beta}\big{]}=|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||(\widetilde{\tau},\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta})||^{-\beta}\cdot p(x)\,\textnormal{d}x\\\
&\quad=|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||(|\widetilde{\tau}|,x+\mathrm{sign}(\tau)\cdot\widetilde{\delta})||^{-\beta}\cdot
p(x)\,\textnormal{d}x\\\
&\quad\lesssim|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||(|\widetilde{\tau}|,x)||^{-\beta}\cdot
p(x)\,\textnormal{d}x.\end{split}$

Let $\beta<d$. Then by (5.3) we get

$\displaystyle{\mathbb{E}}\big{[}||(\tau,\mathrm{sign}(\tau)\cdot
X_{|\tau|}+\delta)||^{-\beta}\big{]}\lesssim|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||x||^{-\beta}\cdot
p(x)\,\textnormal{d}x$
$\displaystyle\quad\lesssim|\tau|^{-\beta/\alpha}\cdot{\mathbb{E}}\big{[}||X_{1}||^{-\beta}\big{]}\lesssim|\tau|^{-\beta/\alpha},$

since negative moments of order $\beta<d$ exist; see Lemma 3.1 in [1].

Let $\beta>d$. Then by (5.3) one has using the volume of a ball with radius
$\widetilde{\tau}$

$\displaystyle{\mathbb{E}}\big{[}||(\tau,\mathrm{sign}(\tau)\cdot
X_{|\tau|}+\delta)||^{-\beta}\big{]}\lesssim|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||(|\widetilde{\tau}|,x)||^{-\beta}\cdot
p(x)\,\textnormal{d}x$
$\displaystyle\quad\leq|\tau|^{-\beta/\alpha}\bigg{(}\int_{\\{||x||<|\widetilde{\tau}|\\}}|\widetilde{\tau}|^{-\beta}\cdot
p(x)\,\textnormal{d}x+\int_{\\{||x||\geq|\widetilde{\tau}|\\}}||x||^{-\beta}\cdot
p(x)\,\textnormal{d}x\bigg{)}$
$\displaystyle\quad\leq|\tau|^{-\beta/\alpha}\bigg{(}|\widetilde{\tau}|^{\,d-\beta}+\int_{\\{|\widetilde{\tau}|\leq||x||\leq
1\\}}||x||^{-\beta}\,\textnormal{d}x+\int_{\\{||x||>1\\}}p(x)\,\textnormal{d}x\bigg{)}$
$\displaystyle\quad\leq|\tau|^{-\beta/\alpha}\bigg{(}|\widetilde{\tau}|^{\,d-\beta}+\int_{|\widetilde{\tau}|}^{1}\int_{S^{d-1}}||ry||^{-\beta}\cdot
r^{d-1}\,\textnormal{d}y\,\textnormal{d}r+1\bigg{)}$
$\displaystyle\quad\lesssim|\tau|^{-\beta/\alpha}\bigg{(}|\widetilde{\tau}|^{\,d-\beta}+\int_{|\widetilde{\tau}|}^{1}r^{d-\beta-1}\,\textnormal{d}r\bigg{)}$
$\displaystyle\quad\lesssim|\tau|^{-\beta/\alpha}\cdot|\widetilde{\tau}|^{\,d-\beta}=|\tau|^{-\beta/\alpha}\cdot|\tau|^{(1-1/\alpha)(d-\beta)}=|\tau|^{(1-1/\alpha)d-\beta}$

This proves (5.1). To prove (5.2) consider the region
$||x||\leq|\widetilde{\delta}|/2$ which yields

$||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\geq\big{|}||x||-||\widetilde{\delta}||\big{|}=||\widetilde{\delta}||-||x||\geq\frac{1}{2}\cdot||\widetilde{\delta}||.$

Thus for the estimates in (5.2) we have

$\displaystyle{\mathbb{E}}\big{[}||(\tau,\mathrm{sign}(\tau)\cdot
X_{|\tau|}+\delta)||^{-\beta}\big{]}=|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||(\widetilde{\tau},\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta})||^{-\beta}\cdot p(x)\,\textnormal{d}x$
$\displaystyle\quad\lesssim||\delta||^{-\beta}+\underbrace{|\tau|^{-\beta/\alpha}\int_{\\{||x||\geq||\widetilde{\delta}||/2,\,||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\geq|\widetilde{\tau}|\\}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\cdot p(x)\,\textnormal{d}x}_{=:I_{1}}$
$\displaystyle\quad\quad+\
\underbrace{|\tau|^{-\beta/\alpha}\int_{\\{||x||\geq||\widetilde{\delta}||/2,\,||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\leq|\widetilde{\tau}|\\}}\widetilde{\tau}^{-\beta}\cdot
p(x)\,\textnormal{d}x}_{=:I_{2}}.$

Now,

$\displaystyle I_{1}$
$\displaystyle=|\tau|^{-\beta/\alpha}\int_{\\{||x||\geq||\widetilde{\delta}||/2,\,||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\geq|\widetilde{\tau}|\\}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\cdot p(x)\,\textnormal{d}x$
$\displaystyle=|\tau|^{-\beta/\alpha}\int_{\\{||x||\geq||\widetilde{\delta}||/2,\,||x+\widetilde{\delta}||\geq|\widetilde{\tau}|,\,||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\geq||\widetilde{\delta}||\\}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\cdot p(x)\,\textnormal{d}x$
$\displaystyle\quad+\
|\tau|^{-\beta/\alpha}\int_{\\{||x||\geq||\widetilde{\delta}||/2,\,||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\geq|\widetilde{\tau}|,\,||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\leq||\widetilde{\delta}||\\}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\cdot p(x)\,\textnormal{d}x$
$\displaystyle\leq||\delta||^{-\beta}+\underbrace{|\tau|^{-\beta/\alpha}\int_{\\{||x||\geq||\widetilde{\delta}||/2,\,||\widetilde{\delta}||\geq||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\geq|\widetilde{\tau}|\\}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\cdot p(x)\,\textnormal{d}x}_{=:I_{3}}.$

By using (1.3) we further have

(5.4)
$\begin{split}I_{3}&=|\tau|^{-\beta/\alpha}\int_{\\{||x||\geq||\widetilde{\delta}||/2,\,||\widetilde{\delta}||\geq||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\geq|\widetilde{\tau}|\\}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\cdot p(x)\,\textnormal{d}x\\\
&\lesssim|\tau|^{-\beta/\alpha}\cdot||\widetilde{\delta}||^{-d-\alpha}\int_{\\{||\widetilde{\delta}||\geq||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\geq|\widetilde{\tau}|\\}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\,\textnormal{d}x\\\
&=|\tau|^{-\beta/\alpha}\cdot||\widetilde{\delta}||^{-d-\alpha}\int_{|\widetilde{\tau}|}^{||\widetilde{\delta}||}r^{d-\beta-1}\,\textnormal{d}r.\end{split}$

For $\alpha\in[1,2)$, $\beta<d$ and $|\tau|\leq||\delta||^{\alpha}$ by (5.4)
we get

$\displaystyle I_{3}$
$\displaystyle\lesssim|\tau|^{-\beta/\alpha}\cdot||\widetilde{\delta}||^{-\alpha-\beta}=|\tau|\cdot||\delta||^{-\alpha-\beta}\lesssim||\delta||^{-\beta},$

whereas for $\alpha\in(0,1]$ and $|\tau|\leq||\delta||$ one has

$\displaystyle
I_{3}\lesssim|\tau|\cdot||\delta||^{-\alpha-\beta}\leq||\delta||^{1-\alpha}\cdot||\delta||^{-\beta}\lesssim||\delta||^{-\beta}.$

For $\alpha\in[1,2)$, $\beta>d$ and $|\tau|\leq||\delta||^{\alpha}$ by (5.4)
one has

$\displaystyle I_{3}$
$\displaystyle\lesssim|\tau|^{-\beta/\alpha}\cdot||\widetilde{\delta}||^{-d-\alpha}\int_{\widetilde{\tau}}^{\infty}r^{-\beta}\cdot
r^{d-1}\,\textnormal{d}r\lesssim|\tau|^{-\beta/\alpha}\cdot||\widetilde{\delta}||^{-d-\alpha}\cdot\widetilde{\tau}^{\,d-\beta}$
$\displaystyle=|\tau|^{d+1-\beta}\cdot||\delta||^{-d-\alpha}\leq||\delta||^{\alpha(d+1-\beta)}\cdot||\delta||^{-d-\alpha}=||\delta||^{(\alpha-1)d-\alpha\beta}.$

Finally, by using (1.3) we get

$\displaystyle I_{2}$
$\displaystyle=|\tau|^{-\beta/\alpha}\int_{\\{||x||\geq||\widetilde{\delta}||/2,\,||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\leq|\widetilde{\tau}|\\}}|\widetilde{\tau}|^{-\beta}\cdot
p(x)\,\textnormal{d}x$
$\displaystyle\lesssim|\tau|^{-\beta/\alpha}\cdot|\widetilde{\tau}|^{-\beta}\int_{||x+\mathrm{sign}(\tau)\cdot\widetilde{\delta}||\leq|\widetilde{\tau}|\\}}||x||^{-d-\alpha}\,\textnormal{d}x$
$\displaystyle\lesssim|\tau|^{-\beta/\alpha}\cdot|\widetilde{\tau}|^{-\beta}\cdot||\widetilde{\delta}||^{-d-\alpha}\cdot|\widetilde{\tau}|^{d}$
$\displaystyle=|\tau|^{d-\beta+1}\cdot||\delta||^{-d-\alpha}$

using the volume of a ball with radius $|\widetilde{\tau}|$ and center
$-\mathrm{sign}(\tau)\cdot\widetilde{\delta}$. Now, $\alpha\in(0,1]$,
$\beta<d$ and $|\tau|\leq||\delta||$ result in

$\displaystyle I_{2}$
$\displaystyle\lesssim|\tau|^{d-\beta+1}\cdot||\delta||^{-d-\alpha}\leq||\delta||^{1-\alpha-\beta}\leq||\delta||^{-\beta}.$

If $\alpha\in[1,2)$, $\beta\leq d$ and $|\tau|\leq||\delta||^{\alpha}$ one has

$\displaystyle I_{2}$
$\displaystyle\lesssim|\tau|^{d-\beta+1}\cdot||\delta||^{-d-\alpha}\leq||\delta||^{(\alpha-1)\cdot
d-\alpha\beta}\leq||\delta||^{-\beta}.$

If $\alpha\in[1,2)$, $\beta\geq d$ and $|\tau|\leq||\delta||^{\alpha}$ one has

$\displaystyle I_{2}$ $\displaystyle\lesssim||\delta||^{(\alpha-1)\cdot
d-\alpha\beta}\quad\text{ and
}\quad||\delta||^{-\beta}\leq||\delta||^{(\alpha-1)\cdot d-\alpha\beta}$

which concludes the proof. ∎

Now, we are able to calculate the lower bound via energy estimates.

######  Theorem 5.7.

Let $T\subseteq\mathbb{R}_{+}$ be a Borel set and $\alpha\in(0,2)$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable Lévy process in
${\mathbb{R}^{d}}$ and
$f:T\rightarrow\\{y\in{\mathbb{R}^{d}}:\|y-x\|\leq\frac{1}{2}\\}$ for fixed
$x\in{\mathbb{R}^{d}}$ be a Borel measurable function. Let
$\varphi_{\alpha}=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$. Then
one $\mathbb{P}$-almost surely has

(5.5)
$\dim\pazocal{G}_{T}(X+f)\geq\begin{cases}\varphi_{1},&\alpha\in(0,1],\\\
\varphi_{\alpha}\ \wedge\
\frac{1}{\alpha}\cdot\varphi_{\alpha}+\big{(}1-\frac{1}{\alpha}\big{)}\cdot
d,&\alpha\in[1,2].\end{cases}$

###### Proof.

We define the increments $\tau:=t-s$ and $\delta:=f(t)-f(s)$ with
$||\delta||\in[0,1]$ and consider the difference kernel
$K^{\beta}(t,x)={\mathbb{E}}\big{[}||(t,\mathrm{sign}(t)\cdot
X_{|t|}+x)||^{-\beta}\big{]}$. We prove that
$\pazocal{E}_{K^{\beta}}(\mu)<\infty$ holds for
$\mu\in\pazocal{M}^{1}(\pazocal{G}_{T}(f))$ from the parabolic version of
Frostman’s lemma in Theorem 5.4 and for every $\beta$ less than the right-hand
side of (5.5). Then the claim follows due to Lemma 5.3. For the energy
integral we have

(5.6)
$\begin{split}&\pazocal{E}_{K^{\beta}}(\mu)=\int\int_{\pazocal{G}_{T}(f)\times\pazocal{G}_{T}(f)}K^{\beta}(t-s,f(t)-f(s))\,\textnormal{d}\mu(s,x)\,\textnormal{d}\mu(t,y)\\\
&\quad\leq\int\int_{\\{|t-s|\in(0,1]\\}}K^{\beta}(\tau,\delta)\,\textnormal{d}\mu\,\textnormal{d}\mu+\int\int_{\\{|t-s|\in(1,\infty)\\}}|t-s|^{-\beta}\,\textnormal{d}\mu\,\textnormal{d}\mu\\\
&\quad\lesssim\int\int_{\\{|\tau|\in(0,1]\\}}{\mathbb{E}}\big{[}||(\tau,\mathrm{sign}(\tau)\cdot
X_{|\tau|}(\omega)+\delta)||^{-\beta}\big{]}\,\textnormal{d}\mu\,\textnormal{d}\mu.\end{split}$

_(i)_ We begin with the case $\alpha\in(0,1]$ and
$\beta=\varphi_{1}-2\varepsilon$ for some arbitrary $\varepsilon>0$. Due to
Lemma 5.6 we have

$\displaystyle\pazocal{E}_{K^{\beta}}(\mu)\lesssim$ $\displaystyle\
\underbrace{\int\int_{\\{|\tau|\in(0,1],\,||\delta||\in[0,\,|\tau|]\\}}|\tau|^{-\beta}\,\textnormal{d}\mu\,\textnormal{d}\mu}_{=:\
I_{1}}+\underbrace{\int\int_{\\{|\tau|\in(0,\,1],\,||\delta||\in(|\tau|,\,1]\\}}||\delta||^{-\beta}\,\textnormal{d}\mu\,\textnormal{d}\mu}_{=:\
I_{2}}.$

We get

$\displaystyle I_{1}\lesssim$ $\displaystyle\
\sum_{k=1}^{\infty}2^{k\beta}\cdot\mu\otimes\mu\,\big{\\{|\tau|\in\big{(}2^{-k},\,2\cdot
2^{-k}\big{]},\,||\delta||\in\big{[}0,2\cdot 2^{-k}\big{]}\big{\\}}}.$

Further,

$\displaystyle I_{2}\lesssim$ $\displaystyle\
\sum_{k=1}^{\infty}2^{k\beta}\cdot\mu\otimes\mu\,\big{\\{|\tau|\in\big{(}0,2\cdot
2^{-k}\big{]},\ ||\delta||\in\big{(}2^{-k},\,2\cdot 2^{-k}\big{]}\big{\\}}}.$

Now we have to calculate the expressions $\mu\otimes\mu\\{\cdot\\}$ for
$I_{1}$ and $I_{2}$. For each $k\in{\mathbb{N}}$ we tile
$\mathbb{R}_{+}\times{\mathbb{R}^{d}}$ by disjoint hypercubes of size
$2^{-k}\times\cdots\times 2^{-k}$ and denote the collection of such hypercubes
by $\pazocal{D}_{k}$. For every $c\in(0,1]$, $\gamma=\varphi_{1}-\varepsilon$
and $\alpha\in(0,1]$ Frostman’s lemma 5.4 yields

$\mu\Bigg{(}[t,\,t+c]\times\prod_{i=1}^{d}\,\big{[}x_{i},\,x_{i}+c\big{]}\Bigg{)}\lesssim
c^{\gamma},$

in particular we have $\mu(Q^{\prime})\lesssim 2^{-k\gamma}$ for each
$Q^{\prime}\in\pazocal{D}_{k}$. In order to estimate $I_{1}$ we define the
following relation on $\pazocal{D}_{k}$. For two hypercubes
$Q,Q^{\prime}\in\pazocal{D}_{k}$ we write $Q\sim Q^{\prime}$ if there exists
$(s,x)\in Q$ and $(t,y)\in Q^{\prime}$ such that
$|\tau|=|t-s|\in\big{(}2^{-k},\,2\cdot 2^{-k}\big{]}$ and
$||\delta||=\|y-x\|\in\big{[}0,\,2\cdot 2^{-k}\big{]}$. Thus

$\displaystyle I_{1}\lesssim$ $\displaystyle\
\sum_{k=1}^{\infty}2^{k\beta}\sum_{\begin{subarray}{c}Q,Q^{\prime}\in\pazocal{D}_{k}\\\
Q\sim Q^{\prime}\end{subarray}}\mu\otimes\mu\,(Q\times
Q^{\prime})=\sum_{k=1}^{\infty}2^{k\beta}\sum_{\begin{subarray}{c}Q,Q^{\prime}\in\pazocal{D}_{k}\\\
Q\sim Q^{\prime}\end{subarray}}\mu(Q)\cdot\mu(Q^{\prime}).$

The number of hypercubes related to a fixed $Q\in\pazocal{D}_{k}$ via $\sim$
is bounded by a universal constant not depending on $k$ and $Q$, hence

$\displaystyle I_{1}$
$\displaystyle\lesssim\sum_{k=1}^{\infty}2^{k\beta}\sum_{Q\in\pazocal{D}_{k}}\sum_{Q\sim
Q^{\prime}}\mu(Q)\cdot\mu(Q^{\prime})\lesssim\sum_{k=1}^{\infty}2^{k(\beta-\gamma)}\cdot\sum_{Q\in\pazocal{D}_{k}}\mu(Q).$

Note that
$\sum_{Q\in\pazocal{D}_{k}}\mu(Q)=\mu\big{(}\bigcup_{Q\in\pazocal{D}_{k}}Q\big{)}=\mu\big{(}\mathbb{R}_{+}\times{\mathbb{R}^{d}}\big{)}=1$
and we conclude

$\displaystyle I_{1}\lesssim$ $\displaystyle\
\sum_{k=1}^{\infty}2^{k(\beta-\gamma)}=\sum_{k=1}^{\infty}2^{-k\varepsilon}<\infty,$

since $\beta=\varphi_{1}-2\varepsilon$ and $\gamma=\varphi_{1}-\varepsilon$.
For the estimation of $I_{2}$ we define a novel relation on $\pazocal{D}_{k}$.
For two hypercubes $Q,Q^{\prime}\in\pazocal{D}_{k}$ we write $Q\approx
Q^{\prime}$ if there exists $(s,x)\in Q$ and $(t,y)\in Q^{\prime}$ such that
$|\tau|\in\big{(}0,\,2\cdot 2^{-k}\big{]}$ and
$||\delta||\in\big{(}2^{-k},\,2\cdot 2^{-k}\big{]}$. Thus

$\displaystyle
I_{2}\lesssim\sum_{k=1}^{\infty}2^{k\beta}\sum_{\begin{subarray}{c}Q,Q^{\prime}\in\pazocal{D}_{k}\\\
Q\approx Q^{\prime}\end{subarray}}\mu(Q)\cdot\mu(Q^{\prime}).$

Again, the number of hypercubes related to some fixed $Q$ via $\approx$ is
bounded by a universal constant. Hence the same calculation as for $I_{1}$
yields

$\displaystyle
I_{2}\lesssim\sum_{k=1}^{\infty}2^{k(\beta-\gamma)}=\sum_{k=1}^{\infty}2^{-k\varepsilon}<\infty.$

_(ii)_ Now we treat the case $\alpha\in[1,2)$ and $\varphi_{\alpha}\leq d$.
Let $\beta=\varphi_{\alpha}-2\alpha\cdot\varepsilon<d$ for some arbitrary
$\varepsilon>0$. Due to Lemma 5.6 we have

$\displaystyle\pazocal{E}_{K^{\beta}}(\mu)\lesssim$ $\displaystyle\
\underbrace{\int\int_{\\{|\tau|\in(0,\,1],\,||\delta||\in[0,\,|\tau|^{1/\alpha}]\\}}|\tau|^{-\beta/\alpha}\,\textnormal{d}\mu\,\textnormal{d}\mu}_{=:\
I_{3}}$
$\displaystyle+\underbrace{\int\int_{\\{|\tau|\in(0,\,1],\,||\delta||\in(|\tau|^{1/\alpha},\,1]\\}}||\delta||^{-\beta}\,\textnormal{d}\mu\,\textnormal{d}\mu}_{=:\
I_{4}}.$

We get

$\displaystyle I_{3}\lesssim$ $\displaystyle\
\sum_{k=1}^{\infty}2^{k\beta/\alpha}\cdot\mu\otimes\mu\,\big{\\{}|\tau|\in\big{(}2^{-k},\,2\cdot
2^{-k}\big{]},\,||\delta||\in\big{[}0,2^{1/\alpha}\cdot
2^{-k/\alpha}\big{]}\big{\\}}.$

Further,

$\displaystyle I_{4}\lesssim$ $\displaystyle\
\sum_{k=1}^{\infty}2^{k\beta/\alpha}\cdot\mu\otimes\mu\,\big{\\{}|\tau|\in\big{(}0,\,2\cdot
2^{-k}\big{]},\ ||\delta||\in\big{(}2^{-k/\alpha},\,2^{1/\alpha}\cdot
2^{-k/\alpha}\big{]}\big{\\}}.$

Now we have to calculate the expressions $\mu\otimes\mu\\{\cdot\\}$ for
$I_{3}$ and $I_{4}$. For each $k\in{\mathbb{N}}$ we tile
$\mathbb{R}_{+}\times{\mathbb{R}^{d}}$ by disjoint $\alpha$-parabolic
cylinders of size $2^{-k}\times 2^{-k/\alpha}\times\cdots\times 2^{-k/\alpha}$
and again denote the collection of such cylinders by $\pazocal{D}_{k}$. For
every $c\in(0,1]$, $\gamma=\varphi_{\alpha}-\alpha\cdot\varepsilon$ and
$\alpha\in[1,2)$ Frostman’s lemma 5.4 yields

$\mu\Bigg{(}[t,\,t+c]\times\prod_{i=1}^{d}\,\big{[}x_{i},\,x_{i}+c^{1/\alpha}\big{]}\Bigg{)}\lesssim
c^{\gamma/\alpha},$

in particular we have $\mu(Q^{\prime})\lesssim 2^{-k\gamma/\alpha}$ for each
$Q^{\prime}\in\pazocal{D}_{k}$. The same technique as in (i) results in

$\displaystyle I_{3},I_{4}\lesssim$ $\displaystyle\
\sum_{k=1}^{\infty}2^{k(\beta-\gamma)/\alpha}\leq\sum_{k=1}^{\infty}2^{-k\varepsilon}<\infty,$

since $\beta=\varphi_{\alpha}-2\alpha\cdot\varepsilon$ and
$\gamma=\varphi_{\alpha}-\alpha\cdot\varepsilon$.

_(iii)_ Finally, we treat the case $\alpha\in[1,2)$ and $\varphi_{\alpha}>d$.
Let $\beta=(1-\frac{1}{\alpha})\cdot
d+\frac{1}{\alpha}\cdot\varphi_{\alpha}-2\varepsilon>d$ for for sufficiently
small $\varepsilon>0$. Due to Lemma 5.6 we have

$\displaystyle\pazocal{E}_{K^{\beta}}(\mu)$
$\displaystyle\lesssim\int\int_{\\{|\tau|\in(0,\,1],\,||\delta||\in[0,\,|\tau|^{1/\alpha}]\\}}|\tau|^{(1-1/\alpha)d-\beta},\,\textnormal{d}\mu\,\textnormal{d}\mu$
$\displaystyle\quad+\int\int_{\\{|\tau|\in(0,\,1],\,||\delta||\in(|\tau|^{1/\alpha},\,1]\\}}||\delta||^{(\alpha-1)d-\alpha\beta}\,\textnormal{d}\mu\,\textnormal{d}\mu$

and the same techniques as in _(ii)_ yield the finiteness of this expression
when choosing $\gamma=\varphi_{\alpha}-\varepsilon$ in Frostman’s lemma 5.4. ∎

##  6\. Range: Upper and Lower Bounds

We give upper and lower bounds for the Hausdorff dimension of the range of a
stable Lévy process with drift.

######  Theorem 6.1.

Let $T\subseteq\mathbb{R}_{+}$ be any set and $\alpha\in(0,2]$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable Lévy process in
${\mathbb{R}^{d}}$ and $f:T\rightarrow{\mathbb{R}^{d}}$ be any function.
Define $\varphi_{\alpha}:=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$.
Then one $\mathbb{P}$-almost surely has

(6.1) $\dim\pazocal{R}_{T}(X+f)\leq\begin{cases}\alpha\cdot\varphi_{\alpha}\
\wedge\ d,&\alpha\in(0,1],\\\ \varphi_{\alpha}\ \wedge\
d,&\alpha\in[1,2].\end{cases}$

###### Proof.

The Gaussian case follows from the proof of Theorem 1.2 in [14] and
Proposition 2.2. Since the Hausdorff dimension of the range never exceeds the
topological dimension of the space a function maps to we always have
$\dim\pazocal{R}_{T}(X+f)\leq d$. In case of $\alpha\in[1,2)$ the claim
directly follows from Theorem 2.4 and Theorem 4.1 which yield

$\dim\pazocal{R}_{T}(X+f)\leq\dim\pazocal{G}_{T}(X+f)\leq\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(X+f)\leq\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)=\varphi_{\alpha}.$

Now, let $\alpha\in(0,1]$, $\beta=\alpha\cdot\varphi_{\alpha}$ and let
$\delta,\,\varepsilon>0$ be arbitrary. Then $\pazocal{G}_{T}(f)$ can be
covered by $\alpha$-parabolic cylinders

$\big{(}\mathsf{P}_{c_{k}}\big{)}_{k\in{\mathbb{N}}}=\Bigg{(}[t_{k},\,t_{k}+c_{k}]\times\prod_{i=1}^{d}\,\big{[}x_{i,k},\,x_{i,k}+c_{k}^{1/\alpha}\big{]}\Bigg{)}_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{\alpha}$

such that
$\sum_{k=1}^{\infty}|\mathsf{P}_{c_{k}}|^{(\beta+\delta)/\alpha}\lesssim\sum_{k=1}^{\infty}c_{k}^{(\beta+\delta)/\alpha}\leq\varepsilon$.
Let $M_{k}(\omega)$ be the random number of a fixed $2^{d}$-nested collection
of hypercubes with sidelength $c_{k}^{1/\alpha}$ that the path $t\mapsto
X_{t}(\omega)$ hits at some time $t\in[t_{k},\,t_{k}+c_{k}]$. As in the proof
of Theorem 4.1 we obtain a random parabolic cover $\bigcup_{k\in{\mathbb{N}}}\
\widetilde{\mathsf{P}}_{c_{k}}(\omega)\supseteq\pazocal{G}_{T}(X(\omega)+f)$
where

$\displaystyle\widetilde{\mathsf{P}}_{c_{k}}(\omega)=[t_{k},\,t_{k}+c_{k}]\times\bigcup_{j=1}^{M_{k}(\omega)}\prod_{i=1}^{d}$
$\displaystyle\left(\big{[}\xi_{i,j,k}(\omega)+x_{i,k},\,\xi_{i,j,k}(\omega)+x_{i,k}+c_{k}^{1/\alpha}\big{]}\right.$
$\displaystyle\cup\ $
$\displaystyle\left.\big{[}\xi_{i,j,k}(\omega)+x_{i,k}+c_{k}^{1/\alpha},\,\xi_{i,j,k}(\omega)+x_{i,k}+2c_{k}^{1/\alpha}\big{]}\right).$

By projection we get the random cover
$\bigcup_{k\in{\mathbb{N}}}\square_{c_{k}^{1/\alpha}}\supseteq\pazocal{R}_{T}(X(\omega)+f)$
with

$\displaystyle\square_{c_{k}^{1/\alpha}}(\omega)=\bigcup_{k=1}^{M_{k}(\omega)}\prod_{i=1}^{d}$
$\displaystyle\left(\big{[}\xi_{i,j,k}(\omega)+x_{i,k},\xi_{i,j,k}(\omega)+x_{i,k}+c_{k}^{1/\alpha}\big{]}\right.$
$\displaystyle\cup\ $
$\displaystyle\left.\big{[}\xi_{i,j,k}(\omega)+x_{i,k}+c_{k}^{1/\alpha},\xi_{i,j,k}(\omega)+x_{i,k}+2c_{k}^{1/\alpha}\big{]}\right).$

This is a union of $M_{k}(\omega)\cdot 2^{d}$ hypercubes with diameter
$|\square_{c_{k}^{1/\alpha}}(\omega)|\asymp c_{k}^{1/\alpha}$. An application
of Pruitt and Taylor’s covering Lemma 6.1 in [15] and Lemma 3.4 in [11] shows
that for all $\delta^{\prime}>0$ we have

${\mathbb{E}}[M_{k}]\lesssim\frac{c_{k}}{{\mathbb{E}}\Big{[}T\Big{(}c_{k}^{1/\alpha}/3,c_{k}\Big{)}\Big{]}}\lesssim
c_{k}^{-\delta^{\prime}/\alpha},$

where $T\big{(}c_{k}^{1/\alpha}/3,c_{k}\big{)}$ denotes the sojourn time of
the process $(X_{t})_{t\in[0,c_{k}]}$ in a ball of radius $c_{k}^{1/\alpha}/3$
centerd at the origin. Hence we get for
$\varepsilon^{\prime}=\delta+\delta^{\prime}>0$

$\displaystyle{\mathbb{E}}\Big{[}\pazocal{H}^{\beta+\varepsilon^{\prime}}(\pazocal{R}_{T}(X+f))\Big{]}\leq{\mathbb{E}}\bigg{[}\sum_{k=1}^{\infty}\big{|}\square_{c_{k}^{1/\alpha}}\big{|}^{\beta+\varepsilon^{\prime}}\bigg{]}\lesssim{\mathbb{E}}\Bigg{[}\sum_{k=1}^{\infty}M_{k}\cdot
2^{d}\cdot c_{k}^{(\beta+\varepsilon^{\prime})/\alpha}\Bigg{]}$
$\displaystyle\quad\lesssim\sum_{k=1}^{\infty}{\mathbb{E}}[M_{k}]\cdot
c_{k}^{(\beta+\varepsilon^{\prime})/\alpha}\lesssim\sum_{k=1}^{\infty}c_{k}^{(\beta+\varepsilon^{\prime}-\delta^{\prime})/\alpha}=\sum_{k=1}^{\infty}c_{k}^{(\beta+\delta)/\alpha}\leq\varepsilon.$

Since $\varepsilon,\varepsilon^{\prime}>0$ are arbitrary, we get for all
$\alpha\in(0,1]$ and $\beta^{\prime}>\beta$

${\mathbb{E}}\Big{[}\pazocal{H}^{\beta^{\prime}}(\pazocal{R}_{T}(X+f))\Big{]}=0,$

hence

$\pazocal{H}^{\beta^{\prime}}(\pazocal{R}_{T}(X+f))=0\quad\mathbb{P}\text{-almost
surely}.$

Since $\beta^{\prime}>\beta$ is also arbitrary, we finally get
$\dim\pazocal{R}_{T}(X+f)\leq\beta=\alpha\cdot\varphi_{\alpha}$
$\mathbb{P}$-almost surely, as claimed. ∎

The lower bound is obtained by the energy method. The stable process $X$ is
transformed into the kernel of the energy integral.

######  Lemma 6.2.

Let $T\subseteq\mathbb{R}_{+}$ be a Borel set and $\alpha\in(0,2)$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic stable Lévy process in
${\mathbb{R}^{d}}$ and $f:\mathbb{R}_{+}\rightarrow{\mathbb{R}^{d}}$ be a
Borel measurable function. Define the difference kernel

$\kappa^{\beta}(t,x):={\mathbb{E}}\big{[}||\mathrm{sign}(t)\cdot
X_{|t|}+x||^{-\beta}\big{]}.$

Then $\operatorname{Cap}_{\kappa^{\beta}}(\pazocal{G}_{T}(f))>0$ implies that
$\mathbb{P}$-almost surely
$\operatorname{Cap}_{\beta}(\pazocal{R}_{T}(X(\omega)+f))>0$ holds. Hence
$\pazocal{E}_{\kappa^{\beta}}(\mu)<\infty$ for some probability measure
$\mu\in\pazocal{M}^{1}(\pazocal{G}_{T}(f))$ implies
$\dim\pazocal{R}_{T}(X+f)\geq\beta$ $\,\mathbb{P}$-almost surely.

###### Proof.

Let $\mu\in\pazocal{M}^{1}(\pazocal{G}_{T}(f))$ and $\pi_{t}$ denote the
projection onto the time component, i.e. $\pi_{t}(t,f(t))=t$. Define the
probability measure $\nu\in\pazocal{M}^{1}(\mathbb{R}_{+})$ as the pushforward
measure $\nu(A)=\mu\big{(}\pi_{t}^{-1}(A)\big{)}$ for Borel sets
$A\subseteq\mathbb{R}_{+}$ and further the random probability measure
$\widetilde{\mu}_{\omega}(R)=\nu((X(\omega)+f)^{-1}(R))$ for every Borel set
$R\subseteq{\mathbb{R}^{d}}$. Then Tonelli’s theorem and the stationarity of
the increments of $X$ yield

$\displaystyle{\mathbb{E}}\big{[}\pazocal{E}_{\beta}(\widetilde{\mu}_{\omega})\big{]}$
$\displaystyle={\mathbb{E}}\bigg{[}\int_{\pazocal{R}_{T}(X(\omega)+f)}\int_{\pazocal{R}_{T}(X(\omega)+f)}||x-y||^{-\beta}\,\textnormal{d}\widetilde{\mu}_{\omega}(x)\,\textnormal{d}\widetilde{\mu}_{\omega}(y)\bigg{]}$
$\displaystyle={\mathbb{E}}\bigg{[}\int_{T}\int_{T}||X_{t}(\omega)+f(t)-(X_{s}(\omega)+f(s))||^{-\beta}\,\textnormal{d}\nu(t)\,\textnormal{d}\nu(s)\bigg{]}$
$\displaystyle=\int_{\pazocal{G}_{T}(f)}\int_{\pazocal{G}_{T}(f)}{\mathbb{E}}\big{[}||X_{t}(\omega)-X_{s}(\omega)+x-y||^{-\beta}\big{]}\,\textnormal{d}\mu(t,x)\,\textnormal{d}\mu(s,y)$
$\displaystyle=\int_{\pazocal{G}_{T}(f)}\int_{\pazocal{G}_{T}(f)}{\mathbb{E}}\big{[}||\mathrm{sign}(t-s)\cdot
X_{|t-s|}(\omega)+x-y||^{-\beta}\big{]}\,\textnormal{d}\mu(t,x)\,\textnormal{d}\mu(s,y)$
$\displaystyle=\pazocal{E}_{\kappa^{\beta}}(\mu).$

By assumption, there exists $\mu\in\pazocal{M}^{1}(\pazocal{G}_{T}(f))$ such
that $\pazocal{E}_{\kappa^{\beta}}(\mu)<\infty$ holds. From that one
$\mathbb{P}$-almost surely has
$\pazocal{E}_{\beta}(\widetilde{\mu}_{\omega})<\infty$ and the final statement
immediately follows by Frostman’s Theorem 5.2 since the range of a Borel set
under a Borel measurable function is a Suslin set, see Section 11 in [8]. ∎

We make use of new kernel estimates.

######  Lemma 6.3.

Let $\alpha\in(0,2)$ and $X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable
Lévy process in ${\mathbb{R}^{d}}$. Let $\beta\in(0,d)$ and
$\tau\in\mathbb{R}$, $\delta\in{\mathbb{R}^{d}}$ be such that
$|\tau|\in(0,1]$, $||\delta||\in[0,1]$. Then appropriate choices for
estimating the difference kernel from Lemma 6.2
$\kappa^{\beta}(\tau,\delta)={\mathbb{E}}\big{[}||\mathrm{sign}(t)\cdot
X_{|t|}+x||^{-\beta}\big{]}$ are

$\kappa^{\beta}(\tau,\delta)\lesssim\begin{cases}|\tau|^{-\beta/\alpha},\\\
||\delta||^{-\beta},&\text{ for }|\tau|\leq||\delta||^{\alpha}.\end{cases}$

###### Proof.

Let $p(x)$ denote the density function of
$X_{1}\overset{\mathrm{d}}{=}|\tau|^{-1/\alpha}X_{|\tau|}$. We define the
rescaled increment $\widetilde{\delta}:=\delta/|\tau|^{1/\alpha}$. The self-
similarity of the stable Lévy process and Lemma 5.5 yield

$\displaystyle{\mathbb{E}}\big{[}||\mathrm{sign}(\tau)\cdot
X_{|\tau|}+\delta||^{-\beta}\big{]}=|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\cdot p(x)\,\textnormal{d}x$
$\displaystyle\quad=|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||x+\mathrm{sign}(\tau)\cdot\widetilde{\delta}||^{-\beta}\cdot
p(x)\,\textnormal{d}x$
$\displaystyle\quad\lesssim|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||x||^{-\beta}\cdot
p(x)\,\textnormal{d}x\lesssim|\tau|^{-\beta/\alpha}\cdot{\mathbb{E}}\big{[}||X_{1}||^{-\beta}\big{]}\lesssim|\tau|^{-\beta/\alpha},$

since negative moments of order $\beta<d$ exist; see Lemma 3.1 in [1]. Now
consider the region $||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\leq||\widetilde{\delta}||/2$ which yields

$||x||=||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}-\widetilde{\delta}||\geq\big{|}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||-||\widetilde{\delta}||\big{|}=||\widetilde{\delta}||-||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\geq\frac{1}{2}\cdot||\widetilde{\delta}||.$

Thus $\beta<d$ and $\tau\leq||\delta||^{\alpha}$ lead to

$\displaystyle{\mathbb{E}}\big{[}||\mathrm{sign}(\tau)\cdot
X_{|\tau|}+\delta||^{-\beta}\big{]}=|\tau|^{-\beta/\alpha}\int_{\mathbb{R}^{d}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\cdot p(x)\,\textnormal{d}x$
$\displaystyle\quad\lesssim||\delta||^{-\beta}+|\tau|^{-\beta/\alpha}\int_{\\{||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||\leq||\widetilde{\delta}||/2\\}}||\mathrm{sign}(\tau)\cdot
x+\widetilde{\delta}||^{-\beta}\cdot p(x)\,\textnormal{d}x$
$\displaystyle\quad\lesssim||\delta||^{-\beta}+|\tau|^{-\beta/\alpha}\cdot||\widetilde{\delta}||^{-d-\alpha}\int_{0}^{||\widetilde{\delta}||}r^{d-\beta-1}\,\textnormal{d}r=||\delta||^{-\beta}+|\tau|^{-\beta/\alpha}\cdot||\widetilde{\delta}||^{-\alpha-\beta}$
$\displaystyle\quad=||\delta||^{-\beta}+|\tau|\cdot||\delta||^{-\alpha-\beta}\lesssim||\delta||^{-\beta},$

where we have used (1.3) to estimate the tail-densities. ∎

We get a lower bound for the Hausdorff dimension of the range of a stable Lévy
process with drift.

######  Theorem 6.4.

Let $T\subseteq\mathbb{R}_{+}$ be a Borel set and $\alpha\in(0,2)$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable Lévy process in
${\mathbb{R}^{d}}$ and
$f:T\rightarrow\\{y\in{\mathbb{R}^{d}}:\|y-x\|\leq\frac{1}{2}\\}$ for fixed
$x\in{\mathbb{R}^{d}}$ be a Borel measurable function. Let
$\varphi_{\alpha}=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$. Then
one $\mathbb{P}$-almost surely has

(6.2) $\dim\pazocal{R}_{T}(X+f)\geq\begin{cases}\alpha\cdot\varphi_{\alpha}\
\wedge\ d,&\alpha\in(0,1],\\\ \varphi_{\alpha}\ \wedge\
d,&\alpha\in[1,2).\end{cases}$

###### Proof.

We consider the difference kernel
$\kappa^{\beta}(t,x)={\mathbb{E}}\big{[}||\mathrm{sign}(t)\cdot
X_{|t|}(\omega)+x||^{-\beta}\big{]}$ from Lemma 6.2. Analogously to the proof
of Theorem 5.7, we can show that $\pazocal{E}_{\kappa^{\beta}}(\mu)$ is finite
for $\mu\in\pazocal{M}^{1}(\pazocal{G}_{T}(f))$ from the parabolic version of
Frostman’s Lemma in Theorem 5.4 and for every $\beta$ less than the right-hand
side of (6.2). Note that the different lower bounds for the dimension of the
range in cases $\alpha\in(0,1]$ and $\alpha\in[1,2)$ occur due to the
different upper bounds in the parabolic version of Frostman’s Lemma. ∎

##  7\. Estimates for the Parabolic Hausdorff Dimension

We first give an estimates for the $\alpha$-parabolic Hausdorff dimension of a
constant function.

######  Lemma 7.1.

Let $T\subseteq\mathbb{R}$ be any set and $\alpha\in(0,\infty)$. Define the
constant function $f_{C}(x)=C\in{\mathbb{R}^{d}}$ for all $x\in T$. Then one
has

$\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{C})\leq(\alpha\vee
1)\cdot\dim T$

###### Proof.

Without loss of generality, let $f_{C}=f_{0}\equiv 0\in\mathbb{R}^{d}$. _(i)_
Let $\alpha\in(0,1]$, $\beta=\dim T$ and let $\delta,\varepsilon>0$ be
arbitrary. Then there exists a cover $\bigcup_{k\in{\mathbb{N}}}\
T_{k}\supseteq T$ with $T_{k}=[t_{k},\,t_{k}+c_{k}]$ and $c_{k}\leq 1$ such
that
$\sum_{k=1}^{\infty}|T_{k}|^{\beta+\delta}=\sum_{k=1}^{\infty}c_{k}^{\beta+\delta}\leq\varepsilon.$
Now, $\pazocal{G}_{T}(f_{0})$ can be covered by $\alpha$-parabolic cylinders

$\big{(}\mathsf{P}_{c_{k}}\big{)}_{k\in{\mathbb{N}}}=\Bigg{(}[t_{k},\,t_{k}+c_{k}]\times\prod_{j=1}^{d}\
[0,c_{k}^{1/\alpha}]\Bigg{)}_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{\alpha}.$

Note that $|\mathsf{P}_{c_{k}}|\asymp c_{k}$. Hence

$\displaystyle\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta+\delta}(\pazocal{G}_{T}(f_{0}))\leq\sum_{k=1}^{\infty}|\mathsf{P}_{c_{k}}|^{\beta+\delta}\lesssim\sum_{k=1}^{\infty}c_{k}^{\beta+\delta}\leq\varepsilon.$

Since $\delta>0$ is arbitrary, for all $\beta^{\prime}>\beta$ we have
$\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta^{\prime}}(\pazocal{G}_{T}(f_{0}))<\infty$
and therefore one has
$\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{0})\leq\beta^{\prime}$.
Since $\beta^{\prime}>\beta$ is also arbitrary, we obtain

$\displaystyle\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{0})\leq\beta=\dim
T.$

_(ii)_ Let $\alpha\in[1,\infty)$, $\beta=\alpha\cdot\dim T$ and let
$\delta,\,\varepsilon>0$ be arbitrary. Then there exists a cover
$\cup_{k\in{\mathbb{N}}}\ T_{k}\supseteq T$ with $T_{k}=[t_{k},\,t_{k}+c_{k}]$
and $c_{k}<1$ for every $k\in{\mathbb{N}}$ such that
$\sum_{k=1}^{\infty}|T_{k}|^{(\beta+\delta)/\alpha}=\sum_{k=1}^{\infty}c_{k}^{(\beta+\delta)/\alpha}\leq\varepsilon$.
Then $\pazocal{G}_{T}(f_{0})$ can be covered by $\alpha$-parabolic cylinders

$\Big{(}\mathsf{P}_{c_{k}^{1/\alpha}}\Big{)}_{k\in{\mathbb{N}}}=\Bigg{(}[t_{k},\,t_{k}+c_{k}]\times\prod_{j=1}^{d}\
[0,c_{k}^{1/\alpha}]\Bigg{)}_{k\in{\mathbb{N}}}\subseteq\mathscr{P}^{\alpha}.$

Since $\big{|}\mathsf{P}_{c_{k}^{1/\alpha}}\big{|}\asymp c_{k}^{1/\alpha}$ it
follows that

$\displaystyle\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta+\delta}(\pazocal{G}_{T}(f_{0}))\leq\sum_{k=1}^{\infty}\big{|}\mathsf{P}_{c_{k}^{1/\alpha}}\big{|}^{\beta+\delta}\lesssim\sum_{k=1}^{\infty}\big{(}c_{k}^{1/\alpha}\big{)}^{\beta+\delta}\lesssim\sum_{k=1}^{\infty}c_{k}^{(\beta+\delta)/\alpha}\leq\varepsilon.$

Since $\delta>0$ is arbitrary, for all $\beta^{\prime}>\beta$ we have
$\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\beta^{\prime}}(\pazocal{G}_{T}(f_{0}))<\infty$
and therefore one has
$\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{0})\leq\beta^{\prime}$.
Since $\beta^{\prime}>\beta$ is also arbitrary, one has

$\displaystyle\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{0})\leq\beta=\alpha\cdot\dim
T,$

as desired. ∎

We can calculate the $\alpha$-parabolic Hausdorff dimension of the graph of an
isotropic $\alpha$-stable Lévy process itself.

######  Theorem 7.2.

Let $T\subseteq\mathbb{R}_{+}$ be a Borel set and $\alpha\in(0,2]$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable Lévy process in
${\mathbb{R}^{d}}$. Then one $\mathbb{P}$-almost surely has

$\displaystyle\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(X)=(\alpha\vee
1)\cdot\dim T.$

###### Proof.

By Theorem 3.2 in [18], Theorem 2.4, Theorem 4.1 and Lemma 7.1 in case
$\alpha\cdot\dim T\geq 1$, i.e. $\alpha\in[1,2]$, and $f_{0}\equiv
0\in\mathbb{R}^{d}$ one $\mathbb{P}$-almost surely has

$\displaystyle\dim T+1-1/\alpha$ $\displaystyle=\dim\pazocal{G}_{T}(X)\leq
1/\alpha\cdot\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(X)+1-1/\alpha$
$\displaystyle\leq
1/\alpha\cdot\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{0})+1-1/\alpha\leq\dim
T+1-1/\alpha.$

In the other cases, Theorem 3.1 in [18] together with the same theorems as
above $\mathbb{P}$-almost surely yield

$\displaystyle(\alpha\vee 1)\cdot\dim
T=\dim\pazocal{G}_{T}(X)\leq\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(X)\leq\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{0})\leq(\alpha\vee
1)\cdot\dim T$

and the claim follows. ∎

######  Remark 7.3.

We can also deduce the Hausdorff dimension of the graph of the fractional
Brownian motion $B^{H}=\big{(}B_{t}^{H}\big{)}_{t\geq 0}$ in
${\mathbb{R}^{d}}$ of Hurst index $1/\alpha=H\in(0,1]$. One
$\mathbb{P}$-almost surely has

$\displaystyle\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}\big{(}B^{H}\big{)}=\frac{\dim
T}{H}=\alpha\cdot\dim T.$

This follows from Theorem 2.1 in [19], Proposition 2.2, Theorem 2.4, Lemma 2.2
in [14] and Lemma 7.1 for $\alpha\cdot\dim T\leq d$ and $f_{0}\equiv
0\in\mathbb{R}^{d}$ which $\mathbb{P}$-almost surely yield

$\displaystyle\alpha\cdot\dim T$
$\displaystyle=\dim\pazocal{G}_{T}\big{(}B^{H}\big{)}\leq\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}\big{(}B^{H}\big{)}$
$\displaystyle=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{0})\leq\alpha\cdot\dim
T.$

In the other cases the same theorems $\mathbb{P}$-almost surely yield

$\displaystyle\dim T+(1-1/\alpha)\cdot
d=\dim\pazocal{G}_{T}\big{(}B^{H}\big{)}\leq\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}\big{(}B^{H}\big{)}/\alpha+(1-1/\alpha)\cdot
d$
$\displaystyle\quad=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{0})/\alpha+(1-1/\alpha)\cdot
d\leq\dim T+(1-1/\alpha)\cdot d$

and the claim follows.

The calculations in the proof of the previous theorem further show:

######  Corollary 7.4.

Let $T\subseteq\mathbb{R}_{+}$ be a Borel set and $\alpha\in(0,\infty)$.
Define the constant function $f_{C}(x)=C\in{\mathbb{R}^{d}}$ for all $x\in T$.
Then one has

$\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{C})=(\alpha\vee
1)\cdot\dim T.$

As a consequence, we recover a well-known result for the range of an isotropic
$\alpha$-stable Lévy process; see [4] and Theorem 3.1 in [11].

######  Corollary 7.5.

Let $T\subseteq\mathbb{R}_{+}$ be a Borel set and $\alpha\in(0,2]$. Let
$X=(X_{t})_{t\geq 0}$ be an isotropic $\alpha$-stable Lévy process. One
$\mathbb{P}$-almost surely has

$\displaystyle\dim\pazocal{R}_{T}(X)=\alpha\cdot\dim T\wedge d.$

###### Proof.

From Theorem 3.2 and Colrollary 7.4 follows

$\displaystyle\dim\pazocal{R}_{T}(X)=(\alpha\wedge
1)\cdot\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f_{0})\wedge
d=\alpha\cdot\dim T\wedge d,$

as claimed. ∎

We can also give some a priori estimates for the $\alpha$-parabolic Hausdorff
dimension of the graph of a function in terms of the genuine Hausdorff
dimension.

######  Theorem 7.6.

Let $T\subseteq\mathbb{R}$ be any set and $f:T\rightarrow{\mathbb{R}^{d}}$ be
any function. Let
$\varphi_{\alpha}=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$. Then
one has

(7.1)
$\varphi_{\alpha}\leq\begin{cases}\varphi_{1}+\big{(}\frac{1}{\alpha}-1\big{)}\cdot
d\ \wedge\ d+1&\alpha\in(0,1],\\\ \varphi_{1}+\alpha-1\ \wedge\
d+1,&\alpha\in[1,\infty)\end{cases}$

and

(7.2) $\varphi_{\alpha}\geq\begin{cases}\varphi_{1}\ \vee\
\frac{1}{\alpha}\cdot\varphi_{1}+1-\frac{1}{\alpha},&\alpha\in(0,1],\\\
\varphi_{1}\ \vee\ \alpha\cdot\varphi_{1}+(1-\alpha)\cdot
d,&\alpha\in[1,\infty).\end{cases}$

Further, if $T\subseteq\mathbb{R}_{+}$ is a Borel set and
$f:T\rightarrow{\mathbb{R}^{d}}$ is a Borel measurable function, then we
obtain

(7.3) $\varphi_{\alpha}\leq\frac{1}{\alpha}\cdot\varphi_{1}\ \wedge\
\varphi_{1}+\bigg{(}\frac{1}{\alpha}-1\bigg{)}\cdot d\ \wedge\
d+1,\quad\alpha\in(0,1].$

###### Proof.

This follows immediately by Theorem 2.4 for (7.1) and (7.2) and Corollary 3.3
for (7.3) and the fact that the Hausdorff dimension never exceeds the
toplogical dimension. ∎

Next we calculate some bounds for the parabolic Hausdorff dimension of
$\beta$-Hölder continuous functions. These are functions
$f:T\rightarrow{\mathbb{R}^{d}}$ that fulfil $||f(t)-f(s)||\leq
C\cdot|t-s|^{\beta}$ for all $s,t\in T$ and some $\beta\in(0,1]$, $C>0$,
denoted as $f\in C^{\beta}(T,{\mathbb{R}^{d}})$. In case of $\alpha=1$, the
following theorem is well-known, see Theorem 10.6 in [9].

######  Theorem 7.7.

Let $T\subseteq\mathbb{R}$ be any set, $\alpha\in(0,\infty)$, $\beta\in(0,1]$
and $f\in C^{\beta}\big{(}T,{\mathbb{R}^{d}}\big{)}$. Define
$\varphi_{\alpha}:=\mathscr{P}^{\alpha}\text{-}\dim\pazocal{G}_{T}(f)$. Then
one has

$\displaystyle\varphi_{\alpha}\leq\begin{cases}\dim
T+d\cdot\big{(}\frac{1}{\alpha}-\beta\big{)}\ \wedge\ \frac{\dim
T}{\alpha\beta}\ \wedge\ d+1,&\alpha\in(0,1],\\\ \alpha\cdot\dim
T+d\cdot(1-\alpha\beta)\ \wedge\ \frac{\dim T}{\beta}\ \wedge\
d+1,&\alpha\in\big{[}1,\frac{1}{\beta}\big{]},\\\ \alpha\cdot\dim T\ \wedge\
d+1,&\alpha\in\big{[}\frac{1}{\beta},\infty\big{)}.\end{cases}$

###### Proof.

Let $\tau>\dim T$ and $\varepsilon>0$ be arbitrary. Then we can cover $T$ by
intervals $(\mathsf{T}_{k})_{k\in{\mathbb{N}}}$ with sidelength
$|\mathsf{T}_{k}|<1$ such that
$\sum_{k=1}^{\infty}|\mathsf{T}_{k}|^{\tau}<\varepsilon$. Since $f\in
C^{\beta}\big{(}T,{\mathbb{R}^{d}}\big{)}$, we can cover $\pazocal{G}_{T}(f)$
by $(\mathsf{B}_{k})_{k\in{\mathbb{N}}}\subseteq\mathbb{R}^{1+d}$ where

$\mathsf{B}_{k}:=\mathsf{T}_{k}\times\prod_{i=1}^{d}\big{[}x_{i,k},\
x_{i,k}+C\cdot|\mathsf{T}_{k}|^{\beta}\big{]}\quad\text{ for every
}k\in{\mathbb{N}}.$

Note that without loss of generality we may assume $C\geq 1$ for the constant
in the definition of Hölder continuity.

_(i)_ Let $\alpha\in(0,1]$. On the one hand, for every $k\in{\mathbb{N}}$ we
can cover $\mathsf{B}_{k}$ by (several) $\alpha$-parabolic cylinders with
sidelength $|\mathsf{T}_{k}|$ in time. Since
$K\cdot|\mathsf{T}_{k}|^{1/\alpha}\geq C\cdot|\mathsf{T}_{k}|^{\beta}$ iff
$K\geq C\cdot|\mathsf{T}_{k}|^{\beta-1/\alpha}$ we find a cover

$\pazocal{G}_{T}(f)\subseteq\bigcup_{k=1}^{\infty}\bigcup_{l=1}^{\big{\lceil}C\cdot|\mathsf{T}_{k}|^{\beta-1/\alpha}\big{\rceil}^{d}}\mathsf{T}_{k}\times\square_{|\mathsf{T}_{k}|^{1/\alpha},\,l}$

with
$\mathsf{T}_{k}\times\square_{|\mathsf{T}_{k}|^{1/\alpha},\,l}\in\mathscr{P}^{\alpha}$
for every $k,l\in{\mathbb{N}}$. Now, for $\gamma=\tau+d\cdot(1/\alpha-\beta)$
we have

$\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\gamma}(\pazocal{G}_{T}(f))\lesssim\sum_{k=1}^{\infty}|\mathsf{T}_{k}|^{d\cdot(\beta-1/\alpha)+\gamma}=\sum_{k=1}^{\infty}|T_{k}|^{\tau}<\varepsilon.$

Since $\tau>\dim T$ is arbitrary, $\varphi_{\alpha}\leq\dim
T+d\cdot(1/\alpha-\beta).$ On the other hand, for every $k\in{\mathbb{N}}$ we
can cover $\mathsf{B}_{k}$ by a single $\alpha$-parabolic cylinder with
sidelength $C^{\alpha}\cdot|\mathsf{T}_{k}|^{\alpha\beta}$ in time. Then
$\pazocal{G}_{T}(f)\subseteq\bigcup_{k\in{\mathbb{N}}}\
\mathsf{P}_{|\mathsf{T}_{k}|^{\alpha\beta}}$ with

$\mathsf{P}_{|\mathsf{T}_{k}|^{\alpha\beta}}:=\big{[}t_{k},t_{k}+C^{\alpha}\cdot|\mathsf{T}_{k}|^{\alpha\beta}\big{]}\times\prod_{i=1}^{d}\big{[}x_{i,k},\,x_{i,k}+C\cdot|\mathsf{T}_{k}|^{\beta}\big{]}\in\mathscr{P}^{\alpha}.$

Now, for $\gamma=\tau/(\alpha\beta)$ we have
$\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\gamma}(\pazocal{G}_{T}(f))\lesssim\sum_{k=1}^{\infty}|\mathsf{T}_{k}|^{\alpha\beta\cdot\gamma}<\varepsilon$.
Since $\tau>\dim T$ is arbitrary this results in
$\varphi_{\alpha}\leq\frac{\dim T}{\alpha\beta}$.
_(ii)_ Let $\alpha\in[1,1/\beta]$. On the one hand, for every
$k\in{\mathbb{N}}$ we can cover $\mathsf{B}_{k}$ by (several)
$\alpha$-parabolic cylinders with sidelength $|\mathsf{T}_{k}|$ in time. Since
$K\cdot|\mathsf{T}_{k}|^{1/\alpha}\geq C\cdot|\mathsf{T}_{k}|^{\beta}$ iff
$K\geq C\cdot|\mathsf{T}_{k}|^{\beta-1/\alpha}$ we find a cover

$\pazocal{G}_{T}(f)\subseteq\bigcup_{k=1}^{\infty}\bigcup_{l=1}^{\big{\lceil}C\cdot|\mathsf{T}_{k}|^{\beta-1/\alpha}\big{\rceil}^{d}}\mathsf{T}_{k}\times\square_{|\mathsf{T}_{k}|^{1/\alpha},\,l}.$

with
$\mathsf{T}_{k}\times\square_{|\mathsf{T}_{k}|^{1/\alpha},\,l}\in\mathscr{P}^{\alpha}$
for every $k,l\in{\mathbb{N}}$. Now, for
$\gamma=\alpha\cdot\tau+d\cdot(1-\alpha\beta)$ we have

$\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\gamma}(\pazocal{G}_{T}(f))\lesssim\sum_{k=1}^{\infty}|\mathsf{T}_{k}|^{d\cdot(\beta-1/\alpha)+\gamma/\alpha}=\sum_{k=1}^{\infty}|T_{k}|^{\tau}<\varepsilon.$

Since $\tau>\dim T$ is arbitrary, this results in
$\varphi_{\alpha}\leq\alpha\cdot\dim T+d\cdot(1-\alpha\beta)$. On the other
hand, for every $k\in{\mathbb{N}}$ we can cover $\mathsf{B}_{k}$ by a single
$\alpha$-parabolic cylinder with sidelength
$C^{\alpha}\cdot|\mathsf{T}_{k}|^{\alpha\beta}$ in time. Then
$\pazocal{G}_{T}(f)\subseteq\bigcup_{k\in{\mathbb{N}}}\
\mathsf{P}_{|\mathsf{T}_{k}|^{\alpha\beta}}$ with

$\mathsf{P}_{|\mathsf{T}_{k}|^{\alpha\beta}}:=\big{[}t_{k},t_{k}+C^{\alpha}\cdot|\mathsf{T}_{k}|^{\alpha\beta}\big{]}\times\prod_{i=1}^{d}\big{[}x_{i,k},\,x_{i,k}+C\cdot|\mathsf{T}_{k}|^{\beta}\big{]}\in\mathscr{P}^{\alpha}.$

Now, for $\gamma=\tau/\beta$ we have
$\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\gamma}(\pazocal{G}_{T}(f))\lesssim\sum_{k=1}^{\infty}|\mathsf{T}_{k}|^{\beta\cdot\gamma}<\varepsilon$.
Since $\tau>\dim T$ is arbitrary, this results in $\varphi_{\alpha}\leq\dim
T/\beta$.
(iii) Let $\alpha\in[1/\beta,\infty)$. On the one hand, for every
$k\in{\mathbb{N}}$ we can cover $\mathsf{B}_{k}$ by (several)
$\alpha$-parabolic cylinders with sidelength $|\mathsf{T}_{k}|^{\alpha\beta}$
in time. Since $K\cdot|\mathsf{T}_{k}|^{\alpha\beta}\geq
C\cdot|\mathsf{T}_{k}|$ iff $K\geq C\cdot|\mathsf{T}_{k}|^{1-\alpha\beta}$ we
find a cover

$\pazocal{G}_{T}(f)\subseteq\bigcup_{k=1}^{\infty}\bigcup_{l=1}^{\big{\lceil}C\cdot|\mathsf{T}_{k}|^{1-\alpha\beta}\big{\rceil}}T_{k,l}\times\square_{|T_{k,l}|^{1/\alpha}}.$

with $T_{k,l}\times\square_{|T_{k,l}|^{1/\alpha}}\in\mathscr{P}^{\alpha}$ for
every $k,l\in{\mathbb{N}}$. Now, for $\gamma=(\tau+\alpha\beta-1)/\beta$ we
have

$\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\gamma}(\pazocal{G}_{T}(f))\lesssim\sum_{k=1}^{\infty}|\mathsf{T}_{k}|^{1-\alpha\beta+\gamma\beta}=\sum_{k=1}^{\infty}|T_{k}|^{\tau}<\varepsilon.$

Since $\tau>\dim T$ is arbitrary, this results in
$\varphi_{\alpha}\leq\frac{1}{\beta}\cdot(\dim T-1)+\alpha$. On the other
hand, for every $k\in{\mathbb{N}}$ we can cover $\mathsf{B}_{k}$ by a single
$\alpha$-parabolic cylinder with length $C^{\alpha}\cdot|\mathsf{T}_{k}|$ in
time. Therefore we find a cover

$\pazocal{G}_{T}(f)\subseteq\bigcup_{k=1}^{\infty}\mathsf{T}^{\prime}_{k}\times\square_{|\mathsf{T}_{k}|^{1/\alpha}}.$

with
$\mathsf{T}^{\prime}_{k}\times\square_{|\mathsf{T}_{k}|^{1/\alpha}}\in\mathscr{P}^{\alpha}$
for every $k\in{\mathbb{N}}$. Now, for $\gamma\geq\alpha\cdot\tau$ we have

$\mathscr{P}^{\alpha}\text{-}\pazocal{H}^{\gamma}(\pazocal{G}_{T}(f))\lesssim\sum_{k=1}^{\infty}|\mathsf{T}_{k}|^{\gamma/\alpha}<\varepsilon.$

Since $\tau>\dim T$ is arbitrary, this results in
$\varphi_{\alpha}\leq\alpha\cdot\dim T$. Finally, $\alpha\cdot\dim
T\leq\frac{1}{\beta}\cdot(\dim T-1)+\alpha$ if and only if $\dim T\leq 1$
since $1/\beta\leq\alpha$. ∎

Let us inspect the important case $\alpha=2$, i.e. we aim to get a bound for
the graph of Brownian motion plus $\beta$-Hölder continuous drift function
over $T$.

######  Corollary 7.8.

Let $T\subseteq\mathbb{R}_{+}$ be any set. Let $B=(B_{t})_{t\geq 0}$ denote
the $d$-dimensional Brownian motion and let $f\in
C^{\beta}\big{(}T,{\mathbb{R}^{d}}\big{)}$ for some $\beta\in(0,1]$. Then one
$\mathbb{P}$-almost surely has

$\dim\pazocal{G}_{T}(B+f)\leq\begin{cases}d+\frac{1}{2},&\beta\leq\frac{\dim
T}{d}\wedge\frac{1}{2}\wedge\dim T-\frac{1}{2},\\\ \dim
T+d\cdot(1-\beta),&\dim T-\frac{1}{2}\leq\beta\leq\frac{\dim
T}{d}\wedge\frac{1}{2},\\\ \frac{\dim T}{\beta},&\frac{\dim
T}{d}\leq\beta\leq\frac{1}{2},\\\ 2\cdot\dim T\wedge\dim
T+\frac{d}{2},&\beta\geq\frac{1}{2}.\end{cases}$

Moreover, one $\mathbb{P}$-almost surely has

$\dim\pazocal{R}_{T}(B+f)\leq\begin{cases}\frac{\dim T}{\beta},&\frac{\dim
T}{d}\leq\beta\leq\frac{1}{2},\\\ 2\cdot\dim T\wedge
d,&\beta\geq\frac{1}{2},\\\ d,&\text{else}.\end{cases}$

###### Proof.

Let $\varphi_{2}=\mathscr{P}^{2}\text{-}\dim\pazocal{G}_{T}(f)$. Corollary 4.2
$\mathbb{P}$-almost surely yields

$\dim\pazocal{G}_{T}(B+f)\leq\varphi_{2}\ \wedge\ \frac{\varphi_{2}+d}{2}$

and Theorem 6.1 $\mathbb{P}$-almost surely yields

$\dim\pazocal{R}_{T}(B+f)\leq\varphi_{2}\ \wedge\ d.$

Finally, by Theorem 7.7 we easily get

$\varphi_{2}\leq\begin{cases}2\cdot\dim T+d\cdot(1-2\beta)\wedge
d+1,&\beta\leq\frac{\dim T}{d}\wedge\frac{1}{2},\\\ \frac{\dim
T}{\beta},&\frac{\dim T}{d}\leq\beta\leq\frac{1}{2},\\\ 2\cdot\dim
T,&\beta\geq\frac{1}{2}\end{cases}$

and the claim follows. ∎

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