# Hitting probabilities for fractional Brownian motion with deterministic
drift

###### Abstract

Let $B^{H}$ be a $d$-dimensional fractional Brownian motion with Hurst index
$H\in(0,1)$, $f:[0,1]\longrightarrow\mathbb{R}^{d}$ a Borel function, and
$E\subset[0,1]$, $F\subset\mathbb{R}^{d}$ are given Borel sets. The focus of
this paper is on hitting probabilities of the fractional Brownian motion
$B^{H}$ with the deterministic drift $f$. It aims to highlight the role of the
regularity properties of the drift $f$ as well as that of the dimension of $E$
in determining the upper and lower bounds of $\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\emptyset\\}$ for $F$ a subset of $\mathbb{R}^{d}$ and also for $F$ a
singleton.

MOHAMED ERRAOUI

Department of mathematics, Faculty of science El jadida,

Chouaïb Doukkali University, Morocco

e-mail<EMAIL_ADDRESS>
YOUSSEF HAKIKI111Supported by National Center for Scientific and Technological
Research (CNRST)

Department of mathematics, Faculty of science Semlalia,

Cadi Ayyad University, 2390 Marrakesh, Morocco

e-mail<EMAIL_ADDRESS>
Keywords: Fractional Brownian motion, Hitting probabilities, Capacity,
Hausdorff measure

Mathematics Subject Classification: 62134, 60J45, 60G17, 28A78

##  1 Introduction

The hitting probability describes the probability that a given process will
ever reach some state or set of states $F$. To find upper and lower bounds for
the hitting probabilities in terms of the Hausdorff measure and the capacity
of the set $F$, is a fundamental problem in probabilistic potential theory.
For $d$-dimensional Brownian motion $B$ the probability that a path, will ever
visit a given set $F\subseteq\mathbb{R}^{d}$, is classically estimated using
the Newtonian capacity of $F$. Kakutani [15] was the first to establish this
result linking capacities and hitting probabilities for Brownian motion.
Precisely, he showed that, for $(d\geq 3)$, a compact set $F$ is hit with
positive probability by $B$ if and only if $F$ has positive Newtonian
capacity. Since then, considerable efforts have been carried out to establish
a series of extensions to other processes. This has given rise to a large and
rapidly growing body of scientific literature on the subject. To cite a few
examples, we refer to Xiao [30] for developments on hitting probabilities of
stationary Gaussian random fields and fractional Brownian motion; to Pruitt
and Taylor [26] and Khoshnevisan [16] for hitting probabilities results for
general stable processes and Lévy processes; to Khoshnevisan and Shi [18] for
hitting probabilities of the Brownian sheet; to Dalang and Nualart [7] for
hitting probabilities for the solution of a system of nonlinear hyperbolic
stochastic partial differential equations; to Dalang, Khoshnevisan and Nualart
[8] and [9], for hitting probabilities for the solution of a non-linear
stochastic heat equation with additive and multiplicative noise respectively;
to Xiao [31] Biermé, Lacaux and Xiao [4] for hitting probabilities of Gaussian
random fields. Finally, we refer to Khoshnevisan [17] for more information on
the latter as well as on potential theory of random fields.

It should be noted that the above characterization is not common to all the
processes and this is generally due to the dependence structures thereof
leading to an upper and lower bounds on hitting probabilities in terms of
capacity and Hausdorff measure. In this context, Chen and Xiao [6] improved
the results established by Xiao (Theorem 7.6 [31]) and by Biermé, Lacaux and
Xiao (Theorem 2.1 [4] ) on hitting probabilities of the
$\mathbb{R}^{d}$-valued Gaussian random field $X$ satisfying conditions
$(C_{1})$ and $(C_{2})$, see Xiao [31] for precise definition, through the
following

$c^{-1}\mathcal{C}_{\rho_{H},d}(E\times F)\leq\mathbb{P}\\{X(E)\cap
F\neq\emptyset\\}\leq c\mathcal{H}_{\rho_{H}}^{d}(E\times F),$ (1.1)

where $E\subseteq[\varepsilon_{0},1]^{N}$, $\varepsilon_{0}\in(0,1)$ and
$F\subseteq\mathbb{R}^{d}$ are Borel sets and $c$ is a finite constant which
depends on $[\varepsilon_{0},1]^{N}$, $F$ and $H$ only. We emphasize that, in
addition to fractional Brownian motion, various processes are part of those
satisfying conditions $(C_{1})$ and $(C_{2})$ namely, fractional Brownian
sheets (Ayache and Xiao [1]), solutions to stochastic heat equation driven by
space-time white noise (Dalang, Khoshnevisan and Nualart [8] and [9], Dalang
and Nualart [7]), Mueller and Tribe [22] and many more. See Xiao [31] for more
examples and further information on conditions $(C_{1})$ and $(C_{2})$.
$\mathcal{C}_{\rho_{H},d}$ and $\mathcal{H}_{\rho_{H}}^{d}$denotes the Bessel-
Riesz type capacity and the Hausdorff measure with respect to the parabolic
metric $\rho_{H}$ of order $d$. Both of these terms are defined next and will
be referred as parabolic capacity and parabolic Hausdorff measure
respectively.

The corresponding problem for $d$-dimensional Brownian motion $B$ with drift
$f$, $d\geq 2$, has been considered by Peres and Souissi [23]. Precisely they
showed that for $f:\mathbb{R}^{+}\longrightarrow\mathbb{R}^{d}$ $(1/2)$-Hölder
continuous function there exists positives constants $\alpha_{1},\alpha_{2}$
such that for all $x\in\mathbb{R}^{d}$ and all closed set
$F\subseteq\mathbb{R}^{d}$

$\alpha_{1}\text{Cap}_{M}\left(F\right)\leq\mathbb{P}_{x}\\{(B+f)(0,\infty)\cap
F\neq\emptyset\\}\leq\alpha_{2}\text{Cap}_{M}\left(F\right),$

where $\text{Cap}_{M}\left(\cdot\right)$ denotes the Martin capacity. At the
heart of their method is the strong Markov property which can’t be used for
fractional Brownian motion. Naturally, this begs the question : can we provide
similar estimate to $\eqref{Chen Xiao Estimate}$ for $d$-dimensional
fractional Brownian motion $B^{H}$ of Hurst index $H$ with drift $f$?

Our first objective in this work is to give an answer to this question. In
fact, we established the desired estimates by adjusting the standard proof,
which relies on the covering argument for the upper bound and the second
moment argument for the lower bound, to take into account the presence of the
$H$-Hölder continuous drift $f$. This allowed us to obtain, this time,
according to the usual Hausdorff measure and Bessel-Riesz capacity the upper
and lower bounds on hitting probabilities of the following type

$\mathbf{\mathfrak{c}}_{1}\,\mathcal{C}_{d-\beta_{1}/H}(F)\leq\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\emptyset\\}\leq\mathbf{\mathfrak{c}}_{2}\,\mathcal{H}^{d-\beta_{2}/H}(F).$

Worthy of special mention is the fact that $\beta_{1}$ and $\beta_{2}$ are two
different constants closely related to the Hausdorff and Minkowski dimensions
of $E$ respectively. In the event that the two dimensions coincide, often this
is a consequence of the existence of a sufficiently regular measure see
condition (S) below, we obtain the above estimates with the same constant
$\beta=\dim(E)$ the Hausdorff dimension of $E$. With these bounds in hand we
draw the conclusion that $\mathbb{P}\\{(B^{H}+f)(E)\cap F\neq\emptyset\\}>0$
(resp. $\mathbb{P}\\{(B^{H}+f)(E)\cap F\neq\emptyset\\}=0$) for any compact
set $F$ such that $\dim(F)>d-\beta/H$ (resp. $\dim(F)<d-\dim(E)/H$). This
leads us to consider the question: is there an $\alpha$-Hölder continuous
function $f$, $\alpha<H$, for which $\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\emptyset\\}>0$?

The idea is then to take $\alpha<H$ such that $\dim(F)>d-\beta/\alpha$ and
$B^{\alpha}$ another fractional Brownian motion with Hurst index $\alpha$
possibly defined on different probability space and thereafter to consider
$B^{H}$ as a drift of $B^{\alpha}$. This induces us to bring them together on
the same space while preserving their distributions. The best way to do this
is to work on the product space and to consider processes on this latter.
Unfortunately, we are unable to have both estimates for the same drift. These
results are proved in Section 2.

A problem related to estimating hitting probabilities for a rondom process $X$
with drift is determining which Borel functions are polar for $X$. Now we
recall the definition of polar function for $X$. A Borel function
$f:\mathbb{R}^{+}\longrightarrow\mathbb{R}^{d}$ is called polar for $X$ if for
any $x\in\mathbb{R}^{d}$,

$\mathbb{P}\left\\{X_{t}+x=f(t)\,\text{for some }t>0\right\\}=0,$

which means that the process $X-f$ does not hit points. The first study of
polar functions for Brownian motion in $2$ dimensions appears in Graversen
[11]. Precisely, he showed that for all $0<\gamma<1/2$, there exists a
$\gamma$-Hölder continuous function
$f:\mathbb{R}^{+}\longrightarrow\mathbb{R}^{2}$ for which $B+f$ hits points.
In [19], Le Gall proved that for any $1/2$-Hölder continuous function $f$, the
process $B+f$ do not hits points and asked, for $d\geq 3$, whether for each
$\gamma<1/d$ there exist $\gamma$-Hölder continuous functions for which $B+f$
hits points. This problem has also been studied for the stable process by
Mountford in [21]. Recently Antunovi$\acute{\text{c}}$, Peres and Vermesi [2]
prove first that, for $d\geq 2$ and for each $\gamma<1/d$ there exist
$\gamma$-Hölder continuous functions for which the range of $B+f$, covers an
open set almost surely, thereby ensuring that $B+f$ hits points. Moreover, for
$d\geq 3$, there exists a $1/d$-Hölder continuous function, accurately the
$d$-dimensional Hilbert curve, such that $B+f$ hits points. Considering this
problem for fractional Brownian motion is our second focus. We begin by
establishing, for a general measurable drift, an upper and lower bounds on
hitting probabilities as follows

$\displaystyle{\textbf{c}}_{1}^{-1}\mathcal{C}_{\rho_{H},d}(Gr_{E}(f))\leq\mathbb{P}\left\\{\exists
t\in
E:(B^{H}+f)(t)=x\right\\}\leq{\textbf{c}}_{1}\,\mathcal{H}_{\rho_{H}}^{d}(Gr_{E}(f)),$
(1.2)

where $Gr_{E}(f)=\\{(t,f(t)):t\in E\\}$ is the graph of $f$ over the set $E$.
The above estimates are aimed first and foremost to seek conditions on the
drift $f$ for which $B^{H}+f$ does or does not hit points. The first
conclusion that we can draw is that functions with a positive parabolic
capacity $\mathcal{C}_{\rho_{H},d}(Gr_{E}(f))$ hit points, on the other hand
those who have parabolic Hausdorff measure
$\mathcal{H}_{\rho_{H}}^{d}(Gr_{E}(f))=0$ does not hit points. As a first
step, we prove that for any $\alpha<\dim(E)/d\wedge H$ there exists a
$\alpha$-Hölder continuous function which is non-polar for $B^{H}$ obtained as
a realization of an independent fractional Brownian motion with Hurst
parameter $\alpha$.

The relationship between lack of regularity and fractal properties for special
classes of functions has been highlighted a long time ago. Frequently graphs
of continuous but sufficiently irregular functions are fractal sets what
connects the lack of regularity of such functions to the Hausdorff dimension
of their graphs. We consider the Weierstrass function as a prototype example
of such functions. Our second step is to show that the one dimensional
fractional Brownian motion with drift given by the Weierstrass function hits
points with positive probability. The two keys ingredients in the proof are a
recent result of Shen [28], which is an improvement on the result of Barański,
Bárány and Romanowska [3] on a long-standing conjecture concerning the
Hausdorff dimension of the graph of the Weierstrass function, giving the exact
value of the latter and a comparaison result for the Hausdorff parabolic
dimensions with different parameters established by the authors in [10]. Among
the properties of Weierstrass nowhere differentiable function most often used
are $\alpha$-Hölder continuity and reverse $\alpha$-Hölder continuity for some
$0<\alpha<1$. Przytycki and Urbański in [27] proved that if
$f:[0,1]\rightarrow\mathbb{R}$ is both $\alpha$-Hölder and reverse
$\alpha$-Hölder for some $0<\alpha<1$, it satisfies $\dim(Gr_{[0,1]}(f))>1$.
Replacing Weierstrass function by such function $f$ we obtain the same result.
We thought and hoped that this result continues to be true for in higher
dimensions, i.e. $d\geq 2$, but it does not. Precisely, we consider a
$d$-dimensional vector-valued function $f$ where each component is the
Weierstrass function for which $\dim_{\rho_{H}}(Gr(f))<d$ leading us to
conclude that $B^{H}+f$ does not hit points. The above mentioned results
constitute the content of Section 3.

##  2 Hitting sets

In this section, we consider the problem on hitting probabilities of
fractional Brownian motion with deterministic drift. Let $H\in(0,1)$ and
$B^{H}_{0}=\left\\{B^{H}_{0}(t),t\geq 0\right\\}$ be a real-valued fractional
Brownian motion of Hurst index $H$ defined on a complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$, i.e. a real valued Gaussian process with
stationary increments and covariance function given by

$\mathbb{E}(B^{H}_{0}(s)B^{H}_{0}(t))=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}).$

Let $B^{H}_{1},...,B^{H}_{d}$ be $d$ independent copies of $B^{H}_{0}$, then
the stochastic process $B^{H}=\left\\{B^{H}(t),t\geq 0\right\\}$ given by

$B^{H}(t)=(B^{H}_{1}(t),....,B^{H}_{d}(t)),$

is called a $d$-dimensional fractional Brownian motion of Hurst index
$H\in(0,1)$.

We consider the following parabolic metric $\rho_{H}$ on
$\mathbb{R}_{+}\times\mathbb{R}^{d}$ defined by

$\rho_{H}((s,x),(t,y))=\max\\{|t-s|^{H},\|x-y\|\\}\quad\forall(s,x),(t,y)\in\mathbb{R}_{+}\times\mathbb{R}^{d},$
(2.1)

where $\|.\|$ denotes the euclidean metric on $\mathbb{R}^{d}$. For $\beta>0$
and $E\subset\mathbb{R}_{+}\times\mathbb{R}^{d}$, the $\beta$-dimensional
Hausdorff measure of $E$ with respect to the metric $\rho_{H}$ is defined by

$\mathcal{H}_{\rho_{H}}^{\beta}(E)=\lim_{\delta\rightarrow
0}\inf\left\\{\sum_{n=1}^{\infty}\left(2r_{n}\right)^{\beta}:E\subseteq\bigcup_{n=1}^{\infty}B_{\rho_{H}}\left(r_{n}\right),r_{n}\leqslant\delta\right\\},$
(2.2)

where $B_{\rho_{H}}(r)$ denotes an open ball of radius $r$ in the metric space
$(\mathbb{R}_{+}\times\mathbb{R}^{d},\rho_{H})$. The Bessel-Riesz type
capacity of order $\alpha$ on the metric space
$(\mathbb{R}_{+}\times\mathbb{R}^{d},\rho_{H})$ is defined by

$\mathcal{C}_{\rho_{H},\alpha}(E)=\left[\inf_{\mu\in\mathcal{P}(E)}\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\varphi_{\alpha}(\rho_{H}(u,v))\mu(du)\mu(dv)\right]^{-1},$
(2.3)

where $\mathcal{P}(E)$ is the family of probability measures carried by $E$
and the function $\varphi_{\alpha}:(0,\infty)\rightarrow(0,\infty)$ is defined
by

$\varphi_{\alpha}(r)=\left\\{\begin{array}[]{ll}{r^{-\alpha}}&{\text{ if
}\alpha>0}\\\ {\log\left(\frac{e}{r\wedge 1}\right)}&{\text{ if }\alpha=0}\\\
{1}&{\text{ if }\alpha<0}\end{array}\right..$ (2.4)

######  Remark 2.1.

Let $\dim_{\rho_{H}}$ be the Hausdorff dimension associated to the measure
$\mathcal{H}_{\rho_{H}}^{\beta}$ which is defined as

$\dim_{\rho_{H}}(E)=\inf\\{\beta>0:\mathcal{H}_{\rho_{H}}^{\beta}(E)=0\\}\text{
for all $E\subset\mathbb{R}_{+}\times\mathbb{R}^{d}$},$

we can verify that $\dim_{\Psi,H}(.)\equiv H\times\dim_{\rho_{H}}(.)$, where
$\dim_{\Psi,H}$ is the $H$-parabolic Hausdorff dimension which was used by
Peres and Sousi in [24] in order to study the Hausdorff dimension of the graph
and the image of $B^{H}+f$.

The usual $\beta$-dimensional Hausdorff measure, $\beta>0$, and Bessel-Riesz
capacity of order $\alpha$ in Euclidean metric $\|.\|$ are denoted by
$\mathcal{H}^{\beta}$ and $\mathcal{C}_{\alpha}$ respectively.
$\mathcal{H}^{\beta}$ is assumed equal to $1$ whenever $\beta\leq 0$. Let
$I=[\varepsilon_{0},1]$, where $\varepsilon_{0}\in(0,1)$ is a fixed constant.
First we have the following result.

######  Theorem 2.2.

Let $\\{B^{H}(t),t\in[0,1]\\}$ be a $d$-dimensional fractional Brownian motion
and $f:[0,1]\rightarrow\mathbb{R}^{d}$ a Hölder continuous function with order
$H$ and constant $K$. If $F\subseteq\mathbb{R}^{d}$ is a compact subset of
$\mathbb{R}^{d}$ and $E$ is a Borel subset of $I$, then

$c_{1}^{-1}\mathcal{C}_{\rho_{H},d}(E\times
F)\leq\mathbb{P}\\{(B^{H}+f)(E)\cap F\neq\emptyset\\}\leq
c_{1}\mathcal{H}_{\rho_{H}}^{d}(E\times F),$ (2.5)

where $c_{1}\geq 1$ is finite constant which depends on $I$, $F$, $H$ and $K$
only.

An important property of the fractional Brownian motion that will serve us
well into the proof is the strong local nondeterminism which follows from
Lemma 7.1 of [25]. Precisely, there exists a constant $0<C<\infty$ such that
for all integers $n\geq 1$ and all $t_{1},...,t_{n},t\in[0,1]$, we have

$\displaystyle
Var\left(B^{H}_{0}(t)/B^{H}_{0}(t_{1}),...,B^{H}_{0}(t_{n})\right)\geq
C\min_{0\leq j\leq n}\left|t-t_{j}\right|^{2H},$ (2.6)

where $Var\left(B^{H}_{0}(t)/B^{H}_{0}(t_{1}),...,B^{H}_{0}(t_{n})\right)$
denotes the conditional variance of $B^{H}_{0}(t)$ given
$B^{H}_{0}(t_{1}),...,B^{H}_{0}(t_{n})$ and $t_{0}=0$.

To prove the above theorem, we will make use of the following two lemmas
proved by Biermé, Lacaux and Xiao in [4] for a general class of Gaussian
processes to which fractional Brownian motion belongs. In fact they will be
used to obtain the upper and lower bounds of (2.5) respectively.

######  Lemma 2.3 (Lemma 3.1,[4]).

Let $\\{B^{H}(t):t\in[0,1]\\}$ be a fractional Brownian motion. For any
constant $M>0$, there exist a positive constants $c_{2}$ and $\delta_{0}$ such
that for all $r\in(0,\delta_{0})$, $t\in I$ and all $x\in[-M,M]^{d}$ we have

$\mathbb{P}\left\\{\inf_{s\in I,|s-t|^{H}\leq r}\|B^{H}(s)-x\|\leqslant
r\right\\}\leqslant c_{2}r^{d}.$ (2.7)

######  Lemma 2.4 (Lemma 3.2,[4] ).

Let $B^{H}$ be a fractional Brownian motion. Then there exists a positive and
finite constants $c_{3}$ and $c_{4}$ such that for all $\epsilon\in(0,1)$,
$s,t\in I$ and $x,y\in\mathbb{R}^{d}$ we have

$\displaystyle\displaystyle\int_{\mathbb{R}^{2d}}e^{-i(\langle\xi,x\rangle+\langle\eta,y\rangle)}\exp\left(-\dfrac{1}{2}(\xi,\eta)\left(\varepsilon
I_{2d}+\text{Cov}(B^{H}(s),B^{H}(t))\right)(\xi,\eta)^{T}\right)d\xi d\eta$
$\displaystyle\leq\dfrac{(2\pi)^{d}}{\left[\text{det}\left(\Gamma_{\varepsilon}(s,t)\right)\right]^{d/2}}\exp\left(-\dfrac{c_{3}}{2}\dfrac{\|x-y\|^{2}}{\text{det}\left(\Gamma_{\varepsilon}(s,t)\right)}\right)$
(2.8)
$\displaystyle\leq\dfrac{c_{4}}{\left(\rho_{H}((s,x),(t,y))\right)^{d}},$

where
$\Gamma_{\varepsilon}(s,t):=I_{2}\varepsilon+\text{Cov}(B^{H}_{0}(s),B^{H}_{0}(t))$,
$I_{2d}$ and $I_{2}$ are the identities matrices of order $2d$ and $2$
respectively, $Cov(B^{H}(s),B^{H}(t))$ and $Cov(B^{H}_{0}(s),B^{H}_{0}(t))$
denote the covariance matrix of the random vectors $(B^{H}(s),B^{H}(t))$ and
$(B^{H}_{0}(s),B^{H}_{0}(t))$ respectively, and $(\xi,\eta)^{T}$ is the
transpose of the row vector $(\xi,\eta)$.

###### Proof of Theorem 2.2.

First of all, we note that the proof of the upper bound in (2.5) is similar to
that of Theorem 2.1 in [6] which relies on the use of a simple covering
argument. Since $F$ is compact set there exists a constant $M_{0}>0$ such that
$F\subset[-M_{0},M_{0}]^{d}$. Let $M_{1}=\sup_{s\in I}\|f(s)\|$ and
$M_{2}=M_{0}+M_{1}$. Then for all $(t,y)\in I\times F$ we have
$y+f(t)\in[-M_{2},M_{2}]^{d}$. Applying Lemma 2.3 with the constant $M_{2}$
leads to the existence of a positive constants $c_{5}$ and $\delta_{1}$ such
that for all $r\in(0,\delta_{1})$ and $(t,y)\in I\times F$ we have

$\displaystyle\mathbb{P}\left\\{\inf_{s\in I,|s-t|^{H}\leq
r}\|B^{H}(s)-(y-f(t))\|\leq r\right\\}\leq c_{5}r^{d},$ (2.9)

where $c_{5}$ and $\delta_{1}$ depend only on $I,H$ and $M_{2}$. Now let us
choose an arbitrary constant $\gamma>\mathcal{H}_{\rho_{H}}^{d}(E\times F)$.
Then there is a covering of $E\times F$ by balls
$\\{B_{\rho_{H}}((t_{i},y_{i}),r_{i}),i\geq 1\\}$ in
$\mathbb{R}_{+}\times\mathbb{R}^{d}$ such that $r_{i}<\delta_{1}/1+K$ for all
$i\geq 1$, where $K$ is the Hölder constant of $f$, and

$E\times
F\subseteq\bigcup_{i=1}^{\infty}B_{\rho_{H}}((t_{i},y_{i}),r_{i})\quad\text{with
}\quad\sum_{i=1}^{\infty}(2r_{i})^{d}\leq\gamma.$ (2.10)

It follows that

$\displaystyle\left\\{(B^{H}+f)(E)\cap F\neq\emptyset\right\\}$
$\displaystyle=\left\\{\,\exists\,(t,y)\in E\times
F\,:\left(B^{H}+f\right)(t)=y\right\\}$
$\displaystyle\subseteq\bigcup_{i=1}^{\infty}\left\\{\left(B^{H}+f\right)\left(\left(t_{i}-r_{i}^{1/H},t_{i}+r_{i}^{1/H}\right)\right)\cap
B(y_{i},r_{i})\neq\emptyset\right\\}.$ (2.11)

As a first step, it is easy to see that

$\left\\{(B^{H}+f)\left(\left(t_{i}-r_{i}^{1/H},t_{i}+r_{i}^{1/H}\right)\right)\cap
B(y_{i},r_{i})\neq\emptyset\right\\}=\left\\{\inf_{|s-t_{i}|^{H}<r_{i}}\|(B^{H}+f)(s)-y_{i}\|<r_{i}\right\\}.$

On the other hand since $f$ is $H$-Hölder continuous then for all
$s\in\left(t_{i}-r_{i}^{1/H},t_{i}+r_{i}^{1/H}\right)$ we have

$\displaystyle\left\\{\|(B^{H}+f)(s)-y_{i}\|<r_{i}\right\\}$
$\displaystyle\subset\left\\{\|B^{H}(s)-(y_{i}-f(t_{i}))\|<(1+K)r_{i}\right\\}.$

This enables us to obtain

$\displaystyle\left\\{(B^{H}+f)\left(\left(t_{i}-r_{i}^{1/H},t_{i}+r_{i}^{1/H}\right)\right)\cap
B(y_{i},r_{i})\neq\emptyset\right\\}\subset\left\\{\inf_{|s-t_{i}|^{H}<c_{6}r_{i}}\|B^{H}(s)-(y_{i}-f(t_{i}))\|<c_{6}r_{i}\right\\}.$

where $c_{6}:=K+1>1$. Combining $\eqref{proba estim}$, $\eqref{Hausdorff
covering}$ and $\eqref{event covering}$ we derive that

$\mathbb{P}\\{(B^{H}+f)(E)\cap F\neq\emptyset\\}\leq c_{7}\gamma,$

where $c_{7}$ depends only on $I,H$ and $K$. Let
$\gamma\downarrow\mathcal{H}_{\rho_{h}}^{d}(E\times F)$, the upper bound in
$(\ref{upper-lower bounds 1})$ follows.

The lower bound in (2.5) can be proved by using a second moment argument. We
assume that $\mathcal{C}_{\rho_{H},d}(E\times F)>0$ otherwise the lower bound
is obvious. We can see easily from (2.3) that there is a probability measure
$\mu$ on $E\times F$ such that

$\mathcal{E}_{\rho_{H},d}(\mu):=\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\frac{\mu(du)\mu(dv)}{(\rho_{H}(u,v))^{d}}\leq\frac{2}{\mathcal{C}_{\rho_{H},d}(E\times
F)}.$ (2.12)

We consider the family of random measures $\left\\{\mu_{n},n\geq 1\right\\}$
on $E\times F$ defined by

$\displaystyle\int_{E\times F}g(s,x)\;\mu_{n}(ds,dx)$
$\displaystyle=\int_{E\times F}(2\pi
n)^{d/2}\exp\left(-\frac{n\|B^{H}(s)+f(s)-x\|^{2}}{2}\right)g(s,x)\,\mu(ds,dx)$
(2.13) $\displaystyle=\int_{E\times
F}\int_{\mathbb{R}^{d}}\exp\left(-\frac{\|\xi\|^{2}}{2n}+i\left\langle\xi,B^{H}(s)+f(s)-x\right\rangle\right)g(s,x)\,d\xi\,\mu(ds,dx),$

thanks to the characteristic function of a Gaussian vector. Here $g$ is an
arbitrary measurable function on $\mathbb{R}_{+}\times\mathbb{R}^{d}$. Our aim
is to show that $\left\\{\mu_{n},n\geq 1\right\\}$ has a subsequence which
converges weakly to a finite measure $\nu$ supported on the set $\\{(s,x)\in
E\times F:B^{H}(s)+f(s)=x\\}$. To carry out this goal, we will start by
establishing the following inequalities

$\displaystyle\mathbb{E}(\|\mu_{n}\|)\geqslant
c_{8},\quad\quad\mathbb{E}(\|\mu_{n}\|^{2})\leqslant
c_{9}\mathcal{E}_{\rho_{H},d}(\mu),$ (2.14)

which constitute together with the Paley-Zygmund inequality the cornerstone of
the proof. Here $\|\mu_{n}\|$ denotes the total mass of $\mu_{n}$. We
emphasize that the positive constants $c_{8}$ and $c_{9}$ are independent of
$\mu$ and $n$. By (2.13), Fubini’s theorem and the use of the characteristic
function of a Gaussian vector we have

$\displaystyle\begin{aligned} \mathbb{E}(\|\mu_{n}\|)&=\int_{E\times
F}\int_{\mathbb{R}^{d}}e^{-i\left\langle\xi,x-f(s)\right\rangle}\exp\left(-\frac{\|\xi\|^{2}}{2n}\right)\mathbb{E}\left(e^{i\left\langle\xi,B^{H}(s)\right\rangle}\right)\,d\xi\,\mu(ds,dx)\\\
&=\int_{E\times
F}\int_{\mathbb{R}^{d}}e^{-i\left\langle\xi,x-f(s)\right\rangle}\exp\left(-\frac{1}{2}\left(\frac{1}{n}+s^{2H}\right)\|\xi\|^{2}\right)\,d\xi\,\mu(ds,dx)\\\
&=\int_{E\times
F}\left(\frac{2\pi}{n^{-1}+s^{2H}}\right)^{d/2}\exp\left(-\frac{\|x-f(s)\|^{2}}{2\left(n^{-1}+s^{2H}\right)}\right)\,\mu(ds,dx)\\\
&\geq\int_{E\times
F}\left(\frac{2\pi}{1+s^{2H}}\right)^{d/2}\exp\left(-\frac{\|x-f(s)\|^{2}}{2s^{2H}}\right)\,\mu(ds,dx)\\\
&\geq c_{8}>0.\end{aligned}$ (2.15)

Since $F$ and $f$ are bounded and $\mu$ is a probability measure we conclude
that $c_{7}$ is independent of $\mu$ and $n$. This gives the first inequality
in $(\ref{esp energy})$.

We will now turn our attention to the second inequality in $(\ref{esp
energy})$. By $(\ref{seq random measure 1})$ and Fubini’s theorem again we
obtain

$\displaystyle\mathbb{E}\left(\|\mu_{n}\|^{2}\right)=$
$\displaystyle\int_{E\times F}\int_{E\times
F}\,\mu(ds,dx)\,\mu(dt,dy)\int_{\mathbb{R}^{2d}}e^{-i(\left\langle\xi,x-f(s)\right\rangle+\left\langle\eta,y-f(t)\right\rangle)}$
$\displaystyle\times\exp\left(-\frac{1}{2}(\xi,\eta)(n^{-1}I_{2d}+\text{Cov}(B^{H}(s),B^{H}(t)))(\xi,\eta)^{T}\right)d\xi
d\eta$ $\displaystyle\leq\int_{E\times F}\int_{E\times
F}\,\frac{(2\pi)^{d}}{\left[\text{det}\left(\Gamma_{1/n}(s,t)\right)\right]^{d/2}}\exp\left(-\frac{c_{3}}{2}\frac{\|x-y+f(s)-f(s)\|^{2}}{\text{det}\left(\Gamma_{1/n}(s,t)\right)}\right)\,\mu(ds,dx)\,\mu(dt,dy),$

where the last inequality follows from Lemma 2.4. We denote by
$I_{n}\left((s,x),(t,y)\right)$ the last integrand. Since
$\|x-y+f(t)-f(s)\|\geq|\|x-y\|-\|f(t)-f(s)\||$, we have that

$\begin{array}[]{ll}I_{n}\left((s,x),(t,y)\right)\leq&\dfrac{(2\pi)^{d}}{\left[\text{det}\left(\Gamma_{1/n}(s,t)\right)\right]^{d/2}}\,\exp\left(-\dfrac{c_{3}}{2}\dfrac{\|x-y\|^{2}}{\text{det}\left(\Gamma_{1/n}(s,t)\right)}\right)\\\
\\\
&\times\exp\left(c_{3}\dfrac{\|x-y\|\|f(s)-f(t)\|}{\text{det}\left(\Gamma_{1/n}(s,t)\right)}\right)\exp\left(-\dfrac{c_{3}}{2}\dfrac{\|f(s)-f(t)\|^{2}}{\text{det}\left(\Gamma_{1/n}(s,t)\right)}\right).\end{array}$

Using the strong local nondeterminism property (2.6) of $B_{0}^{H}$, there
exists a constant $c_{10}$ such that

$\displaystyle\det\Gamma_{1/n}(s,t)$
$\displaystyle\geq\det\text{Cov}\left(B_{0}^{H}(s),B_{0}^{H}(t)\right)=\text{Var}\left(B_{0}^{H}(s)\right)\,\text{Var}\left(B_{0}^{H}(t)|B_{0}^{H}(s)\right)$
$\displaystyle\geq c_{10}\,|t-s|^{2H}.$ (2.16)

Since $f$ is $H$-Hölder continuous we have
$\dfrac{\|f(t)-f(s)\|}{\sqrt{\text{det}\left(\Gamma_{1/n}(s,t)\right)}}\leq\dfrac{K}{\sqrt{c_{10}}}$
for all $s\neq t\in E$. It follows that

$I_{n}\left((s,x),(t,y)\right)\leq\frac{(2\pi)^{d}}{\left[\text{det}\left(\Gamma_{1/n}(s,t)\right)\right]^{d/2}}\exp\left(-\frac{c_{3}}{2}\frac{\|x-y\|^{2}}{\text{det}\left(\Gamma_{1/n}(s,t)\right)}\right)\exp\left(\dfrac{c_{3}\,K}{\sqrt{c_{10}}}\dfrac{\|x-y\|}{\sqrt{\text{det}\left(\Gamma_{1/n}(s,t)\right)}}\right),$

which implies that

$I_{n}\left((s,x),(t,y)\right)\leq\frac{c_{11}\,(2\pi)^{d}}{\left[\text{det}\left(\Gamma_{1/n}(s,t)\right)\right]^{d/2}}\exp\left(-\frac{c_{3}}{4}\frac{\|x-y\|^{2}}{\text{det}\left(\Gamma_{1/n}(s,t)\right)}\right)$

where $c_{11}$ is a positive constant such that $\underset{x\geq
0}{\sup}\,\exp\left(-c_{3}\dfrac{x^{2}}{4}+\dfrac{\,c_{3}\,K}{\sqrt{c_{10}}}\,x\right)\leq
c_{11}$.

It is now straightforward to deduce that there exists a positive constant
$c_{9}$ such that

$I_{n}\left((s,x),(t,y)\right)\leq\frac{c_{9}}{\rho_{H}\left((s,x),(t,y)\right)^{d}}=\frac{c_{9}}{\max\\{|s-t|^{Hd},\|x-y\|^{d}\\}}.$
(2.17)

Indeed, if $\det\Gamma_{1/n}(s,t)\geq\|x-y\|^{2}$ we have

$\displaystyle\begin{aligned}
I_{n}((s,x),(t,y))\leq\frac{(2\pi)^{d}}{(\det\Gamma_{1/n}(s,t))^{d/2}}\leq\frac{c_{9}}{|t-s|^{Hd}},\end{aligned}$
(2.18)

where we use 2.16 for the last the inequality. Otherwise, if
$\det\Gamma_{1/n}(s,t)<\|x-y\|^{2}$ the elementary inequality
$\underset{x>0}{\sup}\,x^{d/2}\,e^{-c_{3}x/4}\leq c_{12}$ enables us to obtain

$I_{n}((s,x),(t,y))\leq\frac{c_{9}}{\|x-y\|^{d}}.$ (2.19)

Combining (2.18) and (2.19) leads to (2.17). Hence the second inequalities in
(2.14) follows immediately.

Plugging the moment estimates of (2.14) into the Paley–Zygmund inequality
(c.f. Kahane [14], p.8), allows us to confirm that $\\{\mu_{n},n\geq 1\\}$ has
a subsequence that converges weakly to a finite measure $\tilde{\mu}$
supported on the set $\\{(s,x)\in E\times F:B^{H}(s)+f(s)=x\\}$, positive with
positive probability and also satisfying the moment estimates of (2.14).
Consequently,

$\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\varnothing\\}\geqslant\mathbb{P}\\{\|\tilde{\mu}\|>0\\}\geqslant\frac{[\mathbb{E}(\|\tilde{\mu}\|)]^{2}}{\mathbb{E}\left[\|\tilde{\mu}\|^{2}\right]}\geqslant
c_{13}\mathcal{C}_{\rho_{H},d}(E\times F),$

where $c_{13}=\dfrac{c_{8}^{2}}{c_{9}}$. All that remains to be done is take
$c_{1}=c_{7}\vee 1/c_{13}$. Thus the lower and upper bounds in (2.5) will
follow immediately which completes the proof. ∎

Recall that in the precise case where $E$ is an interval, Corollary 2.2. in
[6] ensures that there exists a finite constant $c\geq 1$ depending only on
$E$, $F$ and $H$ such that

$c^{-1}\,\mathcal{C}_{d-1/H}(F)\leq\mathbb{P}\left\\{B^{H}(E)\cap
F\neq\emptyset\right\\}\leq c\,\mathcal{H}^{d-1/H}(F),$

for any Borel set $F\subseteq\mathbb{R}^{d}$. Our next goal is to establish
such estimates for $(B^{H}+f)$ and any Borel set $E$ by means of its Hausdorff
measure. However, to achieve our stated goal, we need to make use of the
Minkowski dimension as well. We introduce now the Minkowski dimension of
$E\subset[0,1]$. Let $N(E,r)$ be the smallest number of open intervals of
length $r$ required to cover $E$. The lower and upper Minkowski dimensions of
$E$ are respectively defined as

$\displaystyle\underline{\dim}_{M}(E):=\liminf_{r\rightarrow 0^{+}}\frac{\log
N(E,r)}{\log(1/r)},$
$\displaystyle\overline{\dim}_{M}(E):=\limsup_{r\rightarrow 0^{+}}\frac{\log
N(E,r)}{\log(1/r)}.$

Equivalently, the upper Minkowski dimension of $E$ can be written as

$\overline{\dim}_{M}(E)=\inf\\{\gamma:\exists C<\infty\,\text{ such that
}N(E,r)\leqslant Cr^{-\gamma}\text{ for all }r>0\\}.$ (2.20)

######  Proposition 2.5.

Let $B^{H}$, $f$ and $F$ as in Theorem 2.2. Let $E$ be a subset of $I$ such
that $\dim(E)>0$. Then for any
$0<\beta_{1}<\dim(E)\leq\overline{\dim}_{M}(E)<\beta_{2}<Hd$, we have

$\mathbf{\mathfrak{c}}_{1}\,\mathcal{C}_{d-\beta_{1}/H}(F)\leq\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\emptyset\\}\leq\mathbf{\mathfrak{c}}_{2}\,\mathcal{H}^{d-\beta_{2}/H}(F),$
(2.21)

where $\mathbf{\mathfrak{c}}_{1}$ and $\mathbf{\mathfrak{c}}_{2}$ are two
positive constants which depend on $E$, $F$, $H$, $K$, $\beta_{1}$ and
$\beta_{2}$.

We need the following lemma to establish the lower bound in (2.21).

######  Lemma 2.6.

Let $H\in(0,1)$, $d\geq 1$ and $0<\beta\leq Hd$. Let $\nu$ a Borel probability
measure on $[0,1]$ such that, for all $a\in[0,1]$ and $\delta>0$

$\displaystyle\nu([a,a+\delta])\leq\mathbf{\mathfrak{c}}_{3}\,\delta^{\beta},$
(2.22)

where $\mathbf{\mathfrak{c}}_{3}$ is a positive constant which depends on
$\beta$ only. Then, for all $r>0$, we have

$\sup_{t\in[0,1]}\int_{[0,1]}\frac{\nu(ds)}{\max\\{r^{d},|s-t|^{Hd}\\}}\leq\mathbf{\mathfrak{c}}_{4}\varphi_{d-\beta/H}(r),$
(2.23)

where $\varphi_{d-\beta/H}(.)$ is the function defined in (2.4) and
$\mathbf{\mathfrak{c}}_{4}$ is a positive constant depending only on $\beta$,
$H$ and $d$.

###### Proof.

First, it is worthwhile pointing out that for $r\geq 1$ we have

$\sup_{t\in[0,1]}\int_{[0,1]}\frac{\nu(ds)}{\max\\{r^{d},|s-t|^{Hd}\\}}\leq
r^{-d}\leq\varphi_{d-\beta/H}(r).$ (2.24)

Now we assume that $0<\beta<Hd$. For $r\in(0,1)$, we divide the integral in
(2.23) into two parts $I_{1}$, $I_{2}$ as follows

$I_{1}=\int_{|s-t|<r^{1/H}}\frac{\nu(ds)}{r^{d}}\text{ \,and
\,}I_{2}=\int_{|s-t|\geq r^{1/H}}\frac{\nu(ds)}{|s-t|^{Hd}}.$

By using $\eqref{Frostman}$ we obtain

$I_{1}\leq\mathbf{\mathfrak{c}}_{3}\,2^{\beta}\,\varphi_{d-\beta/H}(r).$
(2.25)

Let us set $k(r):=\min\\{k\geq 0:2^{-k}\leq r^{1/H}\\}$. Then it is easy to
see that

$\displaystyle[r^{1/H},1)\subset\bigcup_{k=1}^{k(r)}[2^{-k},2^{-k+1}).$ (2.26)

A second use of (2.22) gives

$\displaystyle I_{2}$
$\displaystyle\leq\sum_{k=1}^{k(r)}2^{kH\,d}\nu(\\{s\in[0,1]:2^{-k}\leq|s-t|<2^{-k+1}\\})$
$\displaystyle\leq
2\,\mathbf{\mathfrak{c}}_{3}\sum_{k=1}^{k(r)}2^{k(Hd-\beta)}\leq
2\,\mathbf{\mathfrak{c}}_{3}\,\dfrac{2^{(Hd-\beta)}}{2^{(Hd-\beta)}-1}r^{-(d-\beta/H)}=2\,\mathbf{\mathfrak{c}}_{3}\,\dfrac{2^{(Hd-\beta)}}{2^{(Hd-\beta)}-1}\,\varphi_{d-\beta/H}(r).\quad\quad\quad$
(2.27)

Finally, putting it all together enables us to deduce (2.23).

For $\beta=H\,d$ the same techniques as above can give that

$I_{1}\leq 2^{\beta}\,\mathfrak{c}_{3}\,\quad\text{ and }\quad I_{2}\leq
2\,\mathfrak{c}_{3}\,k(r).$

It follows from the definition of $k(r)$ that $r^{1/H}<2^{-k(r)+1}$. Hence, we
have

$I_{1}\leq 2^{\beta}\,\mathfrak{c}_{3}\,\varphi_{0}(r)\,\quad\text{ and }\quad
I_{2}\leq 2\,\mathfrak{c}_{3}(1\vee 1/H\log(2))\,\varphi_{0}(r),$

which ends the proof.

∎

###### Proof of Proposition 2.5.

Using Theorem 2.2 it suffices to prove that that there exists a positive
constant $\mathbf{\mathfrak{c}}_{6}$ such that

$\displaystyle\mathcal{C}_{d-\beta_{1}/H}(F)\leq\mathbf{\mathfrak{c}}_{5}\,\mathcal{C}_{\rho_{H},d}(E\times
F)\quad\text{ and }\quad\mathcal{H}_{\rho_{H}}^{d}(E\times
F)\leq\,\mathbf{\mathfrak{c}}_{6}\mathcal{H}^{d-\beta_{2}/H}(F).$ (2.28)

Indeed for $\beta_{1}\in(0,\dim(E))$, by Frostman’s theorem there is a Borel
probability measure $\nu$ supported on $E$ such that

$\displaystyle\nu([a,a+\delta])\leq\mathbf{\mathfrak{c}}_{7}\,\delta^{\beta_{1}},$
(2.29)

for all $a\in E$ and $\delta>0$, where $\mathbf{\mathfrak{c}}_{7}$ is a
positive constant which depends on $\beta_{1}$ only. Let us suppose that
$\mathcal{C}_{d-\beta_{1}/H}(F)>0$, otherwise there is nothing to prove. It
follows that for all $\gamma\in(0,\mathcal{C}_{d-\beta_{1}/H}(F))$ there is a
probability measure $m$ supported on $F$ such that

$\displaystyle\mathcal{E}_{d-\beta_{1}/H}(m):=\int_{F}\int_{F}\frac{m(dx)m(dy)}{\|x-y\|^{d-\beta_{1}/H}}\leq\gamma^{-1}.$
(2.30)

Since $\nu\otimes m$ is a probability measure on $E\times F$, then applying
Fubini’s theorem and (2.23) of Lemma 2.6 we obtain

$\displaystyle\mathcal{E}_{\rho_{H},d}(\nu\otimes m)=\int_{E\times
F}\int_{E\times F}\frac{\nu\otimes m(du)\nu\otimes
m(dv)}{(\rho_{H}(u,v))^{d}}\leq\mathbf{\mathfrak{c}}_{4}\int_{F}\int_{F}\frac{m(dx)m(dy)}{\|x-y\|^{d-\beta_{1}/H}}\leq\mathbf{\mathfrak{c}}_{4}\,\gamma^{-1}.$
(2.31)

Consequently we have $\mathcal{C}_{\rho_{H},\alpha}(E\times
F)\geq\mathbf{\mathfrak{c}}_{4}^{-1}\,\gamma$. Then we let
$\gamma\uparrow\mathcal{C}_{d-\beta_{1}/H}(F)$ to conclude that the first
inequality in (2.28) holds true.

Now let us prove the second inequality in (2.28). Let
$l>\mathcal{H}^{d-\beta_{2}/H}(F)$ be arbitrary with $d-\beta_{2}/H>0$. Then
there is a covering of $F$ by open balls $B(r_{n})$ of radius $r_{n}$ such
that

$\displaystyle F\subset\bigcup_{n=1}^{\infty}B(r_{n})\quad\text{ and
}\quad\sum_{n=1}^{\infty}(2r_{n})^{d-\beta_{2}/H}\leq l.$ (2.32)

For all $n\geq 1$, let $E_{n,j}$, $j=1,...,N(E,2\,r_{n}^{1/H})$ be a family of
open intervals of length $2\,r_{n}^{1/H}$ covering $E$. It follows that the
family $E_{n,j}\times B(r_{n}),\,j=1,...,N(E,2\,r_{n}^{1/H})$, $n\geq 1$ gives
a covering of $E\times F$ by open balls of radius $r_{n}$ for the parabolic
metric $\rho_{H}$.

It follows from (2.20) that for all $\delta>0$ the number of open intervals of
length $\delta$ needed to cover $E$ satisfies

$\displaystyle N(E,\delta)\leq\mathbf{\mathfrak{c}}_{8}\,\delta^{-\beta_{2}},$
(2.33)

where $\mathbf{\mathfrak{c}}_{8}$ is a positive and finite constant which
depend on $E$ only. Together with the estimates (2.32) and (2.33) that have
been established above, we have

$\displaystyle\sum_{n=1}^{\infty}\sum_{j=1}^{N(E,2\,r_{n}^{1/H})}\left(2r_{n}\right)^{d}\leq\mathbf{\mathfrak{c}}_{8}\,2^{-\beta_{2}(1-1/H)}\,\sum_{n=1}^{\infty}(2r_{n})^{d-\beta_{2}/H}\leq\mathbf{\mathfrak{c}}_{8}\,\,2^{-\beta_{2}(1-1/H)}\,l.$
(2.34)

Then let $l\downarrow\mathcal{H}^{d-\beta_{2}/H}(F)$, the second inequality in
(2.28) follows with
$\mathbf{\mathfrak{c}}_{6}=\mathbf{\mathfrak{c}}_{8}\,\,2^{-\beta_{2}(1-1/H)}$.
∎

It is well known that Hausdorff and Minkowski dimensions agree for many sets
$E$. Often this is linked on the one hand to the geometric properties of the
set, on the other hand it is a consequence of the existence of a sufficiently
regular measure. Among the best known are Ahlfors-David regular sets defined
as follows:

  * (S)

: Let $E\subset I$ and $\beta\in\left]0,1\right]$. We say that $E$ is
$\beta$-regular if there exists a finite positive Borel measure $\nu$
supported on $E$ and positive constant $\mathbf{\mathfrak{c}}_{9}$ and such
that

$\mathbf{\mathfrak{c}}_{9}^{-1}\,\delta^{\beta}\leq\nu([a-\delta,a+\delta])\leq\mathbf{\mathfrak{c}}_{9}\,\delta^{\beta}\,\,\text{
for all $a\in E$, $0<\delta\leq 1$}.$ (2.35)

######  Remark 2.7.

1\. If $E$ is the whole interval $I$ then $\beta$ in the condition (S) should
be equal to $1$. This leads to the conclusion that the measure $\nu$ can be
chosen as the normalized Lebesgue measure on $I$. In this case the above
proposition is simply Corollary 2.2 in [6].

2\. The Cantor set $C(\lambda)$, $0<\lambda<1/2$, subset of $I$ with $\nu$ is
the $\beta$-dimensional Hausdorff measure restricted to $C(\lambda)$ where
$\beta=\dim C(\lambda)=\log(2)/\log(1/\lambda)$. For more details see Theorem
4.14 p.$67$ in Mattila [20]. In general, self similar subsets of $\mathbb{R}$
satisfying the open set condition are standard examples of regular sets, see
[13].

According to Theorem 5.7 p.$80$ in [20], for a set $E$ satisfying the
condition (S) we have

$\beta=\dim(E)=\underline{\dim}_{M}(E)=\overline{\dim}_{M}(E).$

In such case Proposition 2.5 becomes

######  Proposition 2.8.

Let $B^{H}$, $f$ and $F$ as in Theorem 2.2. Let $E$ be a subset of $I$
satisfying the condition (S). Then there is a positive and finite constant
$\mathbf{\mathfrak{c}}_{10}$ which depends on $E$, $F$, $H$, $K$ and $\beta$,
such that

$\mathbf{\mathfrak{c}}_{10}^{-1}\,\mathcal{C}_{d-\beta/H}(F)\leq\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\emptyset\\}\leq\mathbf{\mathfrak{c}}_{10}\,\mathcal{H}^{d-\beta/H}(F).$
(2.36)

###### Proof.

Three cases are to be discussed here: (i) $\beta<Hd$, (ii) $\beta=Hd$ and
(iii) $\beta>Hd$. Let us point out first that for the lower bound, the
interesting cases are (i) and (ii) while for the upper bound it is the case
(i) which requires proof. Indeed we have from (2.4) that
$\mathcal{C}_{\alpha}(\cdot)=1$ for $\alpha<0$ and $\mathcal{H}^{\alpha}(.)$
is assumed to be equal to $1$ whenever $\alpha\leq 0$. In this regard, a close
reading of the proof of Proposition 2.5 is required. Thus, we can clearly see
that it is based on two key estimates, namely (2.29) and (2.33) for the lower
and upper bound respectively. In what follows we will establish such estimates
under the condition (S). The estimation (2.29) with $\beta$ is now a part of
the condition (S). In order to establish (2.33), we will show that for all
$\delta>0$

$\displaystyle N(E,\delta)\leq C\delta^{-\beta},$ (2.37)

where $C$ is a positive and finite constant which depend on $I$ only. Indeed,
let $0<\delta\leq 1$ and $P(E,\delta)$ be the greatest number of disjoint
intervals $I_{j}$ centred in $x_{j}\in E$ with length $\delta$ required to
cover $E$. Condition (S) ensures that

$\mathbf{\mathfrak{c}}_{9}^{-1}\,P(E,\delta)\,(\delta/2)^{\beta}\leq\sum_{j=1}^{P(E,\delta)}\nu(I_{j})=\nu(E)\leq
1.$

Using the fact that

$N(E,2\delta)\leq P(E,\delta),$

we obtain the desired estimation (2.33). The rest of the proof follows closely
the lines of that of Proposition 2.5 especially given that Lemma 2.6 takes
into account the case (ii). ∎

Following the same pattern as above we get the following proposition, which
can be considered also as a corollary of Theorem 2.1 in [6], for the subset
$E$ of $I$ satisfying the condition (S).

######  Proposition 2.9.

Let $B^{H}$ and $F$ as in Theorem 2.2. Let $E$ be a subset of $I$ satisfying
the condition (S). Then there is a positive and finite constant
$\mathbf{\mathfrak{c}}_{11}$ which depends on $E$, $F$, $H$, $K$ and $\beta$,
such that

$\mathbf{\mathfrak{c}}_{11}^{-1}\,\mathcal{C}_{d-\beta/H}(F)\leq\mathbb{P}\\{B^{H}(E)\cap
F\neq\emptyset\\}\leq\mathbf{\mathfrak{c}}_{11}\,\mathcal{H}^{d-\beta/H}(F).$
(2.38)

We would like to point out that, when the drift $f$ is $H$-Hölder continuous
and $E$ satisfies the condition (S), Propositions 2.8 and 2.9 assert that the
hitting probabilities of $(B^{H}+f)$ behave like the ones of $B^{H}$ in the
following sense

$\bullet$ if $\dim(F)<d-\beta/H$ then $\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\emptyset\\}=\mathbb{P}\\{B^{H}(E)\cap F\neq\emptyset\\}=0$,

$\bullet$ if $\dim(F)>d-\beta/H$ then $\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\emptyset\\}>0$ and $\mathbb{P}\\{B^{H}(E)\cap F\neq\emptyset\\}>0$.

This brings us to the following question: when $\dim(F)<d-\beta/H$, is it
possible to get a function, with smaller Hölder order than H, for which
$\mathbb{P}\\{(B^{H}+f)(E)\cap F\neq\emptyset\\}>0$? In order to address this
question, we need to consider $\alpha<H$ such that $d-\beta/\alpha<\dim(F)$
and $(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{P}^{\prime})$ another
probability space on which we define a fractional Brownian motion $B^{\alpha}$
with Hurst parameter $\alpha$. We will work with the mixed process
$Z^{H,\alpha}$ defined on the probability space
$(\Omega\times\Omega^{\prime},\mathcal{F}\times\mathcal{F}^{\prime},\mathbb{P}\otimes\mathbb{P}^{\prime})$
by

$\displaystyle
Z^{H,\alpha}(t,(\omega,\omega^{\prime}))=B^{H}(t,\omega)+B^{\alpha}(t,\omega^{\prime})\text{
for all }t\geq 0\text{ and
}(\omega,\omega^{\prime})\in\Omega\times\Omega^{\prime}.$ (2.39)

It is easy to see that $Z^{H,\alpha}=(Z^{H,\alpha}_{1},...,Z^{H,\alpha}_{d})$
where $Z^{H,\alpha}_{i}$ are independent copies of a real valued Gaussian
process $Z^{H,\alpha}_{0}$ on
$(\Omega\times\Omega^{\prime},\mathcal{F}\times\mathcal{F}^{\prime},\mathbb{P}\otimes\mathbb{P}^{\prime})$
with stationary increment and the covariance function given by

$\widetilde{\mathbb{E}}(Z^{H,\alpha}_{0}(s)Z^{H,\alpha}_{0}(t))=\frac{1}{2}(s^{2H}+t^{2H}-|t-s|^{2H})+\frac{1}{2}(s^{2\alpha}+t^{2\alpha}-|t-s|^{2\alpha}),$

where $\widetilde{\mathbb{E}}$ denote the expectation under the probability
$\mathbb{P}\otimes\mathbb{P}^{\prime}$. The following lemma is about the
strong local nondeterminism property of the process $Z$. Let $I\subset(0,1]$,
be a closed interval, then we have

######  Lemma 2.10.

The real-valued process $\\{Z^{H,\alpha}_{0}(t):t\geq 0\\}$ satisfy the
following

1\. For all $s,t\in[0,1]$,

$\displaystyle|s-t|^{2\alpha}\leq\widetilde{\mathbb{E}}\left(Z^{H,\alpha}_{0}(t)-Z^{H,\alpha}_{0}(s)\right)^{2}\leq
2|s-t|^{2\alpha}.$ (2.40)

2\. There exists a positive constant $C$ depending on $\alpha$, $H$ and $I$
only, such that

$\operatorname{Var}\left(Z^{H,\alpha}_{0}(u)\mid
Z^{H,\alpha}_{0}\left(t_{1}\right),\ldots,Z^{H,\alpha}_{0}\left(t_{n}\right)\right)\geq
C\left[\min_{0\leq k\leq n}\left|u-t_{k}\right|^{2\alpha}+\min_{0\leq k\leq
n}\left|u-t_{k}\right|^{2H}\right],$ (2.41)

for all integers $n\geq 1$, all $u,t_{1},...,t_{n}\in I$ and $t_{0}=0$.

3\. There exists a positive constant $C$ depending on $\alpha$, $H$ and $I$
only. Such that for any $t\in I$ and any $0<r\leq t$,

$\operatorname{Var}\left(Z^{H,\alpha}_{0}(t)\mid\,Z^{H,\alpha}_{0}(s);\mid
s-t\mid\geq r\right)\geq Cr^{2\alpha}$ (2.42)

The proof of the lemma is that of Proposition 4.2. in [10]. Now we are able to
provide an answer to the above question.

######  Theorem 2.11.

Let $E$ be a compact set satisfying the condition (S) and
$F\subset\mathbb{R}^{d}$ be a compact set such that $\dim(F)\leq d-\beta/H$.
Then for all $\alpha<\dfrac{\beta}{d-\dim(F)}$ and for all $\varepsilon>0$
small enough such that $\mathcal{C}_{d-\beta/(\alpha+\varepsilon)}(F)>0$,
there exists a $\alpha$-Hölder continuous function
$f:[0,1]\rightarrow\mathbb{R}^{d}$, satisfying

$\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\emptyset\\}\geq{\textsf{c}}_{1}\,\mathcal{C}_{d-\beta/(\alpha+\varepsilon)}(F),$
(2.43)

where ${\textsf{c}}_{1}$ is a positive constant which depends on
$\alpha,\beta,H,d$ and $\varepsilon$ only.

###### Proof.

Let us set $\alpha^{\prime}:=\alpha+\varepsilon$ with $\varepsilon>0$ small
enough, and consider the stochastic process $Z^{\alpha^{\prime},H}$ stated
above. The previous lemma tells us that $Z^{\alpha^{\prime},H}$ satisfies the
conditions $(C1)$ and $(C2)$ of Theorem 2.1 in [6]. So we have

${\textsf{c}}_{2}^{-1}\,\mathcal{C}_{\rho_{\alpha^{\prime}},d}(E\times
F)\leq\widetilde{\mathbb{P}}\\{Z(E)\cap
F\neq\emptyset\\}\leq{\textsf{c}}_{2}\,\mathcal{C}_{\rho_{\alpha^{\prime}},d}(E\times
F),$

${\textsf{c}}_{2}\geq 1$ is a finite constant which depends on $I$, $F$ and
$H$ only. Since $E$ satisfies the condition (S) we use (2.28) to obtain

${\textsf{c}}_{3}\,\mathcal{C}_{d-\beta/\alpha^{\prime}}(F)\leq\widetilde{\mathbb{P}}\\{Z(E)\cap
F\neq\emptyset\\}\leq{\textsf{c}}_{4}\,\mathcal{H}^{d-\beta/\alpha^{\prime}}(F),$

where ${\textsf{c}}_{3}$ and ${\textsf{c}}_{4}$ are positive constants which
depends on $d$, $\beta$, and $\alpha^{\prime}$ only. We can choose
$\varepsilon$ small enough such that
$\mathcal{C}_{d-\beta/\alpha^{\prime}}(F)>0$. Therefore by Fubini’s theorem we
get

$\mathbb{E}^{\prime}\left(\mathbb{P}\left\\{(B^{H}+B^{\alpha^{\prime}}(\omega^{\prime}))(E)\cap
F\neq\emptyset\right\\}-{\textsf{c}}_{5}\,\mathcal{C}_{d-\beta/\alpha^{\prime}}(F)\right)>0,$

for some fixed positive constant ${\textsf{c}}_{5}\in(0,{\textsf{c}}_{3})$.
The above inequality lead to

$\mathbb{P}^{\prime}\left\\{\mathbb{P}\left\\{(B^{H}+B^{\alpha^{\prime}}(\omega^{\prime}))(E)\cap
F\neq\emptyset\right\\}-{\textsf{c}}_{5}\,\mathcal{C}_{d-\beta/\alpha^{\prime}}(F)>0\right\\}>0.$

We therefore choose the function $f$ among the paths of $B^{\alpha^{\prime}}$
satisfying the above. ∎

######  Remark 2.12.

The same reasoning should also apply to check for all
$\alpha<\dfrac{\beta}{d-\dim(F)}$ and for all $\varepsilon>0$ small enough
such that $\mathcal{H}_{d-\beta/(\alpha+\varepsilon)}(F)<+\infty$, there
exists a $\alpha$-Hölder continuous function
$f:[0,1]\rightarrow\mathbb{R}^{d}$, satisfying

$\mathbb{P}\\{(B^{H}+f)(E)\cap
F\neq\emptyset\\}\leq{\textsf{c}}_{6}\,\mathcal{H}^{d-\beta/(\alpha+\varepsilon)}(F),$
(2.44)

where ${\textsf{c}}_{6}$ is a positive constant which depends on
$\alpha,\beta,H,d$ and $\varepsilon$ only.

##  3 Hitting points

Let’s start with the following fact:

Since $f$ is $H$-Hölder continuous Proposition 2.7. in [10] ensures that
$\dim_{\Psi,H}(Gr_{E}(f))=\dim(E)$ and then according to Theorem 1.2. in [24]
we have

$\dim(B^{H}+f)(E)=\frac{\dim(E)}{H}\wedge d.$

Hence, if $H\,d<\dim(E)$ Theorem 3.2. in [10] implies that
$\mathbb{E}(\lambda_{d}(B^{H}+f)(E))>0$, and a simple application of Fubini’s
theorem leads to

$\displaystyle\lambda_{d}\\{x\in\mathbb{R}^{d}:\mathbb{P}\\{\exists t\in
E:B^{H}(t)+f(t)=x\\}>0\\}>0.$ (3.1)

On the other hand we have $d<\dim(E)/H=\dim_{\rho_{H}}(E\times\\{x\\})$ for
all $x\in\mathbb{R}^{d}$, from which follows that
$\mathcal{C}_{\rho_{H},d}(E\times\\{x\\})>0$. Using Theorem 2.2 we conclude
that $B^{H}+f$ restricted on $E$ hits all points with positive probability
which is stronger than $\eqref{ss}$.

When the function $f$ loses the Hölder property, thus one wonders what about
the less smooth functions? Our goal is to shed some light on this question. In
fact we will need some additional information about the set $Gr_{E}(f)$ in
order to study the hitting probabilities points for the process $B^{H}+f$.
First we provide, for a simply measurable Borel function $f$, the lower and
upper bounds of hitting probabilities of points in terms of the parabolic
capacity of $Gr_{E}(f)$ of order $d$ and the $d$-dimensional parabolic
Hausdorff measure of $Gr_{E}(f)$ respectively. It can be also seen as an
extension of Theorem 2.2 to a measurable drift $f$ and $F=\\{x\\}$.

######  Proposition 3.1.

Let $\\{B^{H}(t):t\in[0,1]\\}$ be a $d$-dimensional fractional Brownian motion
with Hurst index $H\in(0,1)$. Let $f:[0,1]\rightarrow\mathbb{R}^{d}$ be a
bounded Borel measurable function and let $E\subset(0,1]$ be a Borel set. Then
for all $x\in\mathbb{R}^{d}$ there is a finite constant ${\textbf{c}}_{1}\geq
1$ such that

$\displaystyle{\textbf{c}}_{1}^{-1}\mathcal{C}_{\rho_{H},d}(Gr_{E}(f))\leq\mathbb{P}\left\\{\exists
t\in
E:(B^{H}+f)(t)=x\right\\}\leq{\textbf{c}}_{1}\,\mathcal{H}_{\rho_{H}}^{d}(Gr_{E}(f)).$
(3.2)

###### Proof.

we will closely follow the same steps as in the proof of Theorem 2.2. We start
with the upper bound using again the covering argument. Choose an arbitrary
constant $\gamma>\mathcal{H}_{\rho_{H}}^{d}(Gr_{E}(f))$. Then there is a
covering of $Gr_{E}(f)$ by balls $\\{B_{\rho_{H}}((t_{i},y_{i}),r_{i}),i\geq
1\\}$ in $\mathbb{R}_{+}\times\mathbb{R}^{d}$ such that

$Gr_{E}(f)\subseteq\bigcup_{i=1}^{\infty}B_{\rho_{H}}((t_{i},y_{i}),r_{i})\quad\text{
and }\quad\sum_{i=1}^{\infty}(2r_{i})^{d}\leq\gamma.$ (3.3)

It is easy to see that

$\left\\{\exists s\in
E:(B^{H}+f)(s)=x\right\\}\subseteq\bigcup_{i=1}^{\infty}\left\\{\exists\,\left(s,f(s)\right)\in\left(t_{i}-r_{i}^{1/H},t_{i}+r_{i}^{1/H}\right)\times
B(y_{i},r_{i})\text{ s.t. }(B^{H}+f)(s)=x\right\\}.$ (3.4)

Since for every fixed $i\geq 1$ we have

$\left\\{\exists\,\left(s,f(s)\right)\in\left(t_{i}-r_{i}^{1/H},t_{i}+r_{i}^{1/H}\right)\times
B(y_{i},r_{i})\text{ s.t.
}(B^{H}+f)(s)=x\right\\}\subseteq\left\\{\inf_{|s-t_{i}|^{H}<r_{i}}\|B^{H}(s)-x-y_{i}\|\leq
r_{i}\right\\},$ (3.5)

then we get from Lemma 2.3 that

$\displaystyle\mathbb{P}\left\\{\exists\,\left(s,f(s)\right)\in\left(t_{i}-r_{i}^{1/H},t_{i}+r_{i}^{1/H}\right)\times
B(y_{i},r_{i})\text{ s.t. }(B^{H}+f)(s)=x\right\\}$
$\displaystyle\leq\mathbb{P}\left\\{\inf_{|s-t_{i}|^{H}<r_{i}}\|B^{H}(s)-x-y_{i}\|\right\\}$
$\displaystyle\leq{\textbf{c}}_{2}\,r_{i}^{d},$ (3.6)

where $C_{2}$ depends on $H$, $E$ and $f$ only. Combining (3.3), (3.4), (3.5)
and (3.6) we derive that

$\mathbb{P}\left\\{\exists s\in
E:(B^{H}+f)(s)=x\right\\}\leq{\textbf{c}}_{3}\,\gamma,$

where $C_{3}$ depends only on $I,H$ and $f$. Let
$\gamma\downarrow\mathcal{H}_{\rho}^{d}(Gr_{E}(f))$, the upper bound in (3.2)
follows.

The lower bound in(3.2) holds also from the second moment argument. We assume
that $\mathcal{C}_{\rho_{H},d}(Gr_{E}(f))>0$, then let $\sigma$ be a measure
supported on $Gr_{E}(f)$ such that

$\displaystyle\mathcal{E}_{\rho_{H},d}(\sigma)=\int_{Gr_{E}(f)}\int_{Gr_{E}(f)}\frac{d\sigma(s,f(s))d\sigma(t,f(t))}{\rho_{H}((s,f(s)),(t,f(t)))^{d}}\leq\frac{2}{\mathcal{C}_{\rho_{H},d}(Gr_{E}(f))}.$
(3.7)

Let $\nu$ be the measure on $E$ satisfying $\nu:=\sigma\circ P_{1}^{-1}$ where
$P_{1}$ is the projection mapping on $E$, i.e. $P_{1}(s,f(s))=s$. For $n\geq
1$ we consider a family of random measures $\nu_{n}$ on $E$ defined by

$\displaystyle\int_{E}g(s)\nu_{n}(ds)$ $\displaystyle=\int_{E}(2\pi
n)^{d/2}\exp\left(-\frac{n\|B^{H}(s)+f(s)-x\|^{2}}{2}\right)g(s)\nu(ds),$
(3.8)

where $g$ is an arbitrary measurable function on $\mathbb{R}_{+}$. Following
the same steps as in (2.14) we obtain that there exists two positives
constants ${\textbf{c}}_{4}$ and ${\textbf{c}}_{5}$ such that

$\mathbb{E}(\|\nu_{n}\|)\geq\int_{E}\left(\frac{2\pi}{1+s^{2H}}\right)^{d/2}\exp\left(-\frac{\|x-f(s)\|^{2}}{2s^{2H}}\right)\nu(ds)\geq{\textbf{c}}_{4}>0,$
(3.9)

and

$\displaystyle\mathbb{E}\left(\|\nu_{n}\|^{2}\right)$
$\displaystyle=\int_{E}\int_{E}\nu(ds)\nu(dt)\int_{\mathbb{R}^{2d}}e^{-i(\left\langle\xi,x-f(s)\right\rangle+\left\langle\eta,x-f(t)\right\rangle)}$
$\displaystyle\times\exp(-\frac{1}{2}(\xi,\eta)(n^{-1}I_{2d}+Cov(B^{H}(s),B^{H}(t)))(\xi,\eta)^{T})d\xi
d\eta$
$\displaystyle\leqslant{\textbf{c}}_{5}\int_{Gr_{E}(f)}\int_{Gr_{E}(f)}\frac{d\sigma(s,f(s))d\sigma(t,f(t))}{(\max\\{|t-s|^{H},\|f(t)-f(s)\|\\})^{d}}={\textbf{c}}_{5}\,\mathcal{E}_{\rho_{H},d}(\sigma)<\infty.$
(3.10)

where the last inequality is a direct consequence of Lemma 2.4. Using once
again the Paley-Zygmund inequality, we conclude that $(\nu_{n})_{n\geq 1}$
admits a subsequence converging weakly to a finite measure $\bar{\mu}$
supported on the set $\\{(s,x)\in E\times F:B^{H}(s)+f(s)=x\\}$, positive with
positive probability and also satisfying the moment estimates of (2.14). Hence
we have

$\displaystyle\mathbb{P}\left(\exists s\in
E:(B^{H}+f)(s)=x\right)\geq\mathbb{P}\left(\|\bar{\mu}\|>0\right)\geq\frac{\mathbb{E}(\|\bar{\mu}\|)^{2}}{\mathbb{E}(\|\bar{\mu}\|^{2})}\geq\frac{{\textbf{c}}_{4}^{2}}{{\textbf{c}}_{5}\mathcal{E}_{\rho_{H},d}(\sigma)}.$
(3.11)

Combining this with (3.7) yields the lower bound in (3.2). The proof is
completed. ∎

######  Corollary 3.2.

Let $\\{B^{H}(t):t\in[0,1]\\}$ be a $d$-dimensional fractional Brownian motion
of Hurst index $H\in(0,1)$ and $E\subset[0,1]$ be a Borel set. Then for any
$\alpha<\text{dim}(E)/d\wedge H$ there exists a $\alpha$-Hölder continuous
function $f:[0,1]\rightarrow\mathbb{R}^{d}$ such that, for all
$x\in\mathbb{R}^{d}$, we have

$\displaystyle\mathbb{P}\left\\{\exists t\in E:(B^{H}+f)(t)=x\right\\}>0.$
(3.12)

In other words, that the restriction of $f$ to $E$ is non-polar for $B^{H}$.

###### Proof.

Let $\varepsilon>0$ such that
$\alpha^{\prime\prime}=\alpha+\varepsilon<\text{dim}(E)/d\wedge H$ and
$B^{\alpha^{\prime\prime}}$ be the fractional Brownian motion defined above.
It is known from Theorem 2.9 and Corollary 2.11 in [10] that, for
$\alpha^{\prime\prime}<\text{dim}(E)/d\wedge H$, we have

$\dim_{\rho_{H}}(Gr_{E}(B^{\alpha^{\prime\prime}}))=\text{dim}(E)/\alpha^{\prime\prime}\wedge\left(\text{dim}(E)/H+d(1-\alpha^{\prime\prime}/H)\right)>d,\,\mathbb{P}^{\prime}a.s.$

Then for any fixed $x\in\mathbb{R}^{d}$, Proposition 3.1 tells us that for
$\mathbb{P}^{\prime}$ almost all $\omega^{\prime}\in\Omega^{\prime}$ there is
a positive constant $C=C(\omega^{\prime})$ such that

$\displaystyle\mathbb{P}\left\\{\exists s\in
E:(B^{H}+B^{\alpha}(\omega^{\prime}))(s)=x\right\\}\geq
C\,\mathcal{C}_{\rho_{H},d}\left(Gr_{E}(B^{\alpha^{\prime\prime}}(\omega^{\prime}))\right)>0.$

Hence, if we choose $f$ to be one of the trajectories of
$B^{\alpha^{\prime\prime}}$, which is $\alpha$-Hölder continuous, we obtain

$\displaystyle\dim_{\rho_{H}}(Gr_{E}(f))>d.$

Therefore for any fixed $x\in\mathbb{R}^{d}$, $\left\\{(B^{H}+f(s),s\in
E\right\\}$ hits $x$ with positive probability. ∎

######  Remark 3.3.

1\. 2\. We mention that the covering argument used to prove the upper bound in
(3.2) can also serve to show that for any Borel set $F\subset\mathbb{R}^{d}$,
there exists a positive finite constant c such that

$\displaystyle\mathbb{P}\left\\{(B^{H}+f)(E)\cap
F\neq\emptyset\right\\}\leq{\textbf{c}}\,\mathcal{H}_{\widetilde{\rho}_{H}}^{d}(Gr_{E}(f)\times
F).$ (3.13)

Here $\mathcal{H}_{\widetilde{\rho}_{H}}^{\alpha}(\centerdot)$ is the
$\alpha$-dimensional Hausdorff measure on the metric space
$(\mathbb{R}_{+}\times\mathbb{R}^{d}\times\mathbb{R}^{d},\widetilde{\rho}_{H})$,
where the $\widetilde{\rho}_{H}$ is defined by

$\displaystyle\widetilde{\rho}_{H}((s,x,u),(t,y,v)):=\max\\{|t-s|^{H},\|x-y\|,\|u-v\|\\}.$

But we have difficulty in proving the lower band in terms of
$\mathcal{C}_{\widetilde{\rho}_{H},d}(Gr_{E}(f)\times F)$ even when $F$ has
some smooth structure.

Accordingly, in view of the foregoing, can we expect the same result for
others functions with fewer restrictions? This impels us to consider the class
of reverse $\alpha$-Hölder continuous functions whose definition is as follows

######  Definition 3.4.

We say that a continous function $f:[0,1]\rightarrow\mathbb{R}^{d}$ is reverse
$\alpha$-Hölder continuous, for $0<\alpha<1$, if there exists a constant $C>0$
such that for any interval $J\subset[0,1]$, we have

$\displaystyle\sup_{s,t\in J}\|f(s)-f(t)\|\geq C|J|^{\alpha},$ (3.14)

where $|J|$ is the diameter of the interval $J$.

Recall that this notion is closely linked to the geometric properties of the
graph of the function $f$. For $d=1$, a famous example of a function
satisfying the reverse $\alpha$-Hölder continuity condition is the Weierstrass
function given by

$\displaystyle\mathcal{W}_{\tau,\theta}(t)=\sum_{n=0}^{\infty}\tau^{n}\cos\left(2\pi\theta^{n}t\right),\quad
t\in[0,1],$ (3.15)

for $\tau<1<\theta$ and $\tau\,\theta>1$ and
$\alpha=-\log(\tau)/\log(\theta)$. See [5] or [12] for the proof. Recently
Shen [28], improving result of Barański, Bárány and Romanowska [3], proved
that for any integer $\theta\geq 2$ and any $\tau\in(\theta^{-1},1)$, the
Hausdorff dimension of the graph of the Weierstrass function
$\mathcal{W}_{\tau,\theta}$ is equal to
$2+\dfrac{\log\left(\tau\right)}{\log\left(\theta\right)}$. It is worth
pointing out that, for $H=1$, the metric $\rho_{H}$ on
$\mathbb{R}_{+}\times\mathbb{R}$ defined by

$\rho_{H}((s,x),(t,y))=\max\\{|t-s|^{H},|x-y|\\}\quad\forall(s,x),(t,y)\in\mathbb{R}_{+}\times\mathbb{R}$

is nothing but the metric derived from the Maximum norm on $\mathbb{R}^{2}$.
Now using a comparison result for the Hausdorff parabolic dimensions with
different parameters, see Proposition 2.5 in Erraoui and Hakiki [10] (which
remains valid also for $H$ and $1$), one can check that

$1<2+\dfrac{\log\left(\tau\right)}{\log\left(\theta\right)}=\dim(Gr_{[0,1]}(\mathcal{W}_{\tau,\theta}))\leq\dim_{\rho_{H}}(Gr_{[0,1]}(\mathcal{W}_{\tau,\theta})).$

Thus $\dim_{\rho_{H}}(Gr_{[0,1]}(\mathcal{W}_{\tau,\theta}))>1$ and therfore
$\mathcal{C}_{\rho_{H},1}(Gr_{[0,1]}(\mathcal{W}_{\tau,\theta})))>0$. This is
expressed in the following

######  Proposition 3.5.

For any integer $\theta\geq 2$ and any $\tau\in(\theta^{-1},1)$, the
Weierstrass function $\\{\mathcal{W}_{\tau,\theta}(t),\,t\in[0,1]\\}$ is non-
polar for real valued fractional Brownian motion $\\{B^{H}(t),\,t\in[0,1]\\}$
with Hurst index $H\in(0,1)$.

######  Remark 3.6.

It is worth mentioning that the Hölder continuity of order
$\alpha=-\log(\tau)/\log(\theta)$ is also met by the Weierstrass function
$\\{\mathcal{W}_{\tau,\theta}(t),\,t\in[0,1]\\}$, cf. Lemma 5.1.8 in [5].

It is also interesting to note that, in the same context, Theorem 4 in [27]
affirms that, for any $\alpha$-Hölder and reverse $\alpha$-Hölder continuous
function $f$ with $\alpha\in(0,1)$, we have $\dim_{[0,1]}(Gr(f))>1$ which was
the key element in the proof of the above proposition. Therefore, we will come
to the same conclusion as the one for the Weierstrass function stated as
follows

######  Proposition 3.7.

Suppose that $0<\alpha<1$ and $f:[0,1]\rightarrow\mathbb{R}$ is
$\alpha$-Hölder and reverse $\alpha$-Hölder continuous function. Then $f$ is
non-polar for real valued fractional Brownian motion
$\\{B^{H}(t),\,t\in[0,1]\\}$ with Hurst index $H\in(0,1)$.

This raises the question whether the result remains valid in higher
dimensions. However Proposition 3.7 cannot be extended to $d$-dimensional case
as is shown by the following

######  Proposition 3.8.

Let $\\{B^{H}(t):t\in[0,1]\\}$ be a $d$-dimensional fractional Brownian motion
of Hurst index $H\in(0,1)$ such that $H\,d>1$. Then there exists a
$\alpha$-Hölder and reverse $\alpha$-Hölder continuous function
$f:[0,1]\rightarrow\mathbb{R}^{d}$ such that the process
$\\{(B^{H}+f)(t):t\in[0,1]\\}$ does not hit $x$ for all $x\in\mathbb{R}^{d}$,
i.e.

$\mathbb{P}\left\\{\exists t\in[0,1]:(B^{H}+f)(t)=x\right\\}=0.$

###### Proof.

Let $d\geq 2$ and define the function $f$ from $[0,1]$ to $\mathbb{R}^{d}$ by
:
$f(t)=\left(\mathcal{W}_{\tau,\theta}(t),\cdots,\mathcal{W}_{\tau,\theta}(t)\right)$
where $\mathcal{W}_{\tau,\theta}$ is the Weierstrass function given in (3.15).
It is easy to see that $f$ satisfies the $\alpha$-Hölder and reverse
$\alpha$-Hölder conditions with
$\alpha=-\dfrac{\log\left(\tau\right)}{\log\left(\theta\right)}$. Moreover we
have
$\dim_{\rho_{H}}(Gr_{[0,1]}(f))=\dim_{\rho_{H}}(Gr_{[0,1]}(\mathcal{W}_{\tau,\theta}))$.
It follows from Proposition 2.5 in Erraoui and Hakiki [10] that

$\dim_{\rho_{H}}(Gr_{[0,1]}(f))\leq\dim(Gr_{[0,1]}(f))-1+\frac{1}{H}=1+\dfrac{\log\left(\tau\right)}{\log\left(\theta\right)}+\frac{1}{H}.$

Then for any $\theta>1$ and $\tau\in(\theta^{-1},\theta^{d-(1/H)-1}\wedge 1)$
we have $\dim_{\rho_{H}}(Gr_{[0,1]}(f))<d$. The later entails that
$\mathcal{H}_{\rho_{H}}^{d}(Gr_{[0,1]}(f))=0$. Proposition (3.1) will allow us
to achieve the desired outcome. ∎

## References

  * [1] Ayache, A., Xiao, Y.: Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J. Fourier Anal. Appl. 11, 407–439 (2005)
  * [2] Antunović, T., Peres, Y., Vermesi, B.: Brownian motion with variable drift can be space filling. Proc. Amer. Math. Soc. 139, no. 9, 3359-3373 (2011)
  * [3] Barański, K., Bárány, B., Romanowska, J.: On the dimension of the graph of the classical Weierstrass function. Adv. Math. 265, 32-59 (2014)
  * [4] Biermé, H., Lacaux, C., Xiao, Y.: Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields. Bull London Math. Soc. 41, 253-273 (2009)
  * [5] Bishop, C. J., Peres, Y.: Fractals in probability and analysis. Cambridge Studies in Advanced Mathematics, 162. Cambridge University Press, Cambridge, (2017)
  * [6] Chen, Z., Xiao, Y.: On intersections of independent anisotropic Gaussian random fields. Sci. China Math. 55, no. 11, 2217-2232 (2012)
  * [7] Dalang, R. C., Nualart, E.: Potential theory for hyperbolic SPDEs. Ann. Probab. 32, no. 3A, 2099-2148 (2004)
  * [8] Dalang, R.C., Khoshnevisan, D., Nualart, E.: Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. ALEA Lat. Am. J. Probab. Math. Stat. 3 231–271 (2007)
  * [9] Dalang, R.C., Khoshnevisan, D., Nualart, E.: Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise. Probab. Theory Related Fields 144 371– 427 (2009)
  * [10] Erraoui, M., Hakiki, Y.: Images of fractional Brownian motion with deterministic drift: Positive Lebesgue measure and non-empty interior. (Submitted paper)
  * [11] Graversen, S. E.: Polar-functions for Brownian motion. Z. Wahrsch. Verw. Gebiete, 61, no. 2, 261–270, (1982)
  * [12] Hardy, G. H.: Weierstrass’s nondifferentiable function. Trans. Amer. Math. Soc. 17, 301-325 (1916)
  * [13] Hutchinson, J. E.: Fractals and self similarity, Indiana Univ. Math. J., 30, 713-747 (1981)
  * [14] Kahane, J. P.: Some random series of functions, Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, (1985)
  * [15] Kakutani, S.: Two dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo 20, 648-652, (1944)
  * [16] Khoshnevisan, D.: Escape rates for Lévy processes. Studia Sci. Math. Hungar. 33, no. 1-3, 177-183 (1997)
  * [17] Khoshnevisan, D.: Multiparameter Processes: An Introduction to Random Fields. Springer, New York (2002)
  * [18] Khoshnevisan, D., Shi, Z.: Brownian sheet and capacity. Ann. Probab. 27, 1135–1159 (1999)
  * [19] Le Gall, J. F.: Sur les fonctions polaires pour le mouvement brownien. In Séminaire de Probabilités, XXII, volume 1321 of Lecture Notes in Math., pages 186–189. Springer, Berlin, (1988)
  * [20] Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge University Press, Cambridge (1999)
  * [21] Mountford, T. S.: Time inhomogeneous Markov processes and the polarity of single points. Ann. Probab. 17, no. 2, 573-585 (1989)
  * [22] Mueller, C., Tribe, R.: Hitting probabilities of a random string, Electron. J. Probab. 7, 1–29 (2002)
  * [23] Peres, Y., Sousi, P.: Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension. Math. Proc. Cambridge Philos. Soc. 153, no. 2, 215-234 (2012)
  * [24] Peres, Y., Sousi, P.: Dimension of fractional Brownian motion with variable drift. Probab. Theory Related Fields 165, no. 3-4, 771-794 (2016)
  * [25] Pitt, L.D.: Local times for Gaussian vector fields. Indiana Univ. Math. J. 27, 309–330 (1978)
  * [26] Pruitt, W. E., Taylor, S. J.: The potential kernel and hitting probabilities for the general stable process in $R^{N}$. Trans. Amer. Math. Soc. 146, 299-321 (1969)
  * [27] Przytycki, F.; Urbański, M.: On the Hausdorff dimension of some fractal sets. Studia Math. 93, no. 2, 155–186 (1989)
  * [28] Shen, W.: Hausdorff dimension of the graphs of the classical Weierstrass functions. Math. Z. 289, no. 1-2, 223–266 (2018)
  * [29] Taylor, S. J., Watson, N. A.: A Hausdorff measure classification of polar sets for the heat equation. Math. Proc. Cambridge Philos. Soc. 97, no. 2, 325-344 (1985)
  * [30] Xiao, Y.: Packing measure of the sample paths of fractional Brownian motion. Trans. Amer. Math. Soc. 348, no. 8, 3193–3213 (1996)
  * [31] Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. A Minicourse on Stochastic Partial Differential Equations, (D Khoshnevisan and F Rassoul-Agha, editors), Lecture Notes in Math, 1962: 145-212. New York: Springer, (2009)