2. Preliminaries on Hausdorff measures and capacities

In this section, we would like to consider some comparison results that are intended to provide upper bound for the Hausdorff measure (resp. lower bound of the Bessel–Riesz capacity) of a product set $G_1\times G_2$ in terms of the Hausdorff measures (resp. in terms of capacity) of each of the component $G_1$ and $G_2$ with appropriate orders. Such results might be useful in studying the problem of hitting probabilities for $B^H+f$ . First of all, we need to recall definitions of Hausdorff measures as well as Bessel–Riesz capacities in a general metric space.

Let $(\mathscr{X},\rho)$ be a bounded metric space, $\beta \gt 0$ and $G \subset {\mathscr{X}}$ . We define the $\beta$ -dimensional Hausdorff measure of G with respect to the metric $\rho$ as

(2·1) \begin{align} {}\mathcal{H}_{\rho}^{\beta}(G)=\lim_{\delta \rightarrow 0}\inf \left\{\sum_{n=1}^{\infty}\left(2 r_{n}\right)^{\beta}\,:\, G \subseteq \bigcup_{n=1}^{\infty} B_{\rho}\left(r_{n}\right), r_{n} \leqslant \delta\right\}, {}\end{align}

where $B_{\rho}(r)$ denotes an open ball of radius r in the metric space $({\mathscr{X}},\rho)$ . The Hausdorff dimension of G in the metric space $(X,\rho)$ is defined to be

(2·2) \begin{align}\dim_{\rho}(G)=\inf\{\beta \gt 0\,:\, \mathcal{H}_{\rho}^{\beta}(G)=0\}\, \text{ for all $G\subset {\mathscr{X}}$}.\end{align}

The usual $\beta$ -dimensional Hausdorff measure and the Hausdorff dimension in Euclidean metric are denoted by $\mathcal{H}^{\beta}$ and $\dim({\cdot})$ , respectively. Moreover, $\mathcal{H}^{\beta}$ is assumed equal to 1 whenever $\beta\leq 0$ .

We introduce now the Minkowski dimension of $E\subset ({\mathscr{X}},\rho)$ . Let $N_{\rho}(E,r)$ be covering number, that is the smallest number of open balls of radius r required to cover E. The lower and upper Minkowski dimensions of E are respectively defined as

\begin{equation*} {}\begin{aligned} {}&\underline{\dim}_{\rho,M}(E)\,:\!=\, \liminf_{r\rightarrow 0^+}\frac{\log N_{\rho}(E,r)}{\log(1/r)},\\ {}&\overline{\dim}_{\rho,M}(E)\,:\!=\, \limsup_{r\rightarrow 0^+}\frac{\log N_{\rho}(E,r)}{\log(1/r)}. {}\end{aligned} {}\end{equation*}

Equivalently, the upper Minkowski dimension of E can be characterised by

(2·3) \begin{align} {}\overline{\dim}_{\rho,M}(E)=\inf\{\gamma \,:\, \exists\, C\in \mathbb{R}_+\, \text{ such that } N_{\rho}(E,r)\leqslant Cr^{-\gamma} \text{ for all } r \gt 0 \}. {}\end{align}

In the Euclidean case, the upper Minkowski dimension will be denoted by $\overline{\dim}_{M}$ .

Let $\varphi_{\alpha}\,:\,(0,\infty)\rightarrow (0,\infty)$ be the function given by

(2·4) \begin{align} {}\varphi_{\alpha}(r)=\left\{\begin{array}{ll}{r^{-\alpha}} & {\text { if } \alpha \gt 0,} \\ {\log \left(\frac{e}{r \wedge 1}\right)} & {\text { if } \alpha=0,} \\ {1} & {\text { if } \alpha \lt 0.}\end{array}\right. {} {}\end{align}

The Bessel–Riesz type capacity of order $\alpha$ on the metric space $({\mathscr{X}},\rho)$ is defined by

(2·5) \begin{align} {}\mathcal{C}_{\rho}^{\alpha}(G)=\left[\inf _{\mu \in \mathcal{P}(G)} \int_{\mathscr{X}} \int_{\mathscr{X}} \varphi_{\alpha}(\rho(u,v)) \mu(d u) \mu(d v)\right]^{-1}, {}\end{align}

where $\mathcal{P}(G)$ is the family of probability measures carried by G.

We note that for $\alpha \lt 0$ we have $\mathcal{C}_{\rho}^{\alpha }(G)=1$ for any nonempty G. The usual Bessel–Riesz capacity of order $\alpha$ in the Euclidean metric will be denoted by $\mathcal{C}^{\alpha}$ .

Now let $\left({\mathscr{X}_i},\rho_i\right)$ $i=1,2$ be two metric spaces. Let $\rho_3$ be the metric on $\mathscr{X}_1\times \mathscr{X}_2$ given by

$$ {} \rho_3\left((u,x),(v,y)\right)=\max\{\rho_1(u,v),\rho_2(x,y)\}.$$

The following proposition if the main result of this section.

then we have

(2·9) \begin{align} \mathcal{H}_{\rho_3}^{\alpha}(G_1\times G_2)\leq\, \unicode{x1D588}_{4} \mathcal{H}_{\rho_2}^{\alpha-\gamma^{\prime}}(G_2),\end{align}

for some constant $\unicode{x1D588}_4 \gt 0$ .

Similar estimates hold true if we assume that $G_2$ verifies assumptions (i) and (ii). That is there exist positive constants $\unicode{x1D588}_{5}$ and $\unicode{x1D588}_{6}$ such that

(2·10) \begin{align} \mathcal{C}_{\rho_1}^{\alpha-\gamma}(G_{1})\leq \unicode{x1D588}_{5}\,\mathcal{C}_{\rho_3}^{\alpha}(G_1\times G_2), {}\\[-12pt] \nonumber {}\end{align}

(2·11) \begin{align} \mathcal{H}_{\rho_3}^{\alpha}(G_1\times G_2)\leq\, \unicode{x1D588}_{6} \mathcal{H}_{\rho_1}^{\alpha-\gamma^{\prime}}(G_1).\end{align}

The following lemma will help us to establish (2·7) (resp. (2·10)).

Lemma 2·3. Let $(\mathscr{X},\rho)$ be a bounded metric space and $\mu$ a probability measure supported on $\mathscr{X}$ satisfying

(2·12) \begin{align} \mu\left(B_{\rho}(u,r)\right) \leq C_1 r^{\kappa} \quad \text{ for all $\,u\in \mathscr{X}$ and $\,r \gt 0$,}\end{align}

for some positive constants $C_1$ and $\kappa$ . Then for any $\theta \gt 0$ , there exists $C_2 \gt 0$ such that

(2·13) \begin{align} I(r)\,:\!=\,\sup_{v \in \mathscr{X}}\int_{\mathscr{X}}\frac{\mu(du)}{\left(\max\{\rho(u,v),r\}\right)^{\theta}}\leq C_2\, \varphi_{\theta-\kappa}(r),\end{align}

for all $r\in (0,\mathrm{diam}(\mathscr{X}))$ .

Proof of Proposition 2·1. We start by proving (i). Let us suppose that $\mathcal{C}_{\rho_2}^{\alpha-\gamma}(G_2) \gt 0$ , otherwise there is nothing to prove. It follows that for $\eta \in (0,\mathcal{C}_{\rho_2}^{\alpha-\gamma}(G_2))$ there is a probability measure m supported on $G_2$ such that

(2·14) \begin{align}\mathcal{E}_{\rho_2,\alpha-\gamma}(m)\,:\!=\,\int_{G_2}\int_{G_2}\varphi_{\alpha-\gamma}(\rho_2(x,y))\,m(dx)m(dy)\leq \eta^{-1}.\end{align}

Since $\mu \otimes m$ is a probability measure on $G_1 \times G_2$ , then applying Fubini’s theorem and Lemma 2·3 we obtain

(2·15) \begin{align}\begin{aligned} \mathcal{E}_{\rho_3,\alpha}(\mu \otimes m)&= \int_{G_1\times G_2} \int_{G_1\times G_2} \frac{\mu\otimes m(dudx) \mu\otimes m(dvdy)}{(\rho_3\left((u,x),(v,y)\right))^{\alpha}}\\ &\leq C_2\int_{G_2}\int_{G_2}\varphi_{\alpha-\gamma}(\rho_2(x,y))\,m(dx)m(dy)\leq C_2\,\eta^{-1}.\end{aligned}\end{align}

Consequently we have $\mathcal{C}_{\rho_3}^{\alpha}(G_1\times G_2)\geq C_2^{-1}\,\eta$ . Then we let $\eta\uparrow\mathcal{C}_{\rho_2}^{\alpha-\gamma}(G_2)$ to conclude that the inequality in (2·7) holds true.

Now let us prove (ii). Let $\zeta \gt \mathcal{H}_{\rho_2}^{\alpha-\gamma^{\prime}}(G_2)$ be arbitrary. Then there is a covering of $G_2$ by open balls $B_{\rho_2}(x_{n},r_n)$ of radius $r_n$ such that

(2·16) \begin{align}G_2 \subset \bigcup_{n=1}^{\infty} B_{\rho_2}(x_{n},r_{n}) \quad \text { and } \quad \sum_{n=1}^{\infty} (2r_{n})^{\alpha-\gamma^{\prime}} \leq \zeta. {}\end{align}

For all $n\geq 1$ , let $B_{\rho_1}(u_{n,j},r_n)$ , $j=1,...,N_{\rho_1}(G_1,r_n)$ be a family of open balls covering $G_1$ . It follows that the family $B_{\rho_1}(u_{n,j},r_n)\times B_{\rho_2}(x_{n},r_n),\,j=1,...,N_{\rho_1}(G_1,r_n)$ , $n \geq 1$ gives a covering of $G_1\times G_2$ by open balls of radius $r_n$ for the metric $\rho_3$ .

It follows from (2·8) and (2·16) that

(2·17) \begin{align} {} {}\sum_{n=1}^{\infty} \sum_{j=1}^{N_{\rho_1}(G_1, r_n)}\left(2 r_{n}\right)^{\alpha} \leq \unicode{x1D588}_{3}\,2^{\gamma^{\prime}}\, \sum_{n=1}^{\infty} (2r_{n})^{\alpha-\gamma^{\prime}} \leq \unicode{x1D588}_{3}\,\,2^{\gamma^{\prime}}\, \zeta. {} {}\end{align}

Letting $\zeta\downarrow\mathcal{H}_{\rho_2}^{\alpha-\gamma^{\prime}}(G_2)$ , the inequality in (2·9) follows with $\unicode{x1D588}_{4}=\unicode{x1D588}_{3}\,\,2^{\gamma^{\prime}}$ .

In the following we give a sufficient condition ensuring hypotheses (i) and (ii) of Proposition 2·1.

Proposition 2·4. The following condition

$$ 0 \lt \gamma \lt \dim_{\rho_1}(G_1)\leq \overline{\dim}_{\rho_1,M}(G_1) \lt \gamma^{\prime} \lt \alpha,$$

is sufficient to achieve (2·6) and (2·8).

It is well known that Hausdorff and Minkowski dimensions agree for many Borel 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.

Definition 2·5. Let $(\mathscr{X},\rho)$ be a bounded metric space, $\gamma \gt 0$ and $G\subset \mathscr{X}$ . We say that G is $\gamma$ -Ahlfors–David regular if there exists a finite positive Borel measure $\mu$ supported on G and positive constant $\unicode{x1D588}_{\gamma}$ such that

(2·18) \begin{align} {} {}\unicode{x1D588}_{\gamma}^{-1}\,r^{\gamma}\leq\mu\left(B_{\rho}\left(a,r\right)\right)\leq\unicode{x1D588}_{\gamma}\,r^{\gamma}\,\,\ \text{for all a}\ \in G,\ \text{and all} \, \quad 0 \lt r \leq1.\end{align}

For a Borel set $E\subset \mathbb{R}^n$ satisfying the condition (2·18) with $\rho$ is the euclidean metric of $\mathbb{R}^n$ , it is shown in [ Reference Mattila21 , theorem 5·7, p.80] that

$$ \gamma=\dim(E)=\underline{\dim}_M(E)=\overline{\dim}_M(E).$$

This statement still true in a general metric space $(\mathscr{X},\rho)$ , it suffices to go through the same lines of the proof of the euclidean case. Here we provide some examples of such sets.

The following proposition states that when $G_1$ (resp. $G_2$ ) is $\gamma_1$ -Ahlfors–David regular set (resp. $\gamma_2$ -Ahlfors–David regular set), then inequalities (2·7) and (2·9) of Proposition 2·1 are checked for $\gamma=\gamma^{\prime}=\gamma_1$ (resp. for $\gamma=\gamma^{\prime}=\gamma_2$ ).

Similar estimates hold true under the assumption $G_2$ is $\gamma_2$ -Ahlfors–David regular set. Precisely we have

(2·21) \begin{align} \mathcal{C}_{\rho_1}^{\alpha-\gamma_2}(G_{1})\leq \unicode{x1D588}_{9}\,\mathcal{C}_{\rho_3}^{\alpha}(G_1\times G_2), \end{align}

(2·22) \begin{align} \mathcal{H}_{\rho_3}^{\alpha}(G_1\times G_2)\leq\, \unicode{x1D588}_{10} \mathcal{H}_{\rho_1}^{\alpha-\gamma_2}(G_1).\end{align}

Proof. In order to prove (2·19) and (2·20) it is sufficient to check that conditions (2·6) and (2·8) are satisfied with $\gamma=\gamma^{\prime}=\gamma_1$ . Firstly, (2·6) is no other than the right inequality in (2·18). On the other hand, let $0 \lt r\leq 1$ and $P_{\rho_1}(G_1,r)$ be the packing number, that is the greatest number of disjoint balls $B_{\rho_1}(x_j,r)$ with $x_{j}\in G_1$ . The left inequality of (2·18) ensures that

$$ \unicode{x1D588}_{\gamma_1}^{-1}\, P_{\rho_1}(G_1,r)\, r^{\gamma_1}\leq \sum_{j=1}^{P_{\rho_1}(G_1,r)}\mu\left(B_{\rho_1}(x_j,r)\right)=\mu (G_1)\leq 1.$$

Using the well-known fact that $N_{\rho_1}(G_1,2\,r)\leq P_{\rho_1}(G_1,r)$ , we obtain the desired estimation (2·8).

Remark 2·7. Notice that when both $G_i$ , $i=1,2$ are $\gamma_i$ -Ahlfors David regular sets for some constant $\gamma_i \gt 0$ , then there exist two positive constants $\unicode{x1D588}_{11}$ and $\unicode{x1D588}_{12}$ such that

(2·23) \begin{align} & \mathcal{C}_{\rho_3}^{\gamma_1+\gamma_2}(G_1\times G_2)\geq \unicode{x1D588}_{11}\,\mathcal{C}_{\rho_2}^{\gamma_2}(G_{2})\vee \mathcal{C}_{\rho_1}^{\gamma_1}(G_{1}) \nonumber \\ & \quad \text{ and }\quad \mathcal{H}_{\rho_3}^{\gamma_1+\gamma_2}(G_1\times G_2)\leq \unicode{x1D588}_{12}\,\mathcal{H}_{\rho_2}^{\gamma_2}(G_{2})\wedge \mathcal{H}_{\rho_1}^{\gamma_1}(G_{1}).\end{align}

3. Hitting probabilities for fractional Brownian motion with deterministic regular drift

Let $H \in (0,1)$ and $B^{H}_0=\left\lbrace B^{H}_0(t),t \geq 0\right\rbrace $ 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 $B^{H}_0$ . The stochastic process $B^{H}=\left\lbrace B^{H}(t),t\geq 0\right\rbrace $ given by

\[ {}B^{H}(t)=(B^{H}_1(t),....,B^{H}_d(t)), {}\]

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

For $\mathbf{d}$ a metric on $\mathbb{R}_+$ , we define on $\mathbb{R}_+\times\mathbb{R}^d$ the metric $\rho_{\mathbf{d}}$ as follows

(3·1) \begin{align}\rho_{\mathbf{d}}((s,x),(t,y))=\max\{\mathbf{d}\left(t,s\right),\|x-y\|\}\quad \forall (s,x), (t,y)\in \mathbb{R}_+\times\mathbb{R}^d.\end{align}

We denote by $\mathbf{d}_{H}$ the canonical metric of $B^H$ given by

(3·2) \begin{align} \mathbf{d}_{H}(s,t)\,:\!=\,|t-s|^H \, \quad \text{for all } s,t \in \mathbb{R}_+.\end{align}

The associated metric $\rho_{\mathbf{d}_{H}}$ on $\mathbb{R}_+\times \mathbb{R}^d$ is called the parabolic metric.

Let ${\mathbf{I}} = [a, b]$ , where $a \lt b \in (0,1]$ are fixed constants. For $\alpha\in (0,1)$ , $C^{\alpha}(\mathbf{I})$ is the space of $\alpha$ -Hölder continuous function $f \,:\, [a,b] \longrightarrow \mathbb{R}^d $ equipped with the norm

(3·3) \begin{align} {} {}\lVert f \rVert_{\alpha}\,:\!=\, \sup_{s\in \mathbf{I}} \lVert f(s) \rVert +\underset{\substack{s, t \in \mathbf{I} \\ s \neq t}}{\sup}\dfrac{\lVert f(t)-f(s) \rVert}{|t-s|^{\alpha}} \lt +\infty. {}\end{align}

First we state the following result, which is easily deduced from [ Reference Chen5 , theorem 2·6].

Theorem 3·1. Let $\{B^H(t), t\in [0,1]\}$ be a d-dimensional fractional Brownian motion and $f \in C^{H}\left(\mathbf{I}\right)$ . Let $F\subseteq \mathbb{R}^d$ and $E\subset \mathbf{I}$ are two compact subsets. Then there exist a constant $c\geq 1$ depending on $\mathbf{I}$ , F, H and f only, such that

(3·4) \begin{align}c^{-1}\mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}(E\times F)\leq \mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}\leq c\,\,\mathcal{H}_{\rho_{\mathbf{d}_{H}}}^{d}(E\times F).\end{align}

The aim of this section is to provide the hitting probabilities estimates for some particular sets E and F. Such estimates would be helpful in the next section to establish a kind of of sharpness of the H-Hölder regularity of the drift f in Theorem 3·1.

3.1. Hitting probabilities estimates when E is $\beta$ -Ahlfors–David regular set

First let us recall that, when E is an interval, [ Reference Chen and Xiao4 , corollary 2·2] ensures that there exists a constant $ \mathsf{c}\geq 1$ depending only on E, F and H such that

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

for any Borel set $F\subseteq \mathbb{R}^d$ . Our next goal is to establish similar estimates for $(B^H+f)$ when E be a $\beta$ -Ahlfors–David regular set.

Proposition 3·2. Let $B^H$ , f, E and F as in Theorem 3·1 with ${E\subset \left(\mathbf{I},|.| \right)} $ is a $\beta$ -Ahlfors–David regular for some $\beta\in (0,1]$ . Then there is a positive constant $\mathtt{c}_{2}$ which depends on E, F, H, $K_f$ and $\beta$ , such that

(3·5) \begin{align} {} {}\mathtt{c}_{2}^{-1}\, \mathcal{C}^{d-\beta/H}(F)\leq \mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}\leq \mathtt{c}_{2}\,\mathcal{H}^{d-\beta/H}(F). {} {}\end{align}

Proof. Three cases are to be discussed here: (j) $\beta \lt Hd$ , (jj) $\beta =Hd$ and (jjj) $\beta \gt Hd$ . Let us point out first that for the lower bound, the interesting cases are (j) and (jj) while for the upper bound it is the case (j) which requires proof. Using Proposition 2·6 with $X_1=E$ , $\rho_1(s,t)=|t-s|^H$ , $X_2=\mathbb{R}^d$ , $\rho_2(x,y)=\|x-y\|$ , and $\alpha=d$ , we get the first case (j). For the case (jj) we only use Proposition 2·6 (i).

Following the same pattern as above we get the following corollary.

Corollary 3·3. Let $B^H$ , E and F as in Proposition 3·2. Then there is a positive constant $\mathtt{c}_{3}$ which depends on E, F, H and $\beta$ , such that

(3·6) \begin{align}\mathtt{c}_{3}^{-1}\, \mathcal{C}^{d-\beta/H}(F)\leq \mathbb{P}\{B^H(E)\cap F\neq \emptyset \}\leq {} {}\mathtt{c}_{3}\,\mathcal{H}^{d-\beta/H}(F).\end{align}

Remark 3·4.

1. (i) We note that [ Reference Chen and Xiao4 , corollary 2·2] corresponds to the particular case $E=\mathbf{I}$ for which $\beta=1$ .

2. (ii) For less regular set E, with $0 \lt \beta \lt \dim(E)\leq \overline{\dim}_M(E) \lt \beta^{\prime} \lt Hd$ , we can derive, as a consequence of (2·7) and (2·9) of Proposition 2·1, Proposition 2·4 and Theorem 3·1, the following weaker estimates

   (3·7) \begin{align} {} {}\mathtt{c}_1^{-1}\,\mathcal{C}^{d-\beta/H}(F)\leq \mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}\leq \mathtt{c}_1\,\mathcal{H}^{d-\beta^{\prime}/H}(F), {} {}\end{align}
   where $\mathtt{c}_{1}$ is a positive constant depends only on E, F, H, $\mathrm{K}_f$ , $\beta$ and $\beta^{\prime}$ .

3.2. Hitting probabilities estimates when F is $\gamma$ -Ahlfors–David regular

Now we get parallel results to those given in Proposition 3·2, emphasizing regularity properties of F instead of E. If ${F\subset \left( \mathbb{R}^d,\lVert \cdot \rVert\right) } $ is a $\gamma$ -Ahlfors–David regular, we have the following result, which could be proven similarly to Proposition 3·2 by making use of (2·21) and (2·22).

Proposition 3·5. Let $B^H$ , f, E and F as in Theorem 3·1, such that F is a $\gamma$ -Ahlfors–David regular compact subset of $[-M,M]^d$ for some $\gamma \in (0,d]$ . Then there is a positive constant $\mathtt{c}_{5}$ which depends on E, F, H, $\mathrm{K}_f$ and $\gamma$ only, such that

(3·8) \begin{align}\mathtt{c}_{5}^{-1}\, \mathcal{C}^{H(d-\gamma)}(E)\leq \mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}\leq \mathtt{c}_{5}\,\mathcal{H}^{H(d-\gamma)}(E).\end{align}

Corollary 3·6 Let $B^H$ , E and F as in Proposition 3·5. Then there is a positive and constant $\mathtt{c}_{6}$ which depends on E, F, H and $\gamma$ , such that

(3·9) \begin{align}\mathtt{c}_{6}^{-1}\, \mathcal{C}^{H(d-\gamma)}(E)\leq \mathbb{P}\{B^H(E)\cap F\neq \emptyset \}\leq {} {}\mathtt{c}_{6}\,\mathcal{H}^{H(d-\gamma)}(E).\end{align}

Remark 3·7. Similarly to Remark 3·4-(ii), for less regular set F, with $0 \lt \gamma \lt \dim(F)\leq \overline{\dim}_M(F) \lt \gamma^{\prime} \lt d$ , we have

(3·10) \begin{align}\mathtt{c}_{4}^{-1}\, \mathcal{C}^{H(d-\gamma)}(E)\leq \mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}\leq \mathtt{c}_{4}\,\mathcal{H}^{H(d-\gamma^{\prime})}(E),\end{align}

where $\mathtt{c}_{4}$ is a positive constant which depends on E, F, H, $\mathrm{K}_f$ , $\gamma$ and $\gamma^{\prime}$ .

4. Sharpness of the Hölder regularity of the drift:

This subsection brings to light the essential role of the H-Hölder regularity assumed on the drift f in the following sense: the result of Theorem 3·1 fails to hold when the deterministic drift f has a modulus of continuity $\mathsf{w}({\cdot})$ satisfying

$$ r^H=o\left(\mathsf{w}(r)\right) \,\,\text{and}\,\, \mathsf{w}(r)=o\left(r^{H-\iota}\right) \,\,\text{when}\,\, r\rightarrow 0 \,\,\text{for all}\,\, \iota\in (0,H).$$

In this respect, we have to introduce some tools allowing us to reach our target.

Let $\mathscr{L}$ be the class of all continuous functions $\mathsf{w}\,:\,[0, 1] \rightarrow(0, \infty)$ , $\mathsf{w}(0)=0$ , which are increasing on some interval $(0, r_0]$ with $r_0 = r_0(\mathsf{w})\in (0,1)$ . Let $\mathsf{w}\in \mathscr{L}$ be fixed. A continuous function f is said to belong to the space $C^{\mathsf{w}}(\mathbf{I})$ if and only if

$$ \sup _{\substack{s, t \in \mathbf{I} \\ s \neq t}} \frac{|f(s)-f(t)|}{\mathsf{w}(|s-t|)} \lt \infty .$$

It is obvious that the space $C^{\mathsf{w}}(\mathbf{I})$ is a Banach space with the norm

$$ \|f\|_{\mathsf{w}}=\sup_{s \in \mathbf{I}}|f(s)|+\sup _{\substack{s, t \in \mathbf{I} \\ s \neq t}} \frac{|f(s)-f(t)|}{\mathsf{w}(|s-t|)}.$$

For $\alpha\in (0,1)$ and $\mathsf{w}(t)=t^\alpha$ , $\,C^{\mathsf{w}}(\mathbf{I})$ is nothing but the usual space $C^{\alpha}(\mathbf{I})$ .

Let $x_0\in(0,1]$ and $l\,:\,(0, x_0]\rightarrow \mathbb{R}_+$ be a slowly varying function at zero in the sens of Karamata (cf. [ Reference Bingham, Goldie and Teugels2 ]). It is well known that l has the representation

(4·1) \begin{align} l(x)=\exp\left(\eta(x) + \int_{x}^{x_0}\frac{\varepsilon(t)}{t}dt\right), \end{align}

where $\eta,\varepsilon \,:\, [0,x_0)\rightarrow \mathbb{R}$ are Borel measurable and bounded functions such that

$$ \lim_{x\rightarrow 0}\eta(x)=\eta_0\in (0,\infty) \quad \text{ and } \quad \lim_{x\rightarrow 0}\varepsilon(x)=0. $$

An interesting property of slowly varying functions which gives some intuitive meaning to the notion of “slow variation” is that for any $\tau \gt 0$ we have

(4·2) \begin{align} {} {}x^{\tau}\, l(x) \rightarrow 0 \quad \text{ and } x^{-\tau}\,l(x)\rightarrow \infty \quad \text{ as } x\rightarrow 0. \end{align}

It is known from Theorem 1·3·3 and Proposition 1·3·4 in [ Reference Bingham, Goldie and Teugels2 ] and the ensuing discussion that there exists a function $\mathcal{C}^{\infty}$ near zero $\tilde{l}\,:\,(0,x_0]\rightarrow {\mathbb{R}}_+$ such that $l(x)\thicksim \tilde{l}(x)$ when $x\rightarrow 0$ , and $\tilde{l}({\cdot})$ has the following form

(4·3) \begin{align} \tilde{l}(x)=\mathsf{c}\,\exp\left( \int_{x}^{x_0}\frac{\tilde{\varepsilon}(t)}{t}dt\right), \end{align}

for some positive constant $\mathsf{c}$ and $\tilde{\varepsilon}(x) \rightarrow 0$ . Such function is called normalized slowly varying function (Kohlbecker 1958), and in this case

(4·4) \begin{align} \tilde{\varepsilon}(x)=-x\, \tilde{\ell}^{\prime}(x)/\tilde{\ell}(x) \quad \text{for all } \,x\in (0,x_0). \end{align}

A function $\mathsf{v}_{\alpha,{\ell}}\,:\,[0, x_0]\rightarrow \mathbb{R}_+$ is called regularly varying function at zero with index $\alpha \in (0,1)$ if and only if there exists a slowly varying function ${\ell}$ , called the slowly varying part of $\mathsf{v}_{\alpha,\ell}$ , such that

(4·5) \begin{align} \mathsf{v}_{\alpha,{\ell}}(0)=0 \quad \text{ and } \quad \mathsf{v}_{\alpha,{\ell}}(x)=x^{\alpha}\,{\ell}(x),\quad x\in (0,x_0). \end{align}

$\mathsf{v}_{\alpha,\ell}$ is called a normalised regularly varying if its slowly varying part is normalised slowly varying at zero. In the rest of this work, since the value of $x_0$ is unimportant because $\tilde{\ell}(x)$ and $\tilde{\varepsilon}(x)$ may be altered at will for $x\in (x_0,1]$ , one can choose $x_0=1$ without loss of generality. Furthermore, we will only consider normalised regularly/slowly varying function.

Here are some interesting properties of normalized regularly varying functions. Let $\mathsf{v}_{\alpha,\ell}$ be a normalised regularly varying at zero with normalized slowly varying part l.

$$ \mathsf{v}_{\alpha,\ell}^{\prime\prime}(x)= x^{\alpha-2}\, \ell(x)\left[ \left(\alpha-1-\varepsilon(x)\right)\left(\alpha-\varepsilon(x)\right)-x\,\varepsilon^{\prime}(x)\right].$$

(4·2) and hypothesis (4·6) ensure that $\lim_{x\downarrow 0}\mathsf{v}_{\alpha,\ell}^{\prime\prime}(x)=-\infty$ . Then there exists $x_2 \gt 0$ small enough such that $\mathsf{v}_{\alpha,\ell}^{\prime}(x) \gt 0$ and $\mathsf{v}_{\alpha,\ell}^{\prime\prime}(x) \lt 0$ for all $x\in (0,x_2]$ . Thus $\mathsf{v}_{\alpha,\ell}$ is increasingly concave on $(0,x_2]$ .

For the rest, let $x_3 \in (0,x_2]$ and $\mathsf{c} \gt 0$ be arbitrary. Let $s \lt t\in (x_3,x_2)$ and $r \lt x_3$ , then using the monotonicity of $\mathsf{v}_{\alpha,\ell}^{\prime}$ we have for $0 \lt t-s \lt r$ that

(4·7) \begin{align}\frac{\mathsf{v}_{\alpha,\ell}(t)-\mathsf{v}_{\alpha,\ell}(s)}{\mathsf{v}_{\alpha,\ell}(t-s)} \leq \frac{\mathsf{v}_{\alpha,\ell}^{\prime}(x_3)}{\mathsf{v}_{\alpha,\ell}^{\prime}(r)}. \end{align}

Since $\lim_{x\downarrow 0}\mathsf{v}_{\alpha,\ell}^{\prime}(x)=+\infty$ , we can choose $r_0$ to be smallest r guaranteeing that the term $\mathsf{v}_{\alpha,\ell}^{\prime}(x_3)/\mathsf{v}_{\alpha,\ell}^{\prime}(r)$ will be smaller than $\textbf{c}$ . This completes the proof.

Let $\ell$ be a normalised slowly varying function at zero. Now we consider the continuous function $\mathsf{w}_{H,l}$ given by

(4·8) \begin{align} \mathsf{w}_{H,\ell}(0)=0 \quad \text{ and } \quad \mathsf{w}_{H,\ell}(x)= x^H\,\ell(x)\, \log^{1/2}(1/x), \quad x\in (0,1].\end{align}

It is easy to see that $\ell(x)\, \log^{1/2}(1/x)$ stills a normalised slowly varying satisfying (4·3) with $\bar{\varepsilon}({\cdot})=\varepsilon({\cdot})-\log^{-1}(1/\cdot)/2$ . Hence assertion 1 of Lemma 4·1 provides that $\mathsf{w}_{H,\ell}$ is increasing on some interval $(0,x_1]$ with $x_1\in (0,1)$ . Therefore $\mathsf{w}_{H,\ell}\in \mathscr{L}$ .

If we assume in addition that $\ell$ satisfies

(4·9) \begin{align} {}\underset{x\rightarrow 0}{\limsup}\,\,\ell(x)\log^{1/2}(1/x)=+\infty,\end{align}

then the following inclusions hold

(4·10) \begin{align}C^{H}(\mathbf{I})\subsetneq C^{\mathsf{w}_{H,\ell}}(\mathbf{I})\subsetneq \underset{\tau \gt 0}{\bigcap} C^{H-\tau}(\mathbf{I}).\end{align}

Let $\theta_H$ be the normalised regularly varying function defined in (7·1) with a normalized slowly varying part that satisfies;

(4·11) \begin{align} {} {}\left\{ \begin{array}{l} {} {} {}\text{(i)}\quad \underset{x\rightarrow+\infty}{\liminf}\,\,L_{\theta_{H}}(x)=0,\quad \underset{x\rightarrow+\infty}{\limsup}\,\,L_{\theta_{H}}(x) \lt +\infty;\\ {} {} {}\\ \text{(ii)}\,\,\, \underset{x\rightarrow +\infty}{\limsup} \,\,x\, \varepsilon^{\prime}_{\theta}(x)=0. {} {}\end{array}\right. {}\end{align}

Here are some examples of normalised slowly varying functions for which the above conditions are satisfied

\[ L_{\theta_{H}}(x)=\log^{-{\beta}}\left(x\right), \quad L_{\theta_{H}}(x)=\exp\left(-\log^{\alpha}\left(x\right)\right), \quad \alpha \in (0,1) {\text{ and } \beta \gt 0}.\]

In what follows, we will adopt the following notation

(4·12) \begin{align}{ \ell_{\theta_H}({\cdot})\,:\!=\,\mathsf{c}_H^{1/2}\,L^{-1/2}_{\theta_{H}}(1/\cdot)}.\end{align}

Now let us give the main result of this section.

Theorem 4·3. Let $\{B^H(t), t\in [0,1]\}$ be a d-dimensional fractional Brownian motion. Then there exist a function $f\in C^{\mathsf{w}_{H,{\ell_{\theta_H}}}}(\mathbf{I})\setminus C^{H}\left(\mathbf{I}\right)$ , and compact sets $E\subset \mathbf{I}$ and $F\subset \mathbb{R}^d$ such that

(4·13) \begin{align} {} \mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}\left(E\times F\right)=\mathcal{H}_{\rho_{\mathbf{d}_{H}}}^d\left(E\times F\right)=0 \quad \text{ and } \quad \mathbb{P}\left\{(B^H+f)(E)\cap F\neq \varnothing \right\} \gt 0. {} \end{align}

In other words (3·4) fails to hold.

Before drawing up the proof we provide the tools to be used. Let $\delta_{\theta_H}$ be the function given by the representation (7·3). Theorem 7·3·1 in [ Reference Marcus and Rosen20 ] tells us that $\delta_{\theta_H}$ is normalised regularly varying with index H with a slowly varying part $\ell_{\delta_{\theta_H}}$ that satisfies

(4·14) \begin{align} {} {}\ell_{\theta_H}(h)\thicksim \ell_{\delta_{\theta_H}}(h)\quad \text{as $h\rightarrow 0$}.\end{align}

For more details see (7·6). Now we consider another probability space $(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{P}^{\prime})$ on which we define the real valued centered Gaussian process with stationary increments $B_0^{\delta_{\theta_H}}$ , satisfying $B_0^{\delta_{\theta_H}}(0)=0$ a.s. and

(4·15) \begin{align} {}\mathbb{E}^{\prime}\left(B_0^{\delta_{\theta_H}}(t)-B_0^{\delta_{\theta_H}}(s)\right)^2=\,\delta_{\theta_H}^2(|t-s|)\quad \text{ for all $t,s\in [0,1]$}.\end{align}

Proposition 7·1 gives a way to construct this process and Theorem 7·2 provides us its modulus of continuity, that is $B_0^{\delta_{\theta_H}}\in C^{\mathsf{w}_{H,{\ell_{\theta_H}}}}(\mathbf{I})$ $\mathbb{P}^{\prime}$ -almost surely. The d-dimensional version of the process $B^{\delta_{\theta_H}}_0$ is the process $B^{\delta_{\theta_H}}(t)\,:\!=\,(B^{\delta_{\theta_H}}_1(t),....,B^{\delta_{\theta_H}}_d(t))$ , where $B^{\delta_{\theta_H}}_1,...,B^{\delta_{\theta_H}}_d$ are d independent copies of $B^{\delta_{\theta_H}}_0$ . Let Z be the d-dimensional mixed process defined on the product space $(\Omega\times\Omega^{\prime},\mathcal{F}\times\mathcal{F}^{\prime},\mathbb{P}\otimes\mathbb{P}^{\prime})$ by

(4·16) \begin{align} {}Z(t,(\omega,\omega^{\prime}))=B^H(t,\omega)+B^{\delta_{\theta_H}}(t,\omega^{\prime}) \text{ for all } t\in [0,1] \text{ and } (\omega,\omega^{\prime})\in \Omega\times\Omega^{\prime}. \end{align}

It is easy to see that the components of $Z=(Z_1,...,Z_d)$ are independent copies of a real valued Gaussian process $Z_0= B_0^H+ B^{\delta_{\theta_H}}_0$ on $(\Omega\times\Omega^{\prime},\mathcal{F}\otimes\mathcal{F}^{\prime},\mathbb{P}\otimes\mathbb{P}^{\prime})$ . Furthermore we have

$$ \widetilde{\mathbb{E}}\left(Z_0(t)-Z_0(s)\right)^2=\mathsf{v}_{2H,1+\ell_{{\delta_{\theta_H}}}^2}(|t-s|)\,:\!=\,|t-s|^{2H}\left(1+\ell_{{\delta_{\theta_H}}}^2(|t-s|)\right),$$

where $\widetilde{\mathbb{E}}$ denotes the expectation under the probability measure $\widetilde{\mathbb{P}}\,:\!=\,\mathbb{P}\otimes \mathbb{P}^{\prime}$ .

Using the assertion i. of (4·11) and (4·14) we obtain the following

Lemma 4·5. There exists a constant $q \gt 1$ such that

(4·17) \begin{align} {} q^{-1}\,\mathsf{v}_{2H,\, {\ell^2_{\theta_H}}}(h)\leq \widetilde{\mathbb{E}}\left(Z_0(t+h)-Z_0(t)\right)^2 \leq q\,\,\mathsf{v}_{2H,\, {\ell^2_{\theta_H}}}(h),\end{align}

for all $h\in [0,1]$ and $t\in [0,1]$ .

For simplicity, we denote by $\mathbf{d}_{H,\ell_{\delta_{\theta_H}}}$ and $\mathbf{d}_{H,(1+\ell_{\delta_{\theta_H}}^2)^{1/2}}$ the canonical metrics of $B^{\delta_{\theta_H}}$ and Z respectively. A consequence of the previous lemma these canonical metrics are strongly equivalents to the metric $(s,t)\mapsto \mathbf{d}_{H,{\ell_{\theta_H}}}(t,s)\,:\!=\,\mathsf{v}_{H,\,{\ell_{\theta_H}}}(|t-s|)$ , leading to the strong equivalence of the metrics $\rho_{\mathbf{d}_{H,\ell_{\delta_{\theta_H}}}}$ , $\rho_{\mathbf{d}_{H,(1+\ell_{\delta_{\theta_H}}^2)^{1/2}}}$ and $\rho_{\mathbf{d}_{H,{\ell_{\theta_H}}}}$ . Hence, their associated capacities are also equivalents.

Proof of Theorem 4·3. Let us consider the Gaussian process Z stated above. Using condition (4·11) we infer that ${\ell_{\theta_H}({\cdot})}$ satisfies (4·6). Then Lemma 4·1 ensures that $\mathsf{v}_{H,{\ell_{\theta_H}({\cdot})}}$ verifies [13, hypothesis 2·2]. Let $M \gt 0$ , applying [13, theorem 4·1] and the fact that $\rho_{\mathbf{d}_{H,(1+\ell_{\delta_{\theta_H}}^2)^{1/2}}}$ and $\rho_{\mathbf{d}_{H,{\ell_{\theta_H}}}}$ are strongly equivalent, there exist a positive constant $\mathsf{c}_1$ depending only on $\mathbf{I}$ and M such that

(4·18) \begin{align} \widetilde{\mathbb{P}}\left\{Z(E)\cap F\neq \emptyset\right\}\geq \,\mathsf{c}_1\,\mathcal{C}_{\rho_{\mathbf{d}_{H,{\ell_{\theta_H}}}}}^{d}(E\times F),\end{align}

for any compact sets $E\subset \mathbf{I}$ and $F\subset [-M,M]^d$ . Let $ 0 \lt \gamma \lt d$ and fix a $\gamma$ -Ahlfors–David regular set $F_{\gamma}\subset [-M,M]^d$ . Then by using (2·21) with $\rho_1=\mathbf{d}_{H,{\ell_{\theta_H}}}$ and $\rho_3=\rho_{\mathbf{d}_{H,{\ell_{\theta_H}}}}$ and (2·22) with $\rho_1=\mathbf{d}_{H}$ and $\rho_3=\rho_{\mathbf{d}_{H}}$ , we obtain

(4·19) \begin{align}\mathcal{C}_{\rho_{\mathbf{d}_{H,{\ell_{\theta_H}}}}}^{d}(E\times F_{\gamma})\geq\, \mathsf{c}_2^{-1}\, \mathcal{C}_{\mathbf{d}_{H,{\ell_{\theta_H}}}}^{ d-\gamma}(E) \quad \text{ and }\quad \mathcal{H}_{\rho_{\mathbf{d}_{H}}}^d(E\times F_{\gamma})\leq\, \mathsf{c}_{2}\, \mathcal{H}^{H(d-\gamma)}(E),\end{align}

for all compact $\,E\subset \mathbf{I}$ and for some constant $\mathsf{c}_2 \gt 0$ . Now it is not difficult to see that the functions $h(t)\,:\!=\,t^{H(d-\gamma)}$ and

$$ \Phi(t)= 1/\mathsf{v}^{d-\gamma}_{H,{\ell_{\theta_H}}}(t)=1/\mathsf{v}_{H\left(d-\gamma\right),{\ell_{\theta_H}^{\left(d-\gamma\right)}}}(t),$$

satisfy the hypotheses of [ Reference Taylor27 , theorem 4] which allow us to conclude that there exists a compact set $E_{\gamma}\subset \mathbf{I}$ such that

(4·20) \begin{align}\mathcal{H}^{H(d-\gamma)}(E_{\gamma})=0\quad \quad \text{ and } \quad \quad \mathcal{C}_{\mathbf{d}_{H,{\ell_{\theta_H}}}}^{ d-\gamma}(E_{\gamma}) \gt 0.\end{align}

Consequently, combining (4·19) and (4·20), we have

(4·21) \begin{align}\mathcal{H}_{\rho_{\mathbf{d}_{H}}}^d(E_{\gamma}\times F_{\gamma})=0 \quad \text{ and }\quad \widetilde{\mathbb{P}}\left\{Z(E_{\gamma})\cap F_{\gamma}\neq \emptyset\right\}\geq\, (\mathsf{c}_1/\mathsf{c}_2)\, \mathcal{C}_{\mathbf{d}_{H,{\ell_{\theta_H}}}}^{d-\gamma}(E_{\gamma}) \gt 0.\end{align}

Now the remainder of the proof is devoted to the construction of a drift f satisfying (4·13). As a consequence of Fubini’s theorem, we have

\begin{equation*}\mathbb{E}^{\prime}\left(\mathbb{P}\left\{(B^H+B^{\delta_{\theta_H}}(\omega^{\prime}))(E_{\gamma})\cap F_{\gamma}\neq \emptyset \right\}-\mathsf{c}_3\, \mathcal{C}_{\mathbf{d}_{H,{\ell_{\theta_H}}}}^{d-\gamma}(E_{\gamma})\right) \gt 0,\end{equation*}

for some fixed positive constant $\mathsf{c}_3\in (0,\mathsf{c}_1/\mathsf{c}_2)$ , leading to

(4·22) \begin{align}\mathbb{P}^{\prime}\left\{\omega^{\prime}\in \Omega^{\prime}\,:\,\mathbb{P}\left\{(B^H+B^{\delta_{\theta_H}}(\omega^{\prime}))(E_{\gamma})\cap F_{\gamma}\neq \emptyset \right\} \gt \mathsf{c}_3\,\mathcal{C}_{\mathbf{d}_{H,{\ell_{\theta_H}}}}^{ d-\gamma}(E_{\gamma})\right\} \gt 0.\end{align}

We therefore choose the function f among of them. Hence we get the desired result.

5. Hitting points

As mentioned previously in the introduction our goal in this section is to shed some light on the hitting probabilities for general measurable drift. The resulting estimates are given in the following.

Theorem 5·1. 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 $E\subset \mathbf{I}$ be a Borel set. Then for any $M \gt 0$ there exists a constant $ \textbf{c}_1\geq 1$ such that for all $x\in [-M,M]^d$ we have

(5·1) \begin{align} \textbf{c}_1^{-1}\mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}(Gr_E(f))\leq \mathbb{P}\left\lbrace \exists\, t\in E : (B^H+f)(t)=x\right\rbrace\leq \textbf{c}_1\, \mathcal{H}_{\rho_{\mathbf{d}_{H}}}^{d}(Gr_E(f)). {} {}\end{align}

The following lemmas are very valuable to prove Theorem 5·1

Lemma 5·2 [ Reference Biermé, Lacaux and Xiao1 , lemma 3·1]. Let $\{B^H(t)\,:\, t\in [0,1]\}$ be a fractional Brownian motion with Hurst index $H\in (0,1)$ . For any constant $M \gt 0$ , there exists positive constants $ \textbf{c}_2$ and $\varepsilon_0 \gt 0$ such that for all $r\in (0,\varepsilon_0)$ , $t\in \mathbf{I}$ and all $x\in [-M,M]^d$ ,

(5·2) \begin{align} {}\mathbb{P}\left(\inf _{ s\in I,|s-t|^H\leq r}\|B^H(s)-x\| \leqslant r\right) \leqslant \textbf{c}_2\, r^{d}. {} {}\end{align}

Lemma 5·3 [ Reference Biermé, Lacaux and Xiao1 , lemma 3·2]. Let $B^H$ be a fractional Brownian motion with Hurst index $H\in (0,1)$ . Then there exists a positive constant $ \textbf{c}_3$ such that for all $\epsilon \in (0,1)$ , $s,t\in \mathbf{I}$ and $x,y\in \mathbb{R}^d$ we have

(5·3) \begin{align}\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 \nonumber\\& \leq \dfrac{ \textbf{c}_{3}}{\left(\rho_{\mathbf{d}_{H}}((s,x),(t,y))\right)^{d}},\end{align}

where $\Gamma_{\varepsilon}(s,t)\,:\!=\,\varepsilon\,I_2+\text{Cov}(B^H_0(s),B^H_0(t))$ , $I_{2d}$ 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 5·1. We start with the upper bound. Choose an arbitrary constant $\gamma \gt \mathcal{H}_{\rho_{\mathbf{d}_{H}}}^d(Gr_E(f))$ , then there is a covering of $Gr_E(f)$ by balls $\{B_{\rho_{\mathbf{d}_{H}}}((t_i,y_i),r_i), i\geq 1\}$ in $\mathbb{R}_+\times\mathbb{R}^d$ such that

(5·4) \begin{align} {} {}Gr_E(f)\subseteq \bigcup_{i=1}^{\infty}B_{\rho_{\mathbf{d}_{H}}}((t_i,y_i),r_i)\quad \text{ and }\quad \sum_{i=1}^{\infty}(2r_i)^{d} \leq \gamma. {} {}\end{align}

Let $\delta_0$ and M being the constants given in Lemma 5·2. We assume without loss of generality that $r_i \lt \delta_0$ for all $i \geq 1$ . Let $x\in [-M,M]^d$ , it is obvious that

(5·5) \begin{align} {} {}& \left\{ \exists s\in E \,:\, (B^{H}+f)(s)=x \right\} \subseteq \\& \qquad \qquad \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\}.\nonumber {} {}\end{align}

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

(5·6) \begin{align}& \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\} \nonumber\\ {}& \quad \qquad \quad \qquad \quad \qquad \quad \qquad \quad \quad \qquad \subseteq \left\{ \inf_{|s-t_i|^{H} \lt r_i }\|B^H(s)-x+y_i\|\leq r_i \right\}, {} {}\end{align}

then we get from [ Reference Biermé, Lacaux and Xiao1 , lemma 3·1] that

(5·7) \begin{align} {} {}& \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\} \nonumber\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \leq\mathbb{P}\left\{ \inf_{|s-t_{i}|^{H} \lt r_{i}}\|B^{H}(s)-x+y_{i}\|\leq r_i\right\} \nonumber\\ {} {}& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \leq \textbf{c}_{2}\,r_{i}^{d},\end{align}

where $ \textbf{c}_4$ depends on H, $\mathbf{I}$ , M and f only. Combining (5·4)-(5·7) we derive that

$$ \mathbb{P}\left\lbrace \exists s\in E \,:\, (B^H+f)(s)=x\right\rbrace\leq 2^{-d} \textbf{c}_{2}\, \gamma.$$

Let $\gamma\downarrow \mathcal{H}_{{\rho_{\mathbf{d}_{H}}}}^{d}(Gr_E(f))$ , the upper bound in (5·1) follows.

The lower bound in (5·1) holds from the second moment argument. We assume that $\mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}(Gr_E(f)) \gt 0$ , then let $\sigma$ be a measure supported on $Gr_E(f)$ such that

(5·8) \begin{align} {} {}\mathcal{E}_{\rho_{\mathbf{d}_{H}},d}(\sigma)=\int_{Gr_E(f)}\int_{Gr_E(f)}\frac{d\sigma(s,f(s))d\sigma(t,f(t))}{\rho_{\mathbf{d}_{H}}((s,f(s)),(t,f(t)))^{d}}\leq \frac{2}{\mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}(Gr_E(f))}. {} {}\end{align}

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

(5·9) \begin{align} {} {}\int_{E} g(s) \nu_{n}(d s) &=\int_{E} (2 \pi n)^{d / 2} \exp \left(-\frac{n\|B^H(s)+f(s)-x\|^2}{2}\right) g(s) \nu(d s) \nonumber {} {}\\ {} {}&=\int_{E} \int_{\mathbb{R}^{d}} \exp \left(-\frac{\|\xi\|^{2}}{2 n}+i\left\langle\xi, B^H(s)+f(s)-x\right\rangle\right) g(s) d \xi \nu(d s), {} {}\end{align}

where g is an arbitrary measurable function on $\mathbb{R}_+$ . Our aim is to show that $\left\lbrace \nu_n, n\geq 1\right\rbrace $ has a subsequence which converges weakly to a finite measure $\nu_{\infty}$ supported on the set $\{s\in E \,:\, B^H(s)+f(s)=x\}$ . To carry out this goal, we will start by establishing the following inequalities

(5·10) \begin{align} {} {}\mathbb{E}(\|\nu_n\|)\geqslant \textbf{c}_{5},\quad \quad \mathbb{E}(\|\nu_n\|^2)\leqslant \textbf{c}_{3} \mathcal{E}_{\rho_{\mathbf{d}_{H}},d}(\sigma), {} {}\end{align}

which constitute together with the Paley–Zygmund inequality the cornerstone of the proof. Here $\|\nu_n\|$ denotes the total mass of $\nu_n$ . By (5·9), Fubini’s theorem and the use of the characteristic function of a Gaussian vector we have

(5·11) \begin{align} {} {}\mathbb{E}(\|\nu_n\|)& =\int_{E}\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\,\nu(ds)\nonumber\\ {} {}& =\int_{E}\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\,\nu(ds)\nonumber\\ {} {}& =\int_{E}\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)\,\nu(ds)\nonumber\\ {} {}& \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)\nonumber\\ {} {}&\geq \textbf{c}_{5} \gt 0. {} {}\end{align}

Since f is bounded, $x\in [{-}M,M]^d$ and $\nu$ is a probability measure we conclude that $ \textbf{c}_{5}$ is independent of $\nu$ and n. This gives the first inequality in (5·10).

We will now turn our attention to the second inequality in (5·10). By (5·9) and Fubini’s theorem again we obtain

(5·12) \begin{align} {} {}\mathbb{E}\left(\|\nu_{n}\|^{2}\right) & =\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 )} \nonumber\\ {} {}& \quad \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 \nonumber\\ {} {}& \quad \leqslant \textbf{c}_{3}\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}_{3}\,\mathcal{E}_{\rho_{\mathbf{d}_{H}},d}(\sigma) \lt \infty, {} {}\end{align}

where the first inequality is a direct consequence of Lemma 5·3. Plugging the moment estimates of (5·10) into the Paley–Zygmund inequality (c.f. Kahane [ Reference Kahane15 , p.8]), allows us to confirm that there exists an event $\Omega_0$ of positive probability such that, for all $\omega\in \Omega_0$ , $(\nu_n(\omega))_{n\geq 1}$ admits a subsequence converging weakly to a finite positive measure $\nu_{\infty}(\omega)$ supported on the set $\{s\in E \,:\, B^H(\omega,s)+f(s)=x\}$ , satisfying the moment estimates in (5·10). Hence we have

(5·13) \begin{align}\mathbb{P}\left\{\exists s\in E: (B^H+f)(s)=x\right\}\geq \mathbb{P}\left(\|\nu_{\infty}\| \gt 0\right)\geq \frac{\mathbb{E}(\|\nu_{\infty}\|)^2}{\mathbb{E}(\|\nu_{\infty}\|^2)}\geq \frac{ \textbf{c}_5^2}{ \textbf{c}_3\mathcal{E}_{\rho_{\mathbf{d}_{H}},d}(\sigma)}. {} {}\end{align}

Combining this with (5·8) yields the lower bound in (5·1). The proof is completed.

Remark 5·4. We mention that the covering argument used to prove the upper bound in (5·1) can also serve to show that for any Borel set $F\subset\mathbb{R}^d$ , there exists a positive constant $ \textbf{c}$ such that

(5·14) \begin{align}\mathbb{P}\left\lbrace (B^H+f)(E)\cap F\neq \emptyset \right\rbrace\leq \textbf{c}\, \mathcal{H}_{\widetilde{\rho}_H}^d(Gr_E(f)\times F). {} {}\end{align}

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

\begin{align*}\widetilde{\rho}_{\mathbf{d}_{H}}((s,x,u),(t,y,v))\,:\!=\,\max\{|t-s|^H,\|x-y\|,\|u-v\|\}.\end{align*}

But it seems hard to establish a lower bound in terms of $\mathcal{C}_{\widetilde{\rho}_H}^{d}(Gr_E(f)\times F)$ even for Ahlfors–David regular set F.

As a consequence of Theorem 5·1, we obtain a weaker version of [Reference Erraoui and Hakiki9, theorem 3·2]

Proof. Integrating (5·1) of Theorem 5·1 over all cube $[-M,M]^d$ , $M \gt 0$ with respect Lebesgue measure $\lambda_d$ , we obtain that

(5·15) \begin{align} (2M)^d\, \textbf{c}_1^{-1}\mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}(Gr_E(f))& \leq \mathbb{E}\left[\lambda_d\left([-M,M]^d\cap (B^H+f)(E)\right)\right] \nonumber \\ & \leq (2M)^d\, \textbf{c}_1\, \mathcal{H}_{\rho_{\mathbf{d}_{H}}}^{d}(Gr_E(f)). \end{align}

Therefore if $\mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}(Gr_E(f)) \gt 0$ we obtain

$$ \mathbb{E}\left[\lambda_d\left((B^H+f)(E)\right)\right] \gt 0.$$

Hence $\lambda_d\left((B^H+f)(E)\right) \gt 0$ with positive probability, which finishes the proof of (i). On the other hand, if $\mathcal{H}_{\rho_{\mathbf{d}_{H}}}^{d}(Gr_E(f))=0$ we obtain that $\lambda_d\left([-n,n]^d \cap (B^H+f)(E)\right)=0$ a.s. for all $n\in \mathbb{N}^{*}$ . Then we have $\lambda_d\left((B^H+f)(E)\right)=0$ a.s. Hence the proof of (ii) is completed.

6. Application to polarity

Let $E\subset \mathbf{I}$ . We say that a point $x\in \mathbb{R}^d$ is polar for $(B^H+f)\vert_{E}$ , the restriction of $(B^H+f)$ to E, if

(6·1) \begin{align} {}\mathbb{P}\left\{ \exists \, t\in E\,:\, (B^H+f)(t)=x\right\}=0.\end{align}

Otherwise, x is said to be non-polar for $(B^H+f)\vert_{E}$ . In other words, $(B^H+f)\vert_{E}$ hits the point x.

It is noteworthy that, when $f\in C^{H}(\mathbf{I})$ , the hitting probabilities estimates in (5·1) becomes

(6·2) \begin{align} {} \textbf{c}_1^{-1}\mathcal{C}^{Hd}(E)\leq \mathbb{P}\left\lbrace \exists t\in E : (B^H+f)(t)=x\right\rbrace\leq \textbf{c}_1\, \mathcal{H}^{Hd}(E).\end{align}

See also [ Reference Chen5 , corollary 2·8]. Consequently, all points are non-polar (resp. polar) for $(B^H+f)\vert_{E}$ when $\dim(E) \gt Hd$ (resp. $\dim(E) \lt Hd$ ). However, the critical dimensional case, which is the most important and not easy to deal with, is $\dim(E)=Hd$ . This is undecidable in general as illustrated in the following

Proposition 6·1. Let $\{B^{H}(t) : t \in [0,1]\}$ be a d-dimensional fractional Brownian motion of Hurst index $H\in (0,1)$ such that $H d \lt 1$ . Let $f\,:\,[0,1]\rightarrow \mathbb{R}^d$ be a H-Hölder continuous function and $E\subset \mathbf{I}$ be a Borel set. Then there exist two Borel subsets $E_1$ and $E_2$ of $\mathbf{I}$ such that $\dim(E_1)=\dim(E_2)=H\,d$ and for all $x\in \mathbb{R}^d$ we have

$$ {}\mathbb{P}\left\{\exists s\in E_1\,:\, (B^H+f)(s)=x\right\}=0 \quad \text{and} \quad \mathbb{P}\left\{\exists s\in E_2\,:\, (B^H+f)(s)=x\right\} \gt 0. {}$$

The following Lemma is helpful in the proof of Proposition 6·1.

Lemma 6·2 Let $\alpha\in (0,1)$ and $\beta \gt 1$ . Let $E_1$ and $E_2$ are two Borel subsets of $\,\mathbf{I}$ supporting two probability measures $\nu_1$ and $\nu_2$ respectively, that satisfy

(6·3) \begin{align} {}\mathsf{c}_1^{-1} r^{\alpha}\log^{\beta}(e/r)\leq \nu_1\left([a-r,a+r]\right)\leq \mathsf{c}_1 r^{\alpha}\log^{\beta}(e/r)\quad \text{ for all $r\in (0,1)$, $a\in E_1$},\end{align}

and

(6·4) \begin{align} {}\mathsf{c}_2^{-1} r^{\alpha}\log^{-\beta}(e/r)\leq \nu_2\left([a-r,a+r]\right)\leq \mathsf{c}_2 r^{\alpha}\log^{-\beta}(e/r)\quad \text{ for all $r\in (0,1)$, $a\in E_2$},\end{align}

for some positive constants $\mathsf{c}_1$ and $\mathsf{c}_2$ . Then we have

$$ \dim(E_1)=\dim(E_2)=\alpha$$

and

(6·5) \begin{align} {}\mathcal{H}^{\alpha}(E_1)=0 \quad\text{and}\quad \mathcal{C}^{\alpha}(E_2) \gt 0.\end{align}

See Appendix B for examples of such measures $\nu_1$ and $\nu_2$ .

Proof. First, let us start by proving that $\dim(E_1)=\dim(E_2)=\alpha$ . Indeed, for all $t \in [0,1]$ and $n\in \mathbb{N}$ we denote by $I_n(t)$ the nth generation, half open dyadic interval of the form $[\dfrac{j-1}{2^n},\dfrac{j}{2^n})$ containing t. Then, by using (6·3) and (6·4), it is easy to check that

(6·6) \begin{align} {} {}\lim_{n\rightarrow \infty}\frac{\log \nu_1(I_n(t))}{\log(2^{-n})}=\lim_{n\rightarrow \infty}\frac{\log \nu_2(I_n(t))}{\log(2^{-n})}=\alpha. {}\end{align}

Therefore, by Billingsley Lemma [ Reference Bishop and Peres3 , lemma 1·4·1, p. 16] we have $\dim(E_1)=\dim(E_2)=\alpha$ .

Now we are going to look at (6·5). Let $r\in (0,1]$ and $P_{|.|}(E_1,r)$ be the packing number of $E_1$ . The lower bound in (6·3) leads via

$$ {}\mathsf{c}_1^{-1}\,r^{\alpha}\, \log^{\beta}(e/r)\, P_{|.|}(E_1,r)\leq \sum_{j=1}^{P_{|.|}(E_1,r)}\nu_1(I_j)=\nu_1(E_1)=1, {}$$

to

\[ {} {}P_{|.|}(E_{1},r)\leq\mathsf{c}_{1}^{\,}r^{-\alpha}\,\log^{-\beta}(e/r). {} {}\]

So using the well-known fact that $N_{|.|}(E_1,2r)\leq P_{|.|}(E_1,r)$ , we may deduce that

$$ N_{|.|}(E_1,r)\leq \mathsf{c}_1\, 2^{\alpha}\,r^{-\alpha}\, \log^{-\beta}(2e/r),\,\, \forall r\in (0,1).$$

Therefore, it is not hard to make out that

$$ {}\mathcal{H}^{\alpha}(E_1)\leq \, \limsup_{r\rightarrow 0}(2r)^{\alpha}\, N_{|.|}(E_1,r)=0, {}$$

which gives the first outcome of (6·5). On the other hand, to show that $\mathcal{C}^{\alpha}(E_2) \gt 0$ it is sufficient to prove that

$$ \sup_{t\in E_2}{\displaystyle\int}_{E_2}\frac{\nu_2(ds)}{|t-s|^{\alpha}} \lt \infty.$$

Indeed, we first assume without loss of generality that $\kappa=\mathrm{diam}(E_2) \lt 1$ . Now since $\nu_2$ has no atom, then for any $t \in E_2$ we have

(6·7) \begin{align} {} {}& {\displaystyle\int}_{E_2}\frac{\nu_2(ds)}{|t-s|^{\alpha}}=\sum_{j=0}^{\infty} {\displaystyle\int}_{\{s:|t-s|\in (\kappa 2^{-(j+1)},\kappa 2^{-j}]\}}\frac{\nu_2(ds)}{|t-s|^{\alpha}} \nonumber \\ & \quad \leq \sum_{j=0}^{\infty} \kappa^{-\alpha} 2^{\alpha\,(j+1)}\nu_2\left([t-\kappa\,2^{-j},t+\kappa\,2^{-j}]\right)\nonumber\\ {} {}& \quad \leq 2^{\alpha}\mathsf{c}_2\,\sum_{j=0}^{\infty}\frac{1}{\log^{\beta}(e\,2^j/\kappa)\,}, {}\end{align}

which is finite independently of t. Hence $\mathcal{E}_{\alpha}(\nu_2) \lt \infty$ and therefore $\mathcal{C}^{\alpha}(E_2) \gt 0$ .

Against this background, it is worthy to note that, according to (6·2) and Proposition 6·1, the H-Hölder regularity of the drift f is insufficient to guarantee the non-polarity of points for $(B^H+f)\lvert_E$ for a Borel set $E\subset [0,1]$ such that $\dim(E)= Hd$ which implicitly involves the need for a bite of drift roughness. Namely, the drift f will be chosen, as in the previous section, to be ( $H-\varepsilon$ )-Holder continuous for all $\varepsilon \gt 0$ without reaching order H. On the other hand in accordance to Theorem 5·1, for a general measurable drift f, a sufficient condition for $(B^H+f)\vert_{E}$ to hits all points is $\mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}(Gr_E(f)) \gt 0$ . The overall point of what follows is to provide some examples of drifts satisfying this last condition with a Borel set E whose distinctive feature is $\dim(E)=Hd$ .

Given a slowly varying function at zero $\ell:(0,1]\,\rightarrow {\mathbb{R}}_+$ we associate to it the kernel $\Phi_{H,\ell}({\cdot})$ defined as follows

(6·8) \begin{align} {}\Phi_{H,\ell}(r)\,:\!=\,r^{-Hd}\,\ell^{-d}(r)\,\left(1+\log\left(1\vee\ell(r)\right)\right).\end{align}

Let $\pmb{\theta_H}$ be the normalised regularly varying function defined in (7·1) with a normalised slowly varying part $L_{\pmb{\theta_H}}$ . As previously, we consider the normalised regularly varying function at zero $\delta_{\pmb{\theta_H}}$ , with index H and normalised slowly varying part $\ell_{\delta_{\pmb{\theta_H}}}$ , satisfying (7·4) and, on the space $(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{P}^{\prime})$ , the d-dimensional Gaussian process with stationary increments $B^{\delta_{\pmb{\theta_H}}}(t)\,:\!=\,\left( B^{\delta_{\pmb{\theta_H}}}_1(t),....,B^{\delta_{\pmb{\theta_H}}}_d(t)\right), t\in [0,1]$ , where $B^{\delta_{\pmb{\theta_H}}}_1,...,B^{\delta_{\pmb{\theta_H}}}_d$ are d independent copies of $B^{\delta_{\pmb{\theta_H}}}_0$ defined in (7·8). The following lemma will be useful afterwards.

Lemma 6·3. There exists a positive constant ${\mathsf{c}}_3$ such that

(6·9) \begin{align} {} {}\mathbb{E}^{\prime}\left[\left(\max\left\lbrace t^H,\lVert B^{\delta_{\pmb{\theta_H}}}(t)\rVert \right\rbrace \right)^{-d}\right]\leq \mathsf{c}_3\, \Phi_{H,\ell_{\delta_{\pmb{\theta_H}}}}(t), \quad \forall t\in (0,t_0), {}\end{align}

for some $t_0\in (0,1)$ .

Proof. First we note that, since $B^{\delta_{\pmb{\theta_H}}}(t)$ is a d-dimensional Gaussian vector, the term on the left-hand side of (6·9) has the same distribution as $t^{-Hd}\left(\max\left\lbrace 1,\ell_{\delta_{\pmb{\theta_H}}}(t)\lVert N\rVert\right\rbrace \right)^{-d}$ , where N is a d-dimensional standard normal random vector. Due to simple calculations we obtain

\begin{align*} {}\begin{array}{l} {} {}\mathbb{E}^{\prime}\left[\left(\max\left\lbrace 1,\ell_{\delta_{\pmb{\theta_{H}}}}(t)\lVert N\rVert\right\rbrace \right)^{-d}\right] {} {}=\mathbb{P}^{\prime}\left[\lVert N\rVert\leq\ell_{\delta_{\pmb{\theta_{H}}}}^{-1}(t)\right]+\ell_{\delta_{\pmb{\theta_{H}}}}^{-d}(t)\mathbb{E}^{\prime}\left[\dfrac{1}{\lVert N\rVert^{d}}1\!\!1_{\lbrace\lVert N\rVert \gt \ell_{\delta_{\pmb{\theta_{H}}}}^{-1}(t)}\right]\\ {} {}= \left(2\pi\right)^{-d/2}\,\left({\displaystyle\int}_{\left\{ \lVert y\rVert\leq\ell_{\delta_{\pmb{\theta_{H}}}}^{-1}(t)\right\} }\,e^{-\lVert y\rVert^{2}/2}dy+\ell_{\delta_{\pmb{\theta_{H}}}}^{-d}(t){\displaystyle\int}_{\left\{ \lVert y\rVert \gt \ell_{\delta_{\pmb{\theta_{H}}}}^{-1}(t)\right\} }\,\frac{e^{-\lVert y\rVert^{2}/2}}{\lVert y\rVert^{d}}dy\right)\\ {} {}\\ {} {}\leq\mathsf{c}_{4}\,\ell_{\delta_{\pmb{\theta_{H}}}}^{-d}(t)\left(1+{\displaystyle\int}_{\ell_{\delta_{\pmb{\theta_{H}}}}^{-1}(t)}^{\infty}\dfrac{e^{-r^{2}/2}}{r}\,dr\right)\\ {} {}\leq\mathsf{c}_{4}\,\ell_{\delta_{\pmb{\theta_{H}}}}^{-d}(t)\left(1+{\displaystyle\int}_{1\wedge\ell_{\delta_{\pmb{\theta_{H}}}}^{-1}(t)}^{1}r^{-1}\,dr+{\displaystyle\int}_{1}^{\infty}\frac{e^{-r^{2}/2}}{r}\,dr\right)\\ {} {}\\ {} {}\leq\mathsf{c}_{4}\,\ell_{\delta_{\pmb{\theta_{H}}}}^{-d}(t)\left(1-\log\left(1\wedge\ell_{\delta_{\pmb{\theta_{H}}}}^{-1}(t)\right)\right)\leq\mathsf{c}_{4}\,\ell_{\delta_{\pmb{\theta_{H}}}}^{-d}(t)\left(1+\log\left(1\vee\ell_{\delta_{\pmb{\theta_{H}}}}(t)\right)\right). {}\end{array} {}\end{align*}

Lemma 6·4. Let E be a Borel set of [0, 1]. If $\mathcal{C}_{\Phi_{H,\ell_{\delta_{\pmb{\theta_H}}}}}(E) \gt 0$ , then $\mathbb{P}^{\prime}$ -almost surely

$$ \mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}\left(Gr_E(B^{\delta_{\pmb{\theta_H}}})\right) \gt 0. $$

Proof. Firstly, the assumption $\mathcal{C}_{\Phi_{H,\ell_{\delta_{\pmb{\theta_H}}}}}(E) \gt 0$ ensures that there exists a probability measure $\nu$ supported on E with finite energy, i.e.

$$ {}\mathcal{E}_{\Phi_{H,\ell_{\delta_{\pmb{\theta_H}}}}}(\nu)={\displaystyle\int}_E{\displaystyle\int}_E \Phi_{H,\ell_{\delta_{\pmb{\theta_H}}}}(|t-s|)\nu(dt)\nu(ds) \lt \infty. {}$$

Let $\mu_{\omega^{\prime}}$ be the random measure defined on $Gr_E(B^{\delta_{\pmb{\theta_H}}})$ by

$$ {}\mu_{\omega^{\prime}}(G)\,:\!=\,\nu\{s\,:\, (s,B^{\delta_{\theta_0}}(\omega^{\prime},s))\in G\} \quad \text{ for all } G\subset Gr_E(B^{\delta_{\pmb{\theta_H}}}(\cdot,\omega^{\prime})). {}$$

Hence $\mathbb{P}^{\prime}$ -almost surely we have

\[ {}\begin{array}{c} {} {}\mathcal{E}_{\rho_{\mathbf{d}_{H}},d}(\mu_{\omega^{\prime}})\,:\!=\, {\displaystyle\int}_{Gr_{E}(B^{\delta_{\pmb{\theta_{H}}}})}{\displaystyle\int}_{Gr_{E}(B^{\delta_{\pmb{\theta_{H}}}})}\frac{d\mu_{\omega^{\prime}}(s,B^{\delta_{\pmb{\theta_{H}}}}(s))d\mu_{\omega^{\prime}}(t,B^{\delta_{\pmb{\theta_{H}}}}(t))}{\max\left\{ \lvert t-s\lvert^{Hd},\lVert B^{\delta_{\pmb{\theta_{H}}}}(t)-B^{\delta_{\pmb{\theta_{H}}}}(s)\rVert^{d}\right\} }\\ {} {}\\ {} {}={\displaystyle\int}_{E}{\displaystyle\int}_{E}\frac{\nu(ds)\nu(dt)}{\max\left\{ \lvert t-s\lvert^{Hd},\lVert B^{\delta_{\pmb{\theta_{H}}}}(t)-B^{\delta_{\pmb{\theta_{H}}}}(s)\rVert^{d}\right\} }. {}\end{array} {}\]

Therefore, in order to achieve the goal it is sufficient to show that $\mathcal{E}_{\rho_{\mathbf{d}_{H}},d}(\mu_{\omega^{\prime}}) \lt \infty$ for $\mathbb{P}^{\prime}$ -almost surely, which can be done by checking $\mathbb{E}^{\prime}\left[\mathcal{E}_{\rho_{\mathbf{d}_{H}},d}(\mu_{\omega^{\prime}})\right] \lt \infty$ . Indeed, using Fubini’s theorem with the stationarity of increments and Lemma 6·3 we obtain

(6·10) \begin{align} {} {}\begin{aligned} {} {} {}\mathbb{E}^{\prime}\left[ \mathcal{E}_{\rho_{\mathbf{d}_{H}},d}(\mu_{\omega^{\prime}})\right] {} {} {}&= \mathbb{E}^{\prime}\left[{\displaystyle\int}_{E}{\displaystyle\int}_{E}\frac{1}{\max\left\{ \lvert t-s\lvert^{Hd},\lVert B^{\delta_{\pmb{\theta_{H}}}}(t)-B^{\delta_{\pmb{\theta_{H}}}}(s)\rVert^{d}\right\} }\nu(ds)\nu(dt)\right]\\ {} {} {}&={\displaystyle\int}_E{\displaystyle\int}_E \mathbb{E}^{\prime}\left[\frac{1}{\max\left\{\lvert t-s\lvert^{Hd}, \lVert B^{\delta_{\pmb{\theta_{H}}}}(|t-s|)\rVert^d \right\}}\right]\nu(ds)\nu(dt)\\ {} {} {}&\leq \mathsf{c}_3\,\mathcal{E}_{\Phi_{H,\ell_{\delta_{\pmb{\theta_H}}}}}(\nu) \lt \infty. {} {}\end{aligned} {}\end{align}

Thus $\mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}\left(Gr_E(B^{\delta_{\pmb{\theta_H}}})\right) \gt 0$ $\mathbb{P}^{\prime}$ -almost surely.

The following result is the consequence of the two previous lemmas.

Proposition 6·6. Let $\{B^{H}(t) \,:\, t \in [0,1]\}$ be a d-dimensional fractional Brownian motion of Hurst index $H\in (0,1)$ . If $\mathcal{C}_{\Phi_{H,\ell_{\delta_{\pmb{\theta_H}}}}}(E) \gt 0$ , then there exists a continuous function $f\in C^{\mathsf{w}_{H,{\ell_{\theta_H}}}}(\mathbf{I})\setminus C^{H}\left(\mathbf{I}\right)$ such that

(6·11) \begin{align} {} {}\mathbb{P}\left\{\exists t\in E \,:\, (B^{H}+f)(t)=x\right\} \gt 0, {}\end{align}

for all $x\in \mathbb{R}^d$ .

Proof. We start the proof by recalling that Theorem 7·2 provides the modulus of continuity of $B^{\delta_{\pmb{\theta_H}}}$ that is $B^{\delta_{\pmb{\theta_H}}}\in C^{\mathsf{w}_{H, {\ell_{\theta_H}}}}(\mathbf{I})$ , $\mathbb{P}^{\prime}$ -almost surely. Now applying Theorem 5·1 we deduce that for $\mathbb{P}^{\prime}$ -almost surely there is a positive random constant $C=C(\omega^{\prime}) \gt 0$ such that

$$ {}\mathbb{P}\left\{ \exists s\in : (B^H+B^{\delta_{\pmb{\theta_H}}}(\omega^{\prime}))(s)=x \right\}\geq C\, \mathcal{C}_{\rho_{\mathbf{d}_{H}}}^{d}\left(Gr_E\,B^{\delta_{\pmb{\theta_H}}}(\cdot,\omega^{\prime})\right) \gt 0. {}$$

Hence, by choosing f to be one of the trajectories of $B^{\delta_{\pmb{\theta_H}}}$ we get the desired result.

Consequently we have the following outcome:

Corollary 6·8. Let $\{B^{H}(t) : t \in [0,1]\}$ be a d-dimensional fractional Brownian motion of Hurst index $H\in (0,1)$ . Assume that $L_{\pmb{\theta_H}}$ , the normalised slowly varying part of $\pmb{\theta_H}$ , satisfies (4·11). Then there exist a Borel set $E\subset \mathbf{I}$ , such that

$$ \dim(E)=Hd \quad \text{ and }\quad \mathcal{H}^{Hd}(E)=0,$$

and a function $f\in C^{\mathsf{w}_{H,{\ell_{\theta_H}}}}(\mathbf{I})\setminus C^{H}\left(\mathbf{I}\right)$ , such that all points are non-polar for $(B^H+f)\vert_E$ .

Proof of Corollary 6·8. (7·6) with $\underset{x\rightarrow +\infty}{\liminf} \,L_{\pmb{\theta_H}}(x)=0$ imply that

$$ \underset{x\rightarrow 0}{\limsup} \,\ell_{\delta_{\pmb{\theta_H}}}(x)= \underset{x\rightarrow 0}{\limsup} \,\ell_{\pmb{\theta_H}}(x)=+\infty. $$

Thus applying once again [ Reference Taylor27 , theorem 4] with the functions $h(t)\,:\!=\,t^{H d}$ and $\Phi(t)=\Phi_{H,\ell_{\delta_{\pmb{\theta_H}}}}(t)$ we infer that there exists a compact set $E\subset \mathbf{I}$ such that

$$ {}\mathcal{H}^{H d}(E)=0 \quad \text{ and } \quad \mathcal{C}_{\Phi_{H,\ell_{\delta_{\pmb{\theta_H}}}}}(E) \gt 0. {}$$

Finally Proposition 6·6 gives us the function that we are looking for, that is $f\in C^{\mathsf{w}_{H,{\ell_{\theta_H}}}}(\mathbf{I})$ for which $(B^H+f)\vert_E$ hits all points.