2. Preliminaries on parabolic Hausdorff dimension

Let $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

\[\mathbb{E}\left(B^{H}_0(s)B^{H}_0(t)\right)=\frac{1}{2}\left(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\right).\]

Let $B^{H}_1,...,B^{H}_d$ be d independent copies of $B^{H}_0$ , then 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 $H \in (0,1)$ . We start by giving the definition of the parabolic Hausdorff dimension.

Definition 2·1. Let $F\subset \mathbb{R}_{+}\times \mathbb{R}^d$ and $H\in (0,1)$ . For all $\beta>0$ the H-parabolic $\beta$ -dimensional Hausdorff content is defined by

(2·1) \begin{equation} \Psi_{H}^{\beta}(F)=\inf \left\{\sum_{j} \delta_{j}^{\beta}\;:\;F \subseteq \underset{j}{\cup}\left[a_{j}, a_{j}+\delta_{j}\right] \times\left[b_{j, 1}, b_{j, 1}+\delta_{j}^{H}\right] \times \cdots \times\left[b_{j, d}, b_{j, d}+\delta_{j}^{H}\right] \right\}, \end{equation}

where the infimum is taken over all countable covers of F by rectangles of the form given above. The H-parabolic Hausdorff dimension is then defined to be

\[ \dim_{\Psi,\, H}(F)=\inf \left\{\beta\;:\;\Psi_{H}^{\beta}(F)=0\right\}.\]

Remark 2·2. Let $\rho_H$ be the metric defined on $\mathbb{R}_+\times\mathbb{R}^d$ by

(2·2) \begin{equation}\rho_H((s,x),(t,y))=\max\{|s-t|^H,\| x-y \|_{\infty}\} \quad \forall (s,x), (t,y)\in \mathbb{R}_+\times \mathbb{R}^d.\end{equation}

We define the $\beta$ -dimensional Hausdorff content as

(2·3) \begin{equation}\mathcal{H}_{\rho_H}^{\beta}(F)=\inf \left\{\sum_{j} diam(U_{j})^{\beta}\;:\;F \subseteq \underset{j}{\cup} \, U_{j} \right\},\end{equation}

where $\left\lbrace U_{j} \right\rbrace $ is a countable cover of F by any sets and $diam(U_{j}) $ denotes the diameter of a set $U_{j}$ relatively to the metric $\rho_H$ . For any $F\subseteq\mathbb{R}_+\times\mathbb{R}^d$ , the Hausdorff dimension, in the metric $\rho_H$ , of F is defined by

\[\dim_{\rho_H}(F)=\inf \left\{\beta\;:\;\mathcal{H}_{\rho_H}^{\beta}(F)=0\right\}.\]

Since the family $\left(\left[a_{j}, a_{j}+\delta_{j}\right] \times\left[b_{j, 1}, b_{j, 1}+\delta_{j}^{H}\right] \times \ldots \times\left[b_{j, d}, b_{j, d}+\delta_{j}^{H}\right] \right)_{j\geq 1}$ form a particular cover of $F\,$ by sets of diameters $\delta_{j}^{H}$ , then for any $\beta >0$ and $F\subseteq\mathbb{R}_+\times\mathbb{R}^d$ we have

(2·4) \begin{equation}\mathcal{H}_{\rho_H}^{\beta/H}(F)\leq \Psi_H^{\beta}(F).\end{equation}

On the other hand, each set $U_j$ with $\operatorname{diam}\left(U_j\right)=\delta_j$ can be covered by

\[\left(\left[a_{j,k}, a_{j,k}+\delta_j^{1/H}\right] \times\left[b_{j,k, 1}, b_{j,k, 1}+\delta_j\right] \times \ldots \times\left[b_{j,k,d}, b_{j,k, d}+\delta_j\right] \right)_{1\leq k\leq 2^{d+1}}.\]

Therefore, we obtain

\[\Psi_H^{\beta}(F)\leq\, \sum_{j=1}^{\infty}\sum_{k=1}^{2^{d+1}}\delta_j^{\beta/H}=2^{d+1}\sum_{j=1}^{\infty}\delta_j^{\beta/H} ,\]

which implies

(2·5) \begin{equation} \Psi_H^{\beta}(F)\leq 2^{d+1}\mathcal{H}_{\rho_H}^{\beta/H}(F).\end{equation}

Hence, we conclude from (2·4) and (2·5) that

(2·6) \begin{equation}\dim_{\Psi,\,H}(F)=H\times \dim_{\rho_H}(F).\end{equation}

The following proposition relates $\beta$ -dimensional capacity to the H-parabolic Hausdorff dimension.

Proposition 2·3. Let $F\subset \mathbb{R}_+\times\mathbb{R}^d$ be a compact set. Then we have:

(2·7) \begin{equation} \dim_{\Psi,\,H}(F)=\sup\{\beta\;:\;\mathcal{C}_{\rho_H,\beta/H}(F)>0\}=\inf\{\beta\;:\;\mathcal{C}_{\rho_H,\beta/H}(F)=0\}, \end{equation}

where $\mathcal{C}_{\rho_H,\beta}(.)$ is the $\beta$ -capacity on the metric space $(\mathbb{R}_+\times\mathbb{R}^d,\rho_H)$ defined by

(2·8) \begin{equation} \mathcal{C}_{\rho_H,\beta}(F)=\left[\inf _{\mu \in \mathcal{P}(F)} \int_{\mathbb{R}_+\times\mathbb{R}^{d}} \int_{\mathbb{R}_+\times\mathbb{R}^{d}} \frac{\mu(d u) \mu(d v)}{(\rho_H(u,v))^{\beta}} \right]^{-1}. \end{equation}

Here $\mathcal{P}(F)$ is the family of probability measure carried by F.

Proof. The proof is the same as in the Euclidean case, see for example [Reference Bishop and Peres4, theorem 3·4·2] or [Reference Mörters and Peres16, theorem 4·32].

The next theorem is the analogue of Frostman’s theorem for parabolic Hausdorff dimension. The statement can be found in Taylor and Watson [Reference Taylor and Watson21, lemma 4] and the proof follows along the same lines as the proof of usual Frostman’s theorem.

Theorem 2·4. (Frostman’s theorem) Let F a Borel set in $\mathbb{R}_+\times\mathbb{R}^d$ . If $\dim_{\Psi, H}(F)>\kappa$ , then there exists a Borel probability measure $\mu$ supported on F, and a constant $C>0$ , such that

(2·9) \begin{equation} \mu\left([a, a+\delta] \times \prod_{j=1}^d\left[b_{j}, b_{j}+\delta^{H}\right]\right) \leq C \delta^{\kappa}, \end{equation}

for any $\left(a,b_{1},\ldots,b_{d} \right)\in \mathbb{R}_+\times\mathbb{R}^d$ and $\delta > 0$ .

Now we give a comparison result for the Hausdorff parabolic dimensions with different parameters. We note that it is a generalisation of [Reference Balança3, lemma 5·4] to the d-dimensional case. However, there is a flaw (may be a misprint) in the lower bound of Lemma 5·4. The following proposition corrects it and improves the lower and upper bounds.

Proposition 2·5. Let $F\subset \mathbb{R}_+\times \mathbb{R}^d$ and $H,H'\in (0,1)$ such that $H<H'$ . Then we have:

(2·10) \begin{align} \dim_{\Psi,\,H}(F)&\vee \left( \frac{H'}{H}\dim_{\Psi,\,H}(F)+1-\frac{H'}{H}\right) \leq \dim_{\Psi, H'}(F) \nonumber \\&\leq\left( \frac{H'}{H}\dim_{\Psi, H}(F)\right)\wedge \left( \dim_{\Psi, H}(F)+(H'-H)d\right). \end{align}

An equivalent reformulation is

\begin{align*} \frac{H}{H'}\dim_{\rho_{H}}(F)\vee\left(\dim_{\rho_{H}}(F)+\frac{1}{H'}-\frac{1}{H}\right)&\leq\dim_{\rho_{H'}}(F)\\ & \leq\left(\dim_{\rho_{H}}(F)\wedge\left(d+\frac{H}{H'}\left(\dim_{\rho_{H}}(F)-d\right)\right)\right).\end{align*}

Proof. Firstly let us remark that for the infimum in (2·1) does not change if we consider only $\delta_j\leq 1$ . Therefore an immediate consequence of the definition is

(2·11) \begin{equation} \dim_{\Psi,\;H}(F)\leq \dim_{\Psi, H'}(F). \end{equation}

Now let $0<\varepsilon<1$ and $\gamma>{\dim_{\Psi,\,H'}(F)-1}/{H'}$ . Then $\Psi_{H'}^{\gamma H'+1}(F)=0$ , and hence there exists a cover $\left([a_{n},a_{n}+\delta_{n}]\times\underset{j=1}{\overset{d}{\prod}}[b_{n,j},b_{n,j}+\delta_{n}^{H'}]\right)_{n\geq1}$ of the set F, such that

(2·12) \begin{equation}{{\displaystyle \underset{n\geq1}{\sum}\delta_{n}^{\gamma H'+1}\leq\varepsilon.}}\end{equation}

It follows from $(2\cdot12)$ that $\delta_{n}<1$ for all n. Each interval $[a_{n},a_{n}+\delta_{n}]$ can be divided into $\left\lceil \delta_{n}^{1-\frac{H'}{H}}\right\rceil $ intervals of length $\delta_{n}^{H'/H}$ each. In this way we obtain a new cover $\left(\left[a'_{l},a'_{l}+\delta_{l}^{H'/H}\right]\times\underset{j=1}{\overset{d}{\prod}}\left[b'_{l,j},b'_{l,j}+\left(\delta_{l}^{H'/H}\right)^{H}\right]\right)_{l\geq1}$ of the set F which satisfies

(2·13) \begin{equation}{{\displaystyle \underset{l\geq1}{\sum}\,\left(\delta_{l}^{H'/H}\right)^{\gamma H+1}\leq2\underset{n\geq1}{\sum}\,\delta_{n}^{1-\frac{H'}{H}}\,\left(\delta_{n}^{H'/H}\right)^{\gamma H+1}\leq2\varepsilon.}} \end{equation}

From (2·13) we deduce that $\dim_{\Psi,\,H}(F)\leq\gamma H+1$ , which implies ${\dim_{\Psi,\,H}(F)-1}/{H}\leq\gamma$ . Therefore letting $\gamma$ go to ${\dim_{\Psi,\,H'}(F)-1}/{H'}$ we conclude

(2·14) \begin{equation} \dfrac{H'}{H}\dim_{\Psi,\,H}(F)+1-\dfrac{H'}{H}\leq\dim_{\Psi,\,H'}(F).\end{equation}

Combining (2·11) and (2·14), we obtain the lower bound

\[\dim_{\Psi,\,H}(F)\vee \left( \dfrac{H'}{H}\dim_{\Psi,\,H}(F)+1-\dfrac{H'}{H}\right) \leq \dim_{\Psi, H'}(F) .\]

For the second inequality let $\kappa<\dim_{\Psi,\,H'}(F)$ . Then by Frostman’s Theorem 2·4 there exists a probability measure $\mu$ supported on F such that (2·9) is satisfied. Our aim is to show that

(2·15) \begin{align} \mu\left(\left[a, a+\delta\right] \times\prod_{j=1}^d\left[b_{j}, b_{j}+\delta^{H}\right]\right)\leq C \left(\delta^{\kappa+d(H-H')}\wedge \delta^{\kappa H/H'}\right), \end{align}

for any $a\in \mathbb{R}_+$ , $b_1,...,b_d\in \mathbb{R}$ and $0<\delta<1$ . Note first that since $\delta \leq \delta^{H/H'}$ we have

\[\left[a,a+\delta\right]\times\prod_{j=1}^{d}\left[b_{j},b_{j}+\delta^{H}\right]\subset\left[a,a+\delta^{H/H'}\right]\times\prod_{j=1}^d\left[b_{j},b_{j}+\left(\delta^{H/H'}\right)^{H'}\right].\]

Using (2·9) we obtain

(2·16) \begin{align} \mu\left(\left[a, a+\delta\right] \times\prod_{j=1}^d\left[b_{j}, b_{j}+\delta^{H}\right]\right)\leq C \delta^{\kappa\, H/H'}. \end{align}

Now, for any $j=1,...,d$ , the interval $[b_j,b_j+ \delta^{H}]$ can be covered by at most $\delta^{(H-H')}$ interval of length $\delta^{H'}$ . Using once again (2·9) we deduce that

(2·17) \begin{align} \mu\left(\left[a, a+\delta\right] \times\prod_{j=1}^d\left[b_{j}, b_{j}+\delta^{H}\right]\right)\leq C \delta^{\kappa+d(H-H')}. \end{align}

Thus the inequality (2·15) is proved. Since, in the metric space $(\mathbb{R}_+\times\mathbb{R}^d,\rho_H)$ , the diameter of the set $\left[a, a+\delta\right] \times\prod_{j=1}^d\left[b_{j}, b_{j}+\delta^{H}\right]$ is $\delta^{H}$ then the Mass Distribution Principle, (see [Reference Mörters and Peres16, theorem 4·19]), implies that

\[\dim_{\rho_{H}}\!(F)\geq\dfrac{\kappa}{H'}\vee\dfrac{\kappa+d(H-H')}{H}.\]

From (2·6) it follows that

\[\kappa\leq\left(\frac{H'}{H}\dim_{\Psi,\,H}(F)\right)\wedge\left(\dim_{\Psi,\,H}(F)+d(H'-H)\right).\]

Therefore letting $\kappa\uparrow \dim_{\psi,H'}(F)$ the assertion (2·10) follows.

The following proposition looks at the effect of Hölder continuous maps on the Hausdorff dimension of sets $\dim(A)$ and $\dim_{\Psi,\,H}(Gr_A(\kern1.5pt{f}))$

Proposition 2·6. Let f : $[0,1]\rightarrow \mathbb{R}^d$ be a Borel measurable function and A be a Borel subset of [0,1]. Then we have:

(2·18) \begin{equation} \dim (A)\leq \dim_{\Psi, H}(Gr_A(\kern1.5pt{f})). \end{equation}

If in addition the function f is Hölder continuous with exponent $\alpha\leq H$ ( $\alpha$ -Hölder continuous), that is

\[ \exists\,K\geq0\;:\;\lVert f(x)-f(y) \rVert_{\infty}\leq K\vert x-y\vert^{\alpha},\,\,\,\forall x,y\in\left[0,1\right], \]

then we have:

(2·19) \begin{equation} \dim_{\Psi, H}(Gr_A(\kern1.5pt{f})) \leq \left( \frac{H}{\alpha}\dim(A)\right) \wedge \left( \dim(A)+(H-\alpha)d\right) . \end{equation}

Especially, when f is $\left( H-\varepsilon\right) $ -Hölder continuous for all $\varepsilon>0$ then

(2·20) \begin{equation} \dim (A)= \dim_{\Psi, H}(Gr_A(\kern1.5pt{f})). \end{equation}

Proof. We begin by proving (2·18). Let $\gamma>\dim_{\Psi, H}(Gr_A({\kern1.5pt}f))$ . Since $\Psi_{H}^{\gamma}(Gr_A(\kern1.5pt{f}))=0$ then for all $\varepsilon>0$ there exists a cover $\left([a_l,a_l+\delta_l]\times \underset{j=1}{\overset{d}{\prod}}[b_{l,j},b_{l,j}+\delta_l^H]\right) _{l\geq 1}$ of $Gr_A(\kern1.5pt{f})$ such that $\displaystyle\underset{l\geq 1}{\sum}\delta_l^{\gamma}\leq \varepsilon$ . Consequently, in $\mathbb{R}$ with the absolute-value metric, $([a_l,a_l+\delta_l])_{l\geq 1}$ is a covering of A such that

\[\sum_{l\geq1}\left|[a_{l},a_{l}+\delta_{l}]\right|^{\gamma}\leq\varepsilon.\]

Here $\left|[a_{l},a_{l}+\delta_{l}]\right|$ is the diameter of the set $[a_l,a_l+\delta_l]$ . Then we deduce that the $\gamma$ -dimensional Hausdorff content satisfies

(2·21) \begin{align}\mathcal{H}^{\gamma}(A)\;:\!=\;\inf \left\{\sum_{j} \lvert U_{j}\rvert^{\gamma}\;:\;A \subseteq \underset{j}{\cup} \, U_{j} \right\}\leq \varepsilon.\end{align}

Thus $\mathcal{H}^{\gamma}(A)=0$ and therefore $\dim(A)\leq \gamma$ . Now by making $\gamma \downarrow \dim_{\Psi, H}(Gr_A(f))$ we get the desired inequality.

Now let $\alpha\leq H$ and assume that f is $\alpha$ -Hölder continuous. For any $0<\kappa<\dim_{\Psi, H}(Gr_A(f))$ , by resorting again to Frostman’s theorem there exist a measure $\mu$ on $Gr_A(f)$ and a constant $C>0$ such that

(2·22) \begin{equation} \mu\left([a, a+\delta] \times \prod_{j=1}^d\left[b_{j}, b_{j}+\delta^{H}\right]\right) \leq C \delta^{\kappa},\end{equation}

for any $\left(a,b_{1},\ldots,b_{d} \right)\in A\times\mathbb{R}^d$ and $\delta \in \left(0,1 \right] $ . Let $\nu$ be the measure on A satisfying $\nu=\mu \circ P_1^{-1}$ where $P_1$ is the projection mapping on A, i.e. $P_1(s,f(s))=s$ . Our aim is to show that

(2·23) \begin{equation}\nu([a,a+\delta])=\mu(P_{1}^{-1}([a,a+\delta]))\leq C_{1}\left(\delta^{\kappa+d(\alpha-H)}\,\wedge \,\delta^{\kappa\,\alpha/H}\right)\!,\end{equation}

for some constant $C_{1}>0$ and any $\delta \in \left(0,1 \right]$ . Since f is Hölder continuous function with exponent $\alpha$ and constant K, then for $s \in [a,a+\delta]$ we have $f_j(s)\in [{\kern1.5pt}f_j(a)-K \delta^{\alpha},f_j(a)+K \delta^{\alpha}]$ . There are two ways to cover

\[P_1^{-1}\left( [a,a+\delta]\right) =\left\lbrace (s,f(s))\in Gr_A(f) ; s\in [a,a+\delta] \right\rbrace .\]

The first one is : for all $j=1,...,d$ we decompose every interval $[f_j(a)-K \delta^{\alpha},f_j(a)+K \delta^{\alpha}]$ into at most $\lceil 2K\rceil \delta^{(\alpha-H)}$ interval of length $\delta^H$ , then $P_1^{-1}([a,a+\delta])$ is covered by at most $\lceil 2K\rceil^{d}\delta^{d(\alpha-H)}$ sets of the form

\[ [a,a+\delta]\times\prod_{j=1}^{d}[b_j,b_j+\delta^H].\]

With regard to the second way, since $\delta \leq \delta^{\alpha/H}$ then $P_1^{-1}([a,a+\delta])$ may be covered by at most $\lceil 2K\rceil^{d}$ sets of the form

\[[a,a+\delta^{\alpha/H}]\times \prod_{j=1}^{d}[b_j,b_j+\delta^{\alpha}].\]

Now using (2·22) we conclude that there exists a constant $C'>0$ which depends only on K and d, such that

\[ \nu([a,a+\delta])=\sigma(P_1^{-1}([a,a+\delta]))\leq C' \delta ^{\kappa+d(\alpha-H)},\]

for the first cover. For the second one we get

\[ \nu([a,a+\delta])\leq C'' \delta^{\kappa\,\alpha/H},\]

where $C''>0$ is a constant depending only on K and d. Thus inequality (2·23) is proved. Thanks to the mass distribution principle, see [Reference Mörters and Peres16, theorem 4·19], we obtain

\[\kappa \leq \left( \frac{H}{\alpha}\dim(A)\right) \wedge \left( \dim(A)+(H-\alpha)d\right).\]

Therefore letting $\kappa \uparrow \dim_{\Psi, H}(Gr_A(f))$ the assertion (2·19) follows.

Finally it is easy to see from (2·18) and (2·19) that when f is $\left( H-\varepsilon\right) $ -Hölder continuous for all $\varepsilon>0$ we have $ \dim (A)= \dim_{\Psi, H}(Gr_A(f))$ .

A natural question that arises from Proposition 2·6 whether (2·19) is a sharp estimate? The main idea is to use the paths of Hölder continuous stochastic process. Precisely, we have the following result

Theorem 2·8. Let $\alpha\leq H$ , $\{B^{\alpha}(t)\;:\;t \in [0,1]\}$ a d-dimensional fractional Brownian motion of Hurst index $\alpha$ and $A\subset [0,1]$ a Borel set. Then we have:

(2·24) \begin{align} \dim_{\Psi,\,H}(Gr_A(B^{\alpha}))=\left(\left( \frac{H}{\alpha}\dim(A)\right) \wedge(\dim(A)+d(H-\alpha))\right) \text{ a.s.} \end{align}

Moreover, if $\alpha<H$ we have

(2·25) \begin{equation}\dim_{\Psi,\,H}(Gr_A(B^{\alpha}))>\dim(A).\end{equation}

For the proof we need the following lemma.

Lemma 2·9. There exists a constants C such that for all $t \in (0,1]$ we have:

(2·26) \begin{equation}\mathbb{E}\left[\frac{1}{(\max\{t^{H},\lVert B^{\alpha}(t)\rVert_{\infty}\})^{\gamma/H}}\right]\leq\left\{ \begin{array}{lll}C\,t^{-\gamma\alpha/H}\mbox{ } & \text{if} & \gamma<Hd,\\\\C\, t^{d(H-\alpha)-\gamma}\mbox{ } & \text{if} & \gamma>Hd.\end{array}\right.\end{equation}

Proof. We first note that the denominator on the left-hand side of (2·26) has the same distribution as $(\max\{t^{H},t^{\alpha}\lVert N\rVert_{\infty}\})^{\gamma/H}$ , where N is a d-dimensional standard normal random variable.

Now assume that $\gamma<Hd$ . Then we obtain

\[\mathbb{E}\left[\frac{1}{\left(\max\left(t^{H},\lVert B^{\alpha}(t)\rVert_{\infty}\right)\right){}^{\gamma/H}}\right]=\mathbb{E}\left[\frac{1}{\left(\max\left(t^{H},t^{\alpha}\lVert N\rVert_{\infty}\right)\right)^{\gamma/H}}\right]\leq t^{-\alpha\gamma/H}\mathbb{E}\left[\lVert N\rVert_{\infty}^{-\gamma/H}\right].\]

Since $\gamma<Hd$ , we have that $\mathbb{E}\left[\lVert N\rVert_{\infty}^{-\gamma/H}\right]$ is finite and therefore

\[\mathbb{E}\left[\frac{1}{\left(\max\left(t^{H},\lVert B^{\alpha}(t)\rVert_{\infty}\right)\right){}^{\gamma/H}}\right]\leq C\,t^{-\alpha\gamma/H}.\]

Next, let $\gamma>Hd$ . Since $\lVert y\rVert_{\infty}\leq \lVert y\rVert$ for any $ y\in \mathbb{R}^d$ it yields

\begin{align*}\mathbb{E}\left[\dfrac{1}{\left(\max\left(t^{H},t^{\alpha}\lVert N\rVert_{\infty}\right)\right)^{\gamma/H}}\right] & = \dfrac{1}{t^{\gamma}}\mathbb{P}\left[\lVert N\rVert_{\infty}\leq t^{H-\alpha}\right]+\dfrac{1}{t^{\gamma\alpha/H}}\mathbb{E}\left[\dfrac{1}{\lVert N\rVert_{\infty}^{\gamma/H}}1\!\!1_{\{\lVert N\rVert_{\infty}>t^{H-\alpha}\}}\right]\\ & \quad \leq C\left(\dfrac{1}{t^{\gamma}}\displaystyle\int_{\lVert y\rVert_{\infty}\leq t^{H-\alpha}}\,e^{-\lVert y\rVert^{2}/2}dy \right. \\ & \left. \quad +\dfrac{1}{t^{\gamma\alpha/H}}\displaystyle\int_{\lVert y\rVert_{\infty}>t^{H-\alpha}}\,\frac{e^{-\lVert y\rVert^{2}/2}}{\lVert y\rVert_{\infty}^{\gamma/H}}dy\right)\\ & \quad \leq C_1\left(t^{d(H-\alpha)-\gamma}+t^{-\gamma\alpha/H}\displaystyle\int_{t^{H-\alpha}}^{1}r^{d-1-\gamma/H}dr+t^{-\gamma\alpha/H}\right)\\ & \quad \leq C_2\left(t^{d(H-\alpha)-\gamma}+t^{-\gamma\alpha/H}\right)\\ & \quad \leq C_3\,t^{d(H-\alpha)-\gamma},\end{align*}

where the constants $C_j, \,j=1,2,3$ depend only on d, H, $\alpha$ and $\gamma$ .

Proof of Theorem 2·8. The upper bound of $\dim_{\Psi,\,H}(Gr_A(B^{\alpha}))$ follows directly from Lemma 2·6.

Now let us prove the lower bound. First, we consider the case $\dim(A)\leq \alpha d$ . Let $ \gamma<{H}/{\alpha}\dim(A)\leq \dim(A)+d(H-\alpha)$ . By [Reference Mörters and Peres16, theorem 4·32], there exists a probability measure $\nu$ on A such that

(2·27) \begin{align}\mathcal{E}_{\gamma\alpha/H}(\nu)\;:\!=\;\int_A\int_A\frac{1}{|t-s|^{\gamma\alpha/H}}\nu(ds)\nu(dt)<\infty.\end{align}

Let $\widetilde{\mu}$ be the random measure defined by

\[ \widetilde{\mu}(E)=\nu\{s\;:\;(s,B^{\alpha}(s))\in E\},\]

where $E\subset Gr_A(B^{\alpha})$ . We will show that

\[\mathcal{E}_{\rho_{H},\gamma/H}(\widetilde{\mu})\;:\!=\;\int_{\mathbb{R}_+\times\mathbb{R}^{d}} \int_{\mathbb{R}_+\times\mathbb{R}^{d}} \frac{\widetilde{\mu}(d u) \widetilde{\mu}(d v)}{(\rho_H(u,v))^{\gamma/H}}<\infty \text{ a.s.}\]

Taking expectation and using a change of variables and Fubini’s theorem we get

\[\mathbb{E}\left[\mathcal{E}_{\rho_{H},\gamma/H}(\widetilde{\mu})\right]=\int_{A}\int_{A}\mathbb{E}\left[\frac{1}{\left(\max\left(|s-t|^{H},\lVert B^{\alpha}(t)-B^{\alpha}(s)\rVert_{\infty}\right)\right){}^{\gamma/H}}\right]\nu(ds)\nu(dt).\]

By recalling the fact that $B^{\alpha}$ has stationary increments we see that the denominator has the same distribution as $\left(\max\left(|s-t|^{H},\lVert B^{\alpha}(\vert s-t\vert)\rVert_{\infty}\right)\right){}^{\gamma/H}$ . It follows that

\[\mathbb{E}\left[\mathcal{E}_{\rho_{H},\gamma/H}(\widetilde{\mu})\right]=\int_{A}\int_{A}\mathbb{E}\left[\frac{1}{\left(\max\left(|s-t|^{H},\lVert B^{\alpha}(\vert s-t\vert)\rVert_{\infty}\right)\right){}^{\gamma/H}}\right]\nu(ds)\nu(dt).\]

Since $\gamma<Hd$ we deduce from Lemma 2·9 that

\[\mathbb{E}\left[\mathcal{E}_{\rho_H,\gamma/H}(\widetilde{\mu})\right]\leq C\,\int_A\int_A\frac{1}{|t-s|^{\gamma\alpha/H}}\nu(ds)\nu(dt),\]

which is finite from (2·27). Hence $\mathcal{C}_{\rho_H,\gamma/H}(Gr_A(B^{\alpha}))>0$ almost surely and Proposition 2·3 allows that $\dim_{\Psi,\,H}(Gr_A(B^{\alpha}))\geq \gamma\,\,a.s$ .

Now assume that $\dim(A)>\alpha d$ . In this case, we choose $Hd<\gamma<\dim(A)+d\hbox{$(H-\alpha)$}<{H}/{\alpha}\dim(A)$ . We use the same tools, as in the previous case, to prove that $\mathcal{C}_{\rho_H,\gamma/H}(Gr_A(B^{\alpha}))>0\,\,a.s$ via the probability measure $\nu$ satisfying

\[\mathcal{E}_{\gamma-d(H-\alpha)}(\nu)<\infty.\]

Letting $\gamma\uparrow ({H}/{\alpha}\dim(A))\wedge(\dim(A)+d(H-\alpha))$ completes the proof.

As a consequence of (2·24) in Theorem 2·8 and (2·6) we have the following result.

Corollary 2·10. Let $\alpha\leq H$ , $\{B^{\alpha}(t)\;:\;t\in[0,1]\}$ a fractional Brownian motion of Hurst index $\alpha$ and $A\subset[0,1]$ a Borel set. Then we have:

\[ \dim_{\rho_{H}}\left(Gr_{A}(B^{\alpha})\right)>d\,\,a.s\Longleftrightarrow\dim_{\Psi,\,H}\left(Gr_{A}(B^{\alpha})\right)>Hd\,\,a.s\Longleftrightarrow\dim(A)>\alpha d. \]

Proof. The first equivalence comes from (2·6), whereas the second one arises from

\[ \left(\frac{H}{\alpha}\dim(A)\right)\wedge (\dim(A)+d(H-\alpha))>Hd\Longleftrightarrow \left\{\begin{array}{ll} \dfrac{H}{\alpha}\dim(A)>Hd, \\ \\ \dim(A)+d(H-\alpha)>Hd. \end{array}\right. \]

3. Positive Lebesgue measure and non-empty interior of $ (B ^ {H} +f) (A) $

Let $Y=(Y(t))_{t\in[0,1]}$ be an $\mathbb{R}^{d}$ -valued stochastic process and $\nu$ is a positive measure on [0, 1]. The occupation measure of the sample path $[0,1]\ni t\rightarrow Y(t)(w)\in\mathbb{R}^{d}$ is defined by

\[\nu_{Y}(E)\;:\!=\;\nu\left\lbrace t\in[0,1]\;:\;Y(t)\in E\right\rbrace ,\]

where $E\subset\mathbb{R}^{d}$ is a Borel set. We say that Y has an occupation density relative to the Lebesgue measure $\lambda_{d}$ if $\nu_{Y}$ is absolutely continuous with respect to $\lambda_{d}$ almost surely.

Simple modifications of the differentiation’s method, see [Reference Geman and Horowitz7, theorem 21·15], allow to give necessary and sufficient conditions under which $\nu_{Y} $ has an occupation density relative to the Lebesgue measure $\lambda_d$ a.s. Hence we omit the proof. Here is a precise statement:

Proposition 3·1. The following assertions are equivalent:

\begin{equation*} \begin{aligned} (1)\qquad &\nu_{Y} <<\lambda_d \,\,\,\text{with} \,\,\, \frac{d\nu_{Y}}{d\lambda_d}(.)\in L^2(\lambda_d \otimes \mathbb{P});\\ \\ (2)\qquad & \liminf_{r \downarrow 0}\, r^{-d}\int_{A}\int_{A}\mathbb{P}\left\lbrace \lVert Y(s)-Y(t)\rVert <r\right\rbrace \,d\nu(s)\,d\nu(t)<\infty. \end{aligned} \end{equation*}

Let $A \subset [0,1]$ and assume that $\eta\;:\!=\;\dim_{\Psi, H}(Gr_A(f))>Hd$ . It follows from Theorem 2·4 that for $\kappa \in (Hd,\eta)$ there exists a Borel probability measure $\mu$ supported on $Gr_A(f)$ and $C>0$ such that

(3·1) \begin{equation}\mu\left([a, a+\delta] \times \prod_{j=1}^d\left[b_{j}, b_{j}+\delta^{H}\right]\right) \leq C \delta^{\kappa},\end{equation}

for all $a \in A$ , $b_1,...,b_n \in \mathbb{R}^d$ and all $\delta\in \left( 0,1\right] $ . Let $\nu$ be the measure defined on A by $\hbox{$\nu=\mu$} \circ P_1^{-1}$ . We are now ready to state

Proof. To achieve this purpose, it is enough to verify the second assertion of Proposition 3·1 for the process $B^{H}+f$ . Indeed for $s,t \in A$ and $r>0$ , we have

\begin{align*}\mathbb{P}\left\lbrace \lVert(B^{H}+f)(s)-(B^{H}+f)(t)\rVert<r\right\rbrace & =\dfrac{1}{(2\pi)^{d/2}|t-s|^{Hd}}\int_{\lVert y\rVert<r} \\[5pt] & \quad \exp\left(-\frac{\lVert y-f(t)+f(s)\rVert^{2}}{2|t-s|^{2H}}\right)dy.\end{align*}

Then using Fubini’s theorem we obtain, for any fixed $t\in A$ and $r>0$ , that

(3·2) \begin{equation}\begin{aligned}\displaystyle\int_{A}\mathbb{P}&\left \lbrace \lVert(B^{H}+f)(s)-(B^{H}+f)(t)\rVert<r\right\rbrace \,d\nu(s) =\\[5pt] &\dfrac{1}{(2\pi)^{d/2}}\int_{\lVert y\rVert<r}\int_{A}\dfrac{1}{|t-s|^{Hd}}\exp\left(-\frac{\lVert y-f(t)+f(s)\rVert^{2}}{2|t-s|^{2H}}\right)\,d\nu(s)\,dy\\[5pt] & \leq C\,r^{d}\sup_{\lVert y\rVert<r}\underset{=I(y,\left( t,f(t))\right) }{\underbrace{\int_{A}\,\dfrac{1}{|t-s|^{Hd}}\exp\left(-\frac{\lVert y-f(t)+f(s)\rVert^{2}}{2|t-s|^{2H}}\right)\,d\nu(s)}},\end{aligned}\end{equation}

where C is a positive constant depending only on r and d.

For any fixed y in $\left\lbrace \lVert y\rVert<r \right\rbrace $ and $ \left( t,f(t)\right)\in Gr_{A}(f) $ , we have the following decomposition

\[ I\left(y,\left( t,f(t)\right) \right)=I_1\left(y,\left( t,f(t)\right) \right)+I_2\left(y,\left( t,f(t)\right) \right),\]

where

\begin{align*} I_{1}\left(y,\left(t,f(t)\right)\right) &= \displaystyle\int_{\left\lbrace s\in A:\lVert y-f(t)+f(s)\rVert\leq C_{1}\,\vert s-t\vert^{H}\sqrt{\vert\log\vert s-t\vert\vert}\right\rbrace }\\ & \quad \times \frac{1}{|s-t|^{Hd}}\exp\left(-\frac{\lVert y-f(t)+f(s)\rVert^{2}}{2|s-t|^{2H}}\right)\,d\nu(s)\\ &= \displaystyle\int_{\left\lbrace (s,f(s))\in Gr_{A}(f):\lVert y-f(t)+f(s)\rVert\leq C_{1}\,\vert s-t\vert^{H}\sqrt{\vert\log\vert s-t\vert\vert}\right\rbrace }\\ & \quad \times \dfrac{1}{|s-t|^{Hd}}\exp\left(-\frac{\lVert y-f(t)+f(s)\rVert^{2}}{2|s-t|^{2H}}\right)\,d\mu(s,f(s)).\end{align*}

and

\begin{align*} I_{2}\left(y,\left(t,f(t)\right)\right) & =\displaystyle\int_{\left\lbrace s\in A:\lVert y-f(t)+f(s)\rVert>C_{1}\,\vert s-t\vert^{H}\sqrt{\vert\log\vert s-t\vert\vert}\right\rbrace } \\ & \quad \times\frac{1}{|s-t|^{Hd}}\exp\left(-\frac{\lVert y-f(t)+f(s)\rVert^{2}}{2|s-t|^{2H}}\right)\,d\nu(s)\\ & = \displaystyle\int_{\left\lbrace (s,f(s))\in Gr_{A}(f):\lVert y-f(t)+f(s)\rVert>C_{1}\,\vert s-t\vert^{H}\sqrt{\vert\log\vert s-t\vert\vert}\right\rbrace } \\ &\quad \times\frac{1}{|s-t|^{Hd}}\exp\left(-\frac{\lVert y-f(t)+f(s)\rVert^{2}}{2|s-t|^{2H}}\right)\,d\mu(s,f(s)),\end{align*}

where $C_{1}$ is a positive constant which will be chosen later. We first show that

\[\underset{\left( y,\left(t,f(t)\right)\right) \in \left\lbrace \lVert y\rVert<r \right\rbrace \times Gr_{A}(f)}{\sup}I_{1}\left(y,\left( t,f(t)\right) \right)<+\infty.\]

By the inequality (3·1) we can verify that the measure $\nu$ is no atomic. Then we have

\begin{align*}I_1(y,\left( t,f(t))\right)&\leq \sum_{n=1}^{\infty}\,2^{nHd}\,\mu \Big\lbrace \lVert y-f(t)+f(s)\rVert \\ & \leq C_1\,\vert s-t\vert^{H}\sqrt{\vert\log\vert s-t\vert\vert},\,\, 2^{-n}<|s-t|\leq 2^{1-n}\Big\rbrace\\&\leq \sum_{n=1}^{\infty}\,2^{nHd}\,\mu \left\lbrace \lVert y-f(t)+f(s)\rVert \leq C_2 \,2^{-nH}\sqrt{n},\,\, 2^{-n}<|s-t|\leq 2^{1-n}\right\rbrace,\end{align*}

where $C_2=C _1\,2^H\sqrt{\log(2)}$ . Now, for $n\geq1$ , we set

\[S_n\left(y,\left( t,f(t)\right)\right) =\left\lbrace \left( s,f(s)\right)\in Gr_{A}(f)\;:\;\lVert y-f(t)+f(s)\rVert \leq C_2 \,2^{-nH}\sqrt{n},\,\, 2^{-n}<|s-t|\leq 2^{1-n}\right\rbrace .\]

For $(s,f(s))\in S_n\left(y,\left( t,f(t)\right)\right) $ we have $s\in [t-2^{1-n},t+2^{1-n}] $ and

\[f_j(s)\in J_j=\left[ f_j(t)-y_j-C_2 2^{-nH}\sqrt{n},f_j(t)-y_j+C_2 2^{-nH}\sqrt{n}\right]\]

for all $j\in \{1,...,d\}$ . Since $\left[ t-2^{1-n},t+2^{1-n}\right] $ is covered by 4 intervals of length $2^{-n}$ and every $J_j$ is covered by no more than a constant multiple (which depends only on H) of $\sqrt{n}$ intervals of length $2^{-nH}$ , then we can cover $ S_n\left(y,\left( t,f(t)\right)\right) $ by no more than a constant (which depends only on d and H) multiple of $n^{d/2}$ sets of the form $\left[ a,a+2^{-n}\right] \times\prod_{j=1}^{d}\left[ b_j,b_j+2^{-nH}\right] $ . Applying inequality (3·1) allows to have

\[\underset{\left( y,\left(t,f(t)\right)\right) \in \left\lbrace \lVert y\rVert<r \right\rbrace \times Gr_{A}(f)}{\sup}\,\mu( S_n \left(y,\left( t,f(t)\right)\right) \leq C_3 \, n^{d/2} \, 2^{-\kappa n}.\]

Therefore we get

(3·3) \begin{equation}\underset{\left( y,\left(t,f(t)\right)\right) \in \left\lbrace \lVert y\rVert<r \right\rbrace \times Gr_{A}(f)}{\sup}I_{1}\left(y,\left( t,f(t)\right) \right)\leq C_4 \sum_{n=1}^{\infty}\, 2^{-(\kappa-Hd)n}\, n^{d/2}<\infty,\end{equation}

where $C_4$ depends on d and H only. For the second term $I_2\left(y,\left( t,f(t)\right) \right)$ we have

\begin{align*}I_2\left(y,\left( t,f(t)\right) \right)\leq & \sum_{n=1}^{\infty}2^{nHd}\exp\left(- \dfrac{C_1^2}{2}(n-1)\ln2\right) \\ &\,\times\mu \left\lbrace \lVert f(s)-f(t)\rVert > C_1\,\vert s-t\vert^{H}\sqrt{\vert\log\vert s-t\vert\vert},\,\, 2^{-n}<|s-t|\leq 2^{1-n}\right\rbrace.\end{align*}

Thus, for $C_1>\sqrt{2Hd}$ we obtain

(3·4) \begin{equation}\underset{\left( y,\left(t,f(t)\right)\right) \in \left\lbrace \lVert y\rVert<r \right\rbrace \times Gr_{A}(f)}{\sup}I_{1}\left(y,\left( t,f(t)\right) \right)\leq e^{C_1^2\ln2/2} \sum_{n=1}^{\infty}2^{-n\left( C_1^2/2-Hd\right) }<+\infty.\end{equation}

Now putting all this together yields

\[\underset{\left( y,\left(t,f(t)\right)\right) \in \left\lbrace \lVert y\rVert<r \right\rbrace \times Gr_{A}(f)}{\sup}\displaystyle\int_{A}\,\dfrac{1}{|s-t|^{Hd}}\exp\left(-\frac{\lVert y-f(t)+f(s)\rVert^{2}}{2|s-t|^{2H}}\right)\,d\nu(s)<\infty.\]

Thus we get from (3·2) that

\[\liminf_{r \downarrow 0}r^{-d}\int_{A}\int_{A}\, \mathbb{P}\left\lbrace \lVert(B^{H}+f)(s)-(B^{H}+f)(t)\rVert<r\right\rbrace\,d\nu(s)\,d\nu(t)<\infty.\]

We can therefore state from Proposition 3·1 that the occupation measure $\nu_{B^{H}+f}$ is absolutely continuous with respect to the Lebesgue measure $\lambda_{d}$ a.s. Hence $\lambda_{d}\left(B^{H}+f\right)(A)>0$ a.s.

Remark 3·3. In connection with the existence of the occupation density let us mention that another, often easier to apply, criterion due to Berman tells us that, $\nu_{Y}$ has a density $\frac{d\nu_{Y}}{d\lambda_d}(.)\in L^2(\lambda_d \otimes \mathbb{P})$ if and only if

(3·5) \begin{equation}\int_{\mathbb{R}^{d}}\int_{A}\int_{A}\,\mathbb{E}\left(e^{i\langle\theta,(Y(s)-Y(t)\rangle}\right)\,d\nu(s)\,d\nu(t)\,d\theta<\infty.\end{equation}

However, we were unable to apply it to prove the absolute continuity of the occupation measure $\nu_{B^{H}+f}$ with respect to the Lebesgue measure $\lambda_{d}$ a.s. The difficulty stems from the lack of informations on the measure $\nu$ . Indeed, a simple calculation using characteristic function of Gaussian random vector yields

\[ \mathbb{E}\left(e^{i\langle\theta,(B^{H}+f)(s)-(B^{H}+f)(t)\rangle}\right)=e^{i\langle\theta,f(s)-f(t)\rangle}\exp\left(-\frac{|s-t|^{2H}\lVert\theta\rVert^{2}}{2}\right). \]

Now integrating the modulus we have

\[ \int_{\mathbb{R}^{d}}\,\left|\mathbb{E}\left(e^{i\langle\theta,(B^{H}+f)(s)-(B^{H}+f)(t)\rangle}\right)\right|\,d\theta=\frac{\left(2\pi\right)^{d/2}}{|s-t|^{Hd}}, \]

which may be too loose to ensure the finiteness of the energy

\[ \int_{A}\int_{A}\,\frac{\left(2\pi\right)^{d/2}}{|s-t|^{Hd}}\,d\nu(s)\,d\nu(t). \]

This is because the measure $\nu=\mu\circ P_1^{-1}$ does not satisfy necessarily the Frostman’s condition even though it is true for $\mu$ . Hence one cannot use Fubini’s theorem which make the cited criterion unworkable.

According to the conclusion of Theorem 3·2, it may be expected that the interior of $(B^{H}+f)(A)$ is not empty. Recall that Kahane ([Reference Kahane10, theorems 1 and 2, p. 267]) has shown that, for any compact subset $A\subset [0,1]$ , $B^{H}(A)$ is a Salem set almost surely if $\dim (A)<Hd$ and $B^{H} (A)$ has a non-empty interior almost surely if $\dim (A) > Hd$ . That is what prompted us to consider the case $Hd<\dim(A)$ . Let us note that in the light of (2·18), the condition $Hd<\dim(A)$ leads to $\dim_{\Psi, H}(Gr_A(f))>Hd$ and therefore $\lambda_d((B^{H}+f)(A))>0$ almost surely. Our aim is to prove that, under this condition, the set $(B^{H}+f)(A)$ not only has a positive measure but also a non-void interior. A key ingredient is the continuity of the occupation density over A. Obtaining this continuity we will make use of the following.

It follows from [Reference Pitt18, lemma 7·1] that, for any $H\in(0,1)$ , the real-valued fractional Brownian motion $B_{0}^{H}$ has the following important property of strong local nondeterminism: There exists a constant $0<C<\infty$ such that for all integers $n\geq1$ and all $t_{1},\ldots,t_{n},t\in[0,1]$ , we have

\[\text{Var}\left(B_{0}^{H}(t)|B_{0}^{H}(t_{1}),\ldots,B_{0}^{H}(t_{n})\right)\geq\,C\,\underset{0\leq j\leq n}{\min}\,\lvert t-t_{j}\rvert^{2H},\]

where $\text{Var}\left(B_{0}^{H}(t)|B_{0}^{H}(t_{1}),\ldots,B_{0}^{H}(t_{n})\right)$ denotes the conditional variance of $B_{0}^{H}(t)$ given $B_{0}^{H}(t_{1}),\ldots,B_{0}^{H}(t_{n})$ and $t_{0}=0$ , see Appendix for the definition. The following elementary formula allows to estimate the determinant of the covariance matrix of the Gaussian random vector $(Z_{1},\ldots,Z_{n})$

\begin{equation*}\text{det}\text{Cov}(Z_{1},\ldots,Z_{n})=\text{Var}(Z_{1})\underset{k=2}{\overset{n}{\prod}}\text{Var}\left(Z_{k}|Z_{1},\ldots,Z_{k-1}\right).\end{equation*}

For the vector $\left(B_{0}^{H}(t_{1}),\ldots,B_{0}^{H}(t_{n})\right)$ one obtains

(3·6) \begin{equation}\text{det}\text{Cov}\left(B_{0}^{H}(t_{j}),1\leq j\leq p\right)\geq\underset{j=1}{\overset{p}{{\displaystyle \prod}}}\left(\min\,\left\{ \lvert t_{j}-t_{i}\rvert^{2H},0\leq i\leq j-1\right\} \right).\end{equation}

We need also the following lemma which is due to Cuzick and DuPreez [Reference Cuzick and DuPreez5]

Lemma 3·4. Let $Z_{1},...,Z_{n}$ be linearly-independent centered Gaussian variables. If $g\;:\;\mathbb{R}\rightarrow\mathbb{R}$ is a Borel measurable function such that

\[ \int_{-\infty}^{\infty}g(v)e^{-\varepsilon v^{2}}dv<\infty \]

for all $\varepsilon>0$ . Then

\begin{align*} \int_{\mathbb{R}^{n}}g(v_{1}) & \exp\left(-\frac{1}{2}\text{Var}\left(\sum_{j=1}^{n}v_{j}Z_{j}\right)\right)dv_{1}...dv_{n}\\[5pt] & =\frac{(2\pi)^{(n-1)/2}}{(\text{det}\text{Cov}(Z_{1},...,Z_{n}))^{1/2}}\int_{-\infty}^{\infty}g\left(\frac{v}{\sigma_{1}}\right)e^{-v^{2}}dv, \end{align*}

where $\sigma_{1}^{2}=\text{Var}(Z_{1}|Z_{2}...Z_{n})$ is the conditional variance of $Z_{1}$ given $Z_{2},...,Z_{n}$ .

The next result, using the above-mentioned preliminaries, strengthens the result of Theorem 3·2 since it shows that $(B^{H}+f)(A)$ has a non-empty interior.

Proof. The usual Frostman’s theorem ensure that for all $Hd<\kappa <\dim(A)$ there is a constant $C>0$ and a Borel probability measure $\nu$ on A such that

(3·7) \begin{equation} \tau([a,a+\delta])\leq C\, \delta^{\kappa}, \end{equation}

for all $a\in A$ and $0<\delta<1$ . it is easily verified that

\[\displaystyle\int_{\mathbb{R}^{d}}\,\displaystyle\int_{A}\displaystyle\int_{A}\,\mathbb{E}\left(e^{i\langle\theta,(B^{H}+f)(s)-(B^{H}+f)(t)\rangle}\right)\,d\tau(s)\,d\tau(t)\,d\theta\leq (2\pi)^{d/2}\,\displaystyle\int_{A}\displaystyle\int_{A}\dfrac{d\tau(s)d\tau(t)}{|t-s|^{Hd}},\]

which is finite by (3·7) since $\kappa >Hd$ . Hence the condition (3·5) for the process $B^{H}+f$ and the probability measure $\tau$ is checked. Consequently, the occupation measure $\tau_{B^{H}+f}$ has an occupation density relative to the Lebesgue measure $\lambda_d$ a.s denoted by $\vartheta$ . Hence, in order to prove our theorem, it is sufficient to prove that $\vartheta$ has a continuous version.

First of all, let us note that the arguments used to show [Reference Geman and Horowitz7, (25.7)] can be modified to see that, for all $x,y \in \mathbb{R}^d$ and for all even integers $p\geq 2$ , we have

\begin{equation*} \begin{array}{lll} \mathbb{E}\left(\vartheta(x)-\vartheta(y)\right)^{p}= & \left(2\pi\right)^{-pd}\displaystyle\int_{A^{p}}\displaystyle\int_{\mathbb{R}^{pd}}\underset{j=1}{\overset{p}{\prod}}\left(\exp\left(i\langle x,\xi_{j}\rangle\right)-\exp\left(i\langle y,\xi_{j}\rangle\right)\right)\\ \\[-4pt] & \times\,\mathbb{E}\exp\left(i\displaystyle\underset{j=1}{\overset{p}{\sum}}\langle\xi_{j},\left(B^{H}+f\right)(t_{j})\rangle\right)\,d\xi_{1}\ldots d\xi_{p}\,d\tau(t_{1})\ldots d\tau(t_{p}). \end{array} \end{equation*}

Let $p\geq 2$ be a fixed even integer and $0<\gamma<1$ whose value will be determined later. Using the facts that

\[|e^{iu}-1|\leq 2^{1-\gamma} \,\,|u|^{\gamma},\, \,\forall\; u\in \mathbb{R}, \]

and $|a+b|^{\gamma}\leq |a|^{\gamma}+|b|^{\gamma}$ , we have

\[ \underset{j=1}{\overset{p}{\prod}}\vert\exp\left(i\langle x,\xi_{j}\rangle\right)-\exp\left(i\langle y,\xi_{j}\rangle\right)\vert\leq2^{(1-\gamma)p}\,\,\lVert x-y\rVert^{\gamma p}\,\sum\prod_{j=1}^{p}|\xi_{j,k_{j}}|^{\gamma}, \]

where the summation $\sum$ is taken over all $(k_1,...,k_p)\in \{1,...,d\}^{p}$ . It follows that

\[ \begin{array}{ll} \mathbb{E}\left(\vartheta(x)-\vartheta(y)\right)^{p} \leq & \left(2\pi\right)^{-pd}\,\,2^{(1-\gamma)p}\,\, \lVert x-y\rVert^{\gamma p}\, \displaystyle\sum\,\displaystyle\int_{A^{p}}\displaystyle\int_{\mathbb{R}^{pd}}\;\;\prod_{j=1}^{p}|\xi_{j,k_{j}}|^{\gamma}\,\\ \\[-4pt] & \times\exp\left(-\dfrac{1}{2}\text{Var}\left(\displaystyle\overset{p}{\underset{j=1}{\sum}}\left\langle \xi_{j},B^{H}(t_{j})\right\rangle \right)\right)\,d\xi_{1}\ldots d\xi_{p}\,\,d\tau(t_{1})\ldots d\tau(t_{p}). \end{array} \]

Now the generalised Hölder’s inequality leads to

\begin{align*} \mathbb{E}\left(\vartheta(x)-\vartheta(y)\right)^{p} & \leq \left(2\pi\right)^{-pd}\,\,2^{(1-\gamma)p}\,\,\lVert x-y\rVert^{\gamma p}\,{\displaystyle \sum\,\displaystyle\int_{A^{p}}\prod_{j=1}^{p}}\\ &\times\left[\displaystyle\int_{\mathbb{R}^{pd}}\;\;|\xi_{j,k_{j}}|^{p\gamma}\,\exp\left(-\dfrac{1}{2}\text{Var}\left({\displaystyle \overset{p}{\underset{j=1}{\sum}}\left\langle \xi_{j},B^{H}(t_{j})\right\rangle }\right)\right) d\xi_{1}\ldots d\xi_{p}\right]^{1/p} \\ & \times d\tau(t_{1})\ldots d\tau(t_{p}). \end{align*}

Fix a sequence $(k_{1},...,k_{p})\in\left\{ 1,...,d\right\} ^{p}$ , $0<t_{1}<\ldots<t_{p}$ , and consider

\[ J=\prod_{j=1}^{p}\left[\displaystyle\int_{\mathbb{R}^{pd}}\;\;|\xi_{j,k_{j}}|^{p\gamma}\,\exp\left(-\dfrac{1}{2}\text{Var}\left({\displaystyle \overset{p}{\underset{j=1}{\sum}}\left\langle \xi_{j},B^{H}(t_{j})\right\rangle }\right)\right)\,d\xi_{1}\ldots d\xi_{p}\right]^{1/p}. \]

Note that $\left\{ B^{H}_{l},1\leq l\leq d\right\} $ are independent copies of $B^{H}_{0}$ . The strong local nondeterminism of the fractional Brownian motion $B^{H}_{0}$ and (3·6), ensure that the random variables $\left\{ B^{H}_{l}(t_{j}),1\leq l\leq d;1\leq j\leq p\right\} $ are linearly independent. Hence by applying Lemma 3·4, we derive that J is bounded by

(3·8) \begin{align} J \leq & \,\,\dfrac{(2\pi\kern-1pt)^{(pd-1)/2}}{\left[\det\text{Cov}\left(B^{H}_{l}(t_{j}),0\leq l\leq d,1\leq j\leq p\right)\right]^{1/2}}\,\,\displaystyle\int_{\mathbb{R}}|r|^{p\gamma}e^{-r^{2}/2}dr\,\prod_{j=1}^{p}\,\,\dfrac{1}{\sigma_{j}^{\gamma}} \notag\\ \notag\\ \leq &\,\,\dfrac{(dK)^{p}(p!)^{\gamma}}{\left[\det\text{Cov}\left(B^{H}_{0}(t_{j}),1\leq j\leq p\right)\right]^{d/2}}\,\,\underset{j=1}{\overset{p}{\displaystyle\prod}}\,\,\dfrac{1}{\sigma_{j}^{\gamma}}, \end{align}

where $\sigma_j^2$ is the conditional variance of $B^{H}_{k_j}(t_j)$ given $B^{H}_{l}(t_i)$ $(l\neq k_j \text{ or } l=k_j, i\neq j)$ and the last inequality follows from Stirling’s formula. Combining (3·8) with the well-known facts, see for example the proof of [Reference Xiao22, lemma 2·5], that

(3·9) \begin{equation} \underset{j=1}{\overset{p}{\displaystyle\prod}}\dfrac{1}{\sigma_j^{\gamma}} \leq \frac{K^p}{\left[\det\text{Cov}\left(B^{H}_{0}(t_{j}),1\leq j\leq p\right)\right]^{\gamma}} \leq \dfrac{K^{p}}{\underset{j=1}{\overset{p}{{\displaystyle \prod}}}\left(\min\,\left\{ \lvert t_{j}-t_{i}\rvert^{2H\gamma},0\leq i\leq j-1\right\} \right)}, \end{equation}

we conclude

(3·10) \begin{equation} J\leq\dfrac{\left(dK^{2}\right)^{p}(p!)^{\delta}}{\underset{j=1}{\overset{p}{{\displaystyle \prod}}}\left(\min\,\left\{ \lvert t_{j}-t_{i}\rvert ^{H(d+2\gamma)},0\leq i\leq j-1\right\} \right)}. \end{equation}

Now we can simply prove

(3·11) \begin{equation} \displaystyle\int_{A^p}\,\dfrac{d\tau(t_{1})\ldots d\tau(t_{p})}{\underset{j=1}{\overset{p}{{\displaystyle \prod}}}\left(\min\,\left\{ |t_{j}-t_{i}|,0\leq i\leq j-1\right\} \right)^{H(d+2\gamma)}}\leq p!\left( \sup_{s\in [0,1]}\displaystyle\int_{A}\frac{d\tau(t)}{|t-s|^{H(d+2\gamma)}}\right) ^{p}, \end{equation}

which is finite, by using (3·7), if we take

\[0<\gamma<\dfrac{1}{2}(\kappa/H-d).\]

Putting all the above facts together, we arrive at

(3·12) \begin{equation} \mathbb{E}(\vartheta(x)-\vartheta(y))^{p}\leq C_{p,\gamma}\, \lVert x-y\rVert^{\gamma p}, \end{equation}

where $C_{p,\gamma}$ is a constant depends only on p and $\gamma$ . Hence we can apply the Kolmogorov’s continuity theorem ([Reference Khoshnevisan12, theorem 2·3·1, p.158]) to get a continuous version on $\mathbb{R}^d$ of $\vartheta(.)$ . Since A is a compact subset of $\left[0,1\right]$ (closed subset of $\left[0,1\right]$ ) and $\left(B^{H}+f\right)$ is continuous then $\left(B^{H}+f\right)(A)$ is a compact set in $\mathbb{R}^{d}$ . Taking into account that $\vartheta$ has a version which is continuous in x then it follows from Pitt [Reference Pitt18, p. 324] or Geman and Horowitz [Reference Geman and Horowitz7, p. 12] that $\left\{ x\;:\;\vartheta(x)>0\right\} $ is open, non-empty and contained in $\left(B^{H}+f\right)(A)$ almost surely. This completes the proof.

Remark 3·6. It is worth noting that the condition $\dim_{\Psi,\,H}(Gr_{A}(f))>H\,d$ in Theorem 3·2 is a weaker than $\dim(A)>H\,d$ in Theorem 3·5. Indeed let us take A the classical middle thirds Cantor set which Hausdorff dimension $\dim(A)={\ln\left(2\right)}/{\ln\left(3\right)}$ . For $0<\alpha\,d<{\ln\left(2\right)}/{\ln\left(3\right)}<H\,d$ , we have from $(2\cdot24)$ that $\dim_{\Psi,\,H}(Gr_{A}(B^{\alpha}))>Hd$ . Then by choosing f as a sample function of $B^{\alpha}$ we obtain

\[\dim(A)<H\,d<\dim_{\Psi,\,H}(Gr_{A}(f)).\]

Moreover, as can be seen from the proofs of Theorems 3·2 and 3·5, we were able to prove the existence of an occupation density through the conditions $\dim_{\Psi,\,H}(Gr_{A}(f))>H\,d$ and $\dim(A)>H\,d$ using respectively two different measures namely $\nu$ and $\tau$ . It should be noted that these two measures are constructed by Frostman’s lemma. But it was simpler and more practical to work with $\tau$ thanks to the advantageous properties that presents, which are consequences of the condition $\dim(A)>H\,d$ , and who have played a key role in the fact that $(B^{H}+f)(A)$ has a non-empty interior. We are therefore naturally led to ask the following question that we have not solved: can we establish the result of Theorem 3·5 by assuming only that $\dim_{\Psi,\,H}(Gr_{A}(f))>H\,d$ ?

4. Hölder functions for which $(B^{H}+f)(A)$ has a non-empty interior a.s

The aim of this section is to consider the case $\dim(A)\leq Hd$ . Precisely, we seek to construct Hölder functions for which the range $(B^{H}+f)(A)$ has non-empty interior. Let us make this precise in the following statement.

The remainder of this section is devoted to the proof of this theorem. To that end, we will make some preparations. Let $(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{P}^{\prime})$ be another probability space and $B^{\alpha^{\prime}}$ be a fractional Brownian motion with Hurst parameter $\alpha^{\prime}\in \left( 0,\dim(A)/d\right)$ defined on it. Let us consider the d-dimensional centred Gaussian process Z defined on $(\Omega\times\Omega^{\prime},\mathcal{F}\times\mathcal{F}^{\prime},\mathbb{P}\otimes\mathbb{P}^{\prime})$ by

(4·1) \begin{align}Z(t,(\omega,\omega^{\prime}))=B^{H}(t,\omega)+B^{\alpha^{\prime}}(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 $Z=(Z_1,...,Z_d)$ where $Z_i$ are independent copies of the real-valued Gaussian process $Z_{0}=B^{H}_{0}+B^{\alpha^{\prime}}_{0}$ with the covariance function given by

\begin{align*} \widetilde{\mathbb{E}}(Z_0(s)Z_0(t))& =\mathbb{E}(B^{H}_0(s)B^{H}_0(t))+\mathbb{E}^{\prime}(B^{\alpha^{\prime}}_{0}(s)B^{\alpha^{\prime}}_{0}(t))\\ & =\dfrac{1}{2}(s^{2H}+t^{2H}-|t-s|^{2H})+\dfrac{1}{2}(s^{2\alpha^{\prime}}+t^{2\alpha^{\prime}}-|t-s|^{2\alpha^{\prime}}),\end{align*}

where $\widetilde{\mathbb{E}}$ and $\mathbb{E}^{\prime}$ denote the expectation under the probability $\widetilde{\mathbb{P}}=\mathbb{P}\otimes \mathbb{P}^{\prime}$ and $\mathbb{P}^{\prime}$ respectively.

The following proposition is about the local nondeterminism property of the process Z. Let $I\subset (0,1]$ be a closed interval, then we have

Proof.

1. (1) For any $s,t \in I$ , we have

   \[\begin{array}{lll}\widetilde{\mathbb{E}}\left(Z_{0}(t)-Z_{0}(s)\right)^{2} & = & \mathbb{E}\left(B^{H}_{0}(t)-B^{H}_{0}(s)\right)^{2}+\mathbb{E}^{\prime}\left(B^{\alpha^{\prime}}_{0}(t)-B^{\alpha^{\prime}}_{0}(s)\right)^{2}\\\\& = & \vert t-s\vert^{2H}+\vert t-s\vert^{2\alpha^{\prime}}.\end{array}\]
   Since $\alpha \leq H$ it follows that
   \[\vert t-s\vert^{2\alpha^{\prime}}\leq\begin{array}{lll}\widetilde{\mathbb{E}}\left(Z_{0}(t)-Z_{0}(s)\right)^{2} & \leq & \,2\,\vert t-s\vert^{2\alpha^{\prime}}.\end{array}\]

2. (2) By definition we can write

   \begin{align*}{}Var\left(Z_{0}(u)\vert Z_{0}(t_{1}),...,Z_{0}(t_{n})\right) & = \underset{a_{j}\in\mathbb{R},1\leq j\leq n}{\inf}\widetilde{\mathbb{E}}\left(Z_{0}(u)-\displaystyle\overset{p}{\underset{j=1}{\sum}}a_{j}\,Z_{0}(t_{j})\right)^{2}\\& = \underset{a_{j}\in\mathbb{R},1\leq j\leq n}{\inf}[\,\,\mathbb{E}\left(B^{H}_{0}(u)-\displaystyle\overset{p}{\underset{j=1}{\sum}}a_{j}\,B^{H}_{0}(t_{j})\right)^{2}\\ & \quad+\mathbb{E}^{\prime}\left(B^{\alpha^{\prime}}_{0}(u)-\displaystyle\overset{p}{\underset{j=1}{\sum}}a_{j}\,B^{\alpha^{\prime}}_{0}(t_{j})\right)^{2}]\\ & \quad\geq \underset{a_{j}\in\mathbb{R},1\leq j\leq n}{\inf}\mathbb{E}\left(B^{H}_{0}(u)-\displaystyle\overset{p}{\underset{j=1}{\sum}}a_{j}\,B^{H}_{0}(t_{j})\right)^{2}\\ & \quad +\underset{b_{j}\in\mathbb{R},1\leq j\leq n}{\inf}\mathbb{E}^{\prime}\left(B^{\alpha^{\prime}}_{0}(u)-\displaystyle\overset{p}{\underset{j=1}{\sum}}b_{j}\,B^{\alpha^{\prime}}_{0}(t_{j})\right)^{2}\\& = Var\left(B^{H}_{0}(u)\vert B^{H}_{0}(t_{1}),...,B^{H}_{0}(t_{n})\right)\\ & \quad +Var\left(B^{\alpha^{\prime}}_{0}(u)\vert B^{\alpha^{\prime}}_{0}(t_{1}),...,B^{\alpha^{\prime}}_{0}(t_{n})\right).\end{align*}

By the property of strong local nondeterminism of fractional Brownian motion there exists a positive constant C depending on $\alpha$ , H and I only, such that for all integers $n\geq 1$ , all u, $t_1,...,t_n \in I$ ,

\[Var\left(Z_{0}(u)\vert Z_{0}(t_{1}),...,Z_{0}(t_{n})\right)\geq C\left[\min_{0\leq k\leq n}\vert u-t_{k}\vert^{2\alpha^{\prime}}+\min_{0\leq k\leq n}\vert u-t_{k}\vert^{2H}\right].\]

(3) We note that, in the Hilbert space setting, the conditional variance in (4·4) is the squared distance of $Z_{0}(t)$ from the linear subspace spanned by $\left\lbrace Z_0(s), \vert s-t\vert\geq r\right\rbrace $ in $L^{2}(\widetilde{\mathbb{P}})$ . Hence it is sufficient to show that there exists a positive constant C depending on $\alpha^{\prime}$ and H only, such that for all integers $n\geq 1$ , $a_{j}\in \mathbb{R}$ and $s_j \in I$ satisfying $\vert s_{j}-t\vert \geq r,\, (j=1,\ldots,n)$ ,

\[\widetilde{\mathbb{E}}\left(Z_{0}(t)-\displaystyle\overset{p}{\underset{j=1}{\sum}}a_{j}\,Z_{0}(s_{j})\right)^{2}\geq \,C\, r^{2\alpha^{\prime}}.\]

What follows from (4·3). This completes the proof.

The following is a direct consequence of [Reference Shieh and Xiao20, corollary 3·3].

Proof of Theorem 4·1. Let $\alpha^{\prime}\in \left( \alpha,\dim(A)/d\right)$ . It follows from Proposition 4·3 that $Z(A)=(B^{H}+B^{\alpha^{\prime}})(A)$ has $\mathbb{P}\otimes\mathbb{P}^{\prime}-a.s.$ a non-void interior. Therefore, there exists a $\mathbb{P}^{\prime}$ -negligible set $N^{\prime}$ such that, for any $\omega^{\prime}\in N^{\prime\,c}$ , there exists a $\mathbb{P}$ -negligible set N with for any $\omega\in N^{c}$ the set $(B^{H}(\omega)+B^{\alpha^{\prime}}(\omega^{\prime}))(A)$ has a non-void interior. It is a well known fact that almost all trajectories of $B^{\alpha^{\prime}}$ are $\alpha$ -Hölder continuous, and hence the desired drift can be chosen as a sample function of $B^{\alpha^{\prime}}$ .