revtex4-1Repair the float

# An optical Ising spin glass simulator with tuneable short range couplings

Louis Delloye  Gianni Jacucci  Laboratoire Kastler Brossel, Sorbonne
Université, Ecole Normale Supérieure-Paris Sciences et Lettres (PSL) Research
University, Centre National de la Recherche Scientifique (CNRS) UMR 8552,
Collège de France, 24 rue Lhomond, 75005 Paris, France  Raj Pandya
Laboratoire Kastler Brossel, Sorbonne Université, Ecole Normale Supérieure-
Paris Sciences et Lettres (PSL) Research University, Centre National de la
Recherche Scientifique (CNRS) UMR 8552, Collège de France, 24 rue Lhomond,
75005 Paris, France  Cavendish Laboratory, University of Cambridge, JJ Thomson
Avenue, Cambridge, CB3 0HE, UK  Davide Pierangeli  Institute for Complex
System, National Research Council (ISC-CNR), 00185 Rome, Italy  Physics
Department, Sapienza University of Rome, 00185 Rome, Italy  Claudio Conti
Physics Department, Sapienza University of Rome, 00185 Rome, Italy  Sylvain
Gigan  Laboratoire Kastler Brossel, Sorbonne Université, Ecole Normale
Supérieure-Paris Sciences et Lettres (PSL) Research University, Centre
National de la Recherche Scientifique (CNRS) UMR 8552, Collège de France, 24
rue Lhomond, 75005 Paris, France

###### Abstract

Non-deterministic polynomial-time (NP) problems are ubiquitous in almost every
field of study. Recently, all-optical approaches have been explored for
solving classic NP problems based on the spin-glass Ising Hamiltonian.
However, obtaining programmable spin-couplings in large-scale optical Ising
simulators, on the other hand, remains challenging. Here, we demonstrate
control of the interaction length between user-defined parts of a fully-
connected Ising system. This is achieved by exploiting the knowledge of the
transmission matrix of a random medium and by using diffusers of various
thickness. Finally, we exploit our spin-coupling control to observe replica-
to-replica fluctuations and its analogy to standard replica symmetry breaking.

##  I Introduction

color=red]update abstract+intro to fit the flow inversion done in last section
Non-deterministic polynomial-time problems (NP-problems) are important in many
fields of study from the physical to social sciences Parisi (2006); Takeda
_et al._ (2017); Maccari  _et al._ (2019); Das and Chakrabarti (2008); Goto
_et al._ (2019); Lenstra and Kan (1975); Lucas (2014). However, they often
have intractable solve-times with classical computers as their solve-time
scales exponentially with the size of the input Garey and Johnson (1990). An
archetypal NP-problem is finding the ground-state of an Ising spin-system
Barahona (1982); Bachas (1984); Nishimori (2001); Fu and Anderson (1986);
Sherrington and Kirkpatrick (1975). This is of particular interest as many
wide-spread NP-problems can be analytically mapped Lucas (2014); Jacucci  _et
al._ (2022) onto an Ising Hamiltonian (c.f. Equation 1). Solving any of these
particular NP problem thus reduces to finding the ground state of the
corresponding Ising system.

Recently, Ising models have been experimentally simulated in a number of ways,
both by using classical and quantum systems Pierangeli  _et al._ (2019, 2021,
2020); Tommasi  _et al._ (2016); Ghofraniha  _et al._ (2015); Johnson  _et
al._ (2011); Boixo  _et al._ (2014); Berloff  _et al._ (2017); Kalinin  _et
al._ (2020); Harris  _et al._ (2018); Hamerly  _et al._ (2019); Pierangeli
_et al._ (2017); Jacucci  _et al._ (2022); Waidyasooriya  _et al._ (2022);
Marandi  _et al._ (2014); Haribara  _et al._ (2016); McMahon  _et al._ (2016);
Inagaki  _et al._ (2016a); Böhm  _et al._ (2019); Takesue  _et al._ (2020);
Honjo  _et al._ (2021); Nixon  _et al._ (2013); Hamerly  _et al._ (2020);
Calvanese Strinati  _et al._ (2021); Gershenzon  _et al._ (2020); Edri  _et
al._ (2021); Inagaki  _et al._ (2016b). A class of very promising systems are
optical Ising simulators based on optical parametric oscillators (OPOs)
Hamerly  _et al._ (2020); Inagaki  _et al._ (2016b); Marandi  _et al._ (2014);
Haribara  _et al._ (2016); McMahon  _et al._ (2016); Inagaki  _et al._
(2016a); Böhm  _et al._ (2019); Takesue  _et al._ (2020); Honjo  _et al._
(2021); Nixon  _et al._ (2013); Hamerly  _et al._ (2020); Calvanese Strinati
_et al._ (2021); Gershenzon  _et al._ (2020); Edri  _et al._ (2021); Honjo
_et al._ (2022); Cen  _et al._ (2022) and photonic annealers based on
wavefront shaping Sun  _et al._ ; Kumar  _et al._ (a, b); Huang  _et al._ ;
Fang  _et al._ . The latter uses propagation in complex media whereas the
other exploits OPOs and time-multiplexing. OPO-based methods have a high
tunability but lack scalability. On the other hand, WS-based methods present a
high scalability and connectivity but are limited in their tunability. The
degree of customization of WS-based simulators can be increased by leveraging
the knowledge of the transmission matrix Jacucci  _et al._ (2022).

In this Letter, we demonstrate the control of the spin interaction length by
using thin diffusive media. By tuning the interaction length we observe
replica-to-replica fluctuations analogous to replica symmetry breaking (RSB)
in spin glasses Mézard  _et al._ (1984); Mezard  _et al._ . Our results
represent a relevant step toward the realization of fully-programmable Ising
machines on free-sapce optical-platforms, capable of solving complex spin-
glass Hamiltonians on a large scale.

Figure 1: Optical spin-glass simulator. A laser is collimated on a spatial
light modulator (SLM) and passes through a thin diffuser to then reach the
imaging plane. The surface of the SLM is directly imaged (via a 4f-system) on
the surface of the diffuser. The light is focused on two output modes. The
inset shows the two rows of the transmission matrix $(t_{im})_{im}$ summed and
then reshaped to SLM dimensions. The light reaching the pink (resp. yellow)
focus can only come from certain pixels within the pink (resp. yellow) cone on
the SLM plane ($\sigma_{i}$), of which the extent is given by the distance $d$
and/or the diffuser’s matrix $t_{im}$, corresponding to solving an Ising
Hamiltonian with local couplings. Figure 2: Tuning the spin correlation length
by optical propagation. a) Schematic representation of the setup in two
different distance configurations: $d_{1}$ in red and $d_{2}$ in orange, where
the colored cones represent where the light reaching one CCD pixel comes from
on the SLM. b) Simulation (first col) and experimental (second col) spin maps
at three different imaging plane CCD-DIFF distances. The further away the
plane, the wider the correlation. c) The amplitudes of the experimental
transmission matrices at these same three distances.

A system of $N$ coupled spins is described by the Hamiltonian in Equation 1.
For the Sherrington-Kirkpatrick model Nishimori (2001), the couplings
($J_{ij}$) are all-to-all random couplings, and are drawn from a Gaussian
distribution. Finding the ground-state of Equation 1 is an NP-hard problem.
This problem can be mapped onto a physical hardware thanks to coherent light-
propagation in disordered medium. In this configuration, one can show that the
transmitted intensity after a multiply-scattering medium takes the form
Pierangeli  _et al._ (2021):

$H=-\sum_{i,j=1}^{N}J_{ij}\sigma_{i}\sigma_{j}=-I$ (1)

with $J_{ij}=-\sum_{m}^{M}\Re{\overline{t_{im}}t_{jm}}$, where $m$ runs on the
output modes and $t_{im}$ and $t_{jm}$ are transmission matrix ($T$) elements
of the random medium. The matrix $T$ links the input modes on the spatial
light modulator (SLM) ($E_{in}$) and the output modes on the camera
($E_{out}$) via $E_{out}=TE_{in}$ Popoff  _et al._ (2010). The spins are
encoded on the input modes (pixels) with a binary phase of $0$ and $\pi$,
corresponding to $\pm 1$ spin states, respectively Pierangeli  _et al._
(2021); Leonetti  _et al._ (2021). Thanks to the physical system described in
Figure 1 and using an optimization algorithm one can find the ground state of
the Ising problem Pierangeli  _et al._ (2019).

Figure 3: Controlling the interaction between spin clusters. a) Simulation
(first columns) and experimental (second columns) spin ground states. b) TM
amplitudes obtained summing all rows of the measured TM. Two isolated regions
of magnetized spins (when optimizing over to two foci at the CCD plane) as
displayed on the SLM (first row). As the distance between CCD and diffuser
increases the extent of the magnetized regions also increases, the regions
start to overlap and therefore interact (second row).

color=yellow]fig 3 the red looks faded compared to fig2

The couplings distribution ($J_{ij}$) can be tuned by using the knowledge of
the transmission matrix $T$ Jacucci  _et al._ (2022). In the present work, we
use thin diffusive media Hobbs (2022); Sebbah (2001). These allow us —in
contrast to thick scattering media Pierangeli  _et al._ (2021, 2019, 2020)— to
control the extent of spins contributing to the Hamiltonian. The further the
camera is from the diffuser, the more the speckle pattern will spatially
expand and the more the light coming from various input modes is mixed. In
terms of couplings, it means that for a given input mode, the number of
connected other input modes will increase as we get further from the medium.
This effect can be used to control the spatial extent of the spins’ couplings.
The insets in Figure 2 show a graphical representation of the amount of pixels
contributing to the intensity on one single CCD pixel at two different
distances. In terms of spin couplings, it means that for a given spin, the
number of connected spins will increase as we get further from the medium.
This effect can be used to control the spatial extent of the spin interaction.
Remarkably, there exists two extreme points. The first one being when the
distance is such that the speckle is fully developed, leading to an all-to-all
coupled spin system (e.g. at position $d_{2}$ in Figure 2). The second one is
when the distance is such that the spins are not coupled at all (e.g. at
position $d_{1}$ in Figure 2). In this situation, the distance is so small
that the light reaching the specific output can only come from one input mode
(or SLM macro pixel). In other words, it is an Ising system of N decoupled
spins.

Figure 4: Controlling frustration and replica symmetry breaking. a) Two
regions of fixed size are increasingly getting closer on the SLM plane. Each
point corresponds to ten replicas for one given distance. The graph shows the
maximum of the probability distribution of the correlation between replicas
$q_{max}$ as a function of the distance between the regions (in px on the CCD
image). Replica-to-replica fluctuations vanish when the correlation is close
to 1. b) and c) Histograms of the ground-state correlation for different
replicas at maximum and minimum distance respectively.

In Reference Jacucci  _et al._ (2022) we demonstrate the ability to tune the
Hamiltonian of the system such that we obtain magnetized ground states for a
given set of couplings. As shown in Figure 2, we apply the same technique to
thin diffusers to tune the couplings such that we obtain a ground state with a
localized magnetization. Figure 2a shows, for both simulations and
experiments, three ground states for three different (increasing) distances
between diffuser and CCD. As shown in Figure 2, we observe that the magnetized
region is localized and has a finite size, whereas fully-magnetized ground
states are expected for a thick medium Jacucci  _et al._ (2022). Figure 2c
offers a preview of the amplitude of the transmission matrix coefficients
($|T|$) for the three previous distances. It is evident here that the light
that reaches this given CCD pixel (chosen for the optimization) comes from
only a subset of the input modes (i.e. spins). Secondly, one can note that the
interaction length between the spins grows with the distance to the diffuser
(c.f. Figure 2c).

The ability to tune the spatial extent of the region where spins are mutually
coupled allows us to create several clusters and explore the interactions
between them by selecting two pixels at some distance, and optimizing over the
sum of their intensities. We define the interaction area as the zone where the
two regions overlap. The spins within this area contribute to both foci at the
two different camera-pixels corresponding respectively to the two regions.
Figure 3 displays an example of two regions, varying their distance and the
interaction length. In detail, Figure 3a shows the ground states, for both
simulations and experiments, obtained for the whole system (both regions) and
for two different distances between diffuser and CCD. The first column shows
the case where the two regions are not overlapping, the second column shows
the case where the two regions are overlapping. Figure 3b shows the
transmission matrix amplitudes for the two previous cases. We can also tune
the overlap between spin clusters by keeping the distance $d$ fixed and by
translating the two pixels on the CCD — effectively optimizing for two closer
pixels.

We finally investigate the probability of the two interaction regions being
magnetized in an uncorrelated way as their distance varies. When regions are
not coupled, they are independent and the magnetization is thus the same
$50\%$ of the time. When they get closer, their magnetization tends to be
correlated. This is due the fact that the regions are not independent anymore,
as there is an overlap between them, corresponding to a coupling between the
two regions. To quantify the correlations between various replica, we
defined—in analog manner as the Parisi order parameter Pierangeli  _et al._
(2017); Ghofraniha  _et al._ (2015); Mézard and Young (1992); Mézard  _et al._
(1984); Parisi (1983)—the following metric for fully-magnetized spin
configurations:

$q_{\alpha\beta}=1-\frac{|\sum_{i=1}^{N}X^{\alpha}_{i}-X^{\beta}_{i}|}{N}$

where $\alpha$ and $\beta$ refer to two replicas, $N$ is the number of spins
in the region of interest and $X^{\alpha,\beta}_{i}\in\\{-1,1\\}$ are spin
configurations. The zone of interest in which the correlation between replica
ground states is calculated is a smaller region than the full input modes mask
(c.f. green square in inset schematics of Figure 4a).

Figure 4a shows the correlation between replicas as a function of the distance
between the two regions. The correlation is defined as the maximum of the
probability distribution of the correlation between replicas $q_{max}$ (which
is defined as $q_{\text{max}}=\text{argmax}_{q}P(q)$, where $P$ represents the
distributioncolor=red]i need to renormalize for it to be a probability (1 must
be the max) of the degree of correlation $q$ between ground states). The
correlation is close to unity when replica-to-replica fluctuations drop to
$0$. Figure 4b and Figure 4c show the histograms of the correlation between
ground states of different replicas when the two regions are the furthest and
the closest respectively. We observe that by increasing the overlap between
the two areas —and therefore their interaction— the correlation increases, in
analogy with the replica symmetry breaking transition, typical of random spin
systems (c.f. Figure 4c). A simulation of our system can be found in the
Appendix C, describing some conditions on the overlap for which one observes
RSB-like behavior and what the assumptions on the couplings amplitude
distribution are.

In short, we have observed and demonstrated experimental control over the
interaction length of an Ising spin-glass system based on free-space optics
and disordered media. We have also shown that we can control the interaction
between two regions of spins and induce replica symmetry breaking.

The proposed system is a step towards encoding more complex Hamiltonians in
hope of solving more complex NP-problems. It is also a new platform for
studying replica symmetry breaking. Moreover, the degeneracy changing when
regions interact means that the systems seems to develop long-range couplings
from short-range couplings. This effect could be leveraged as a new type of
annealing approach. Indeed, one could anneal a subset of the system and then —
driven by the interaction — the whole system would anneal.

Furthermore, one could even consider generalizing this Ising system to a
Hopfield system Leonetti  _et al._ (2021) as our experimental setup is
algorithm-agnostic and the spins are defined with a continuous encoding,
therefore a continuous orientation of spins could be explored. Other platforms
could also be envisioned, such as using multiple SLMs or using non-linear or
more complex media to obtain more complex couplings distributions (bimodal,
multimodal, etc).

###### Acknowledgements.

L.D., G.J., R.P., D.P., C.C., and S.G. designed the project. L.D. carried out
experiments and data analysis, G.J. and R.P. numerical simulations. L.D. and
D.P. wrote the paper with contributions from all the authors. This project was
funded by the European Research Council under the grant agreement No. 724473
(SMARTIES). R.P. thanks Clare College, University of Cambridge for funding via
a Junior Research Fellowship.

## Methods

Experimental setup. We evaluated experimentally our approach for controlling
the couplings of the spin simulator (c.f. Figure 1) as well as its extension
with the additions described below. A laser (Coherent Sapphire SF 532,
$\lambda=532nm$) is directed onto a reflective phase-only, liquid-crystal SLM
(Meadowlark Optics HSP192-532, $1920\times 1152$ pixels, aggregated into
$N=256$ macro-pixels) divided into N macro-pixels (spins). The Fourier
transform of the modulated light is projected on the objective back focal-
plane (OBJ1, $10\times$, $\mathrm{NA}=0.1$) and focused on a scattering medium
(DIFF). As a scattering, medium we used a surface-diffuser that is
commercially available (Edmund, 12.5mm, 25°). In practice, using a thin
volumetric diffuser or combining a surface diffuser and free-space propagation
are equivalent in our scheme. The scattered light is then collected by a
second objective (OBJ2, $20\times$, $\mathrm{NA}=0.4$) and the transmitted
intensity is detected by a CCD camera (Basler acA2040-55$\mu$m, $2048\times
1536$ pixels). The spins and bias (from the TM Jacucci  _et al._ (2022)) are
encoded by a spatial light modulator (SLM) in a phase pattern whose binary
part is sequentially updated until the ground-state is reached. Note that for
the optimization any algorithm can be used, i.e., the setup is algorithm
agnostic, as the advantage of the aforementioned simulator resides in the
parallel measurements of the energy Pierangeli  _et al._ (2021).

Transmission matrix calculation and ground-state search. The transmission
matrix of the scattering medium was estimated as in Popoff  _et al._ (2010).
In detail, each row of the TM can be reconstructed by monitoring how the
intensity on a given CCD pixel changes when a phase modulation is applied to
the input patterns on the SLM. Those interferometry measurements provide the
TM. Taking the phase conjugate of this matrix gives the SLM mask necessary for
proper focusing Jacucci  _et al._ (2022). The TM is sensitive to translations
and rotations of the scattering medium as well as to the input and detection
hardware. In this work, we define the stability-time as a variation within
$10\%$ of it original value. This time (typically $\sim 120$ minutes) is long
enough to run our experiments but for larger systems one would need more
stable architectures. The ground-state search is conducted sequentially by
means of the recurrent digital feedback. Computation starts from a random
configuration of N binary macro-pixels (spins) on the SLM. The measured
intensity distribution determines the feedback signal. At each iteration, an
arbitrary batch of spins is randomly flipped if it increases the intensity at
a chosen output mode. The batch size decreases over the optimization
procedure, starting from $12\%$ of the pixels to a single pixel for the last
$\sim 600$ iterations.

Numerical methods. The numerical model used in this work is a generalization
of Pierangeli  _et al._ (2021). The optical SG is numerically simulated by
forming N pixel blocks (SLM plane). The initial optical field has a constant
amplitude, and its phase is a random configuration of N binary phases,
$\phi_{i}=0,\pi$. A gaussian i.i.d. transmission matrix T with random complex
numbers is generated. At each iteration, a randomly selected single spin is
flipped. The input phase is updated if the output total intensity increased
after the linear propagation of the field. The bias in the numerical framework
is calculated as in the experiment by starting from the knowledge of T.
Numerical evaluation of $I_{T}$ corresponds to a measurement with a detector
in a noiseless system. In general, within this scheme, $\sim 10N$ iterations
are sufficient for a good convergence, i.e., when focus intensity reaches a
plateau. All codes are implemented in MATLAB on an Intel processor with 14
cores running at 3.7 GHz and supported by 64 GB ram.

## References

  * Parisi (2006) G. Parisi, Proceedings of the National Academy of Sciences 103, 7948 (2006), https://www.pnas.org/doi/pdf/10.1073/pnas.0601120103 .
  * Takeda  _et al._ (2017) Y. Takeda, S. Tamate, Y. Yamamoto, H. Takesue, T. Inagaki, and S. Utsunomiya, Quantum Science and Technology 3, 014004 (2017).
  * Maccari  _et al._ (2019) I. Maccari, L. Benfatto, and C. Castellani, Phys. Rev. B 99, 104509 (2019).
  * Das and Chakrabarti (2008) A. Das and B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061 (2008).
  * Goto  _et al._ (2019) H. Goto, K. Tatsumura, and A. R. Dixon, Science Advances 5, eaav2372 (2019), https://www.science.org/doi/pdf/10.1126/sciadv.aav2372 .
  * Lenstra and Kan (1975) J. K. Lenstra and A. H. G. R. Kan, Journal of the Operational Research Society 26, 717 (1975), https://doi.org/10.1057/jors.1975.151 .
  * Lucas (2014) A. Lucas, Frontiers in Physics 2, 5 (2014).
  * Garey and Johnson (1990) M. R. Garey and D. S. Johnson,  _Computers and Intractability; A Guide to the Theory of NP-Completeness_ (W. H. Freeman & Co., USA, 1990).
  * Barahona (1982) F. Barahona, Journal of Physics A: Mathematical and General 15, 3241 (1982).
  * Bachas (1984) C. P. Bachas, Journal of Physics A: Mathematical and General 17, L709 (1984).
  * Nishimori (2001) N. Nishimori,  _Statistical Physics of Spin Glasses and In- formation Processing: An Introduction_ (Clarendon Press-Oxford, 2001).
  * Fu and Anderson (1986) Y. Fu and P. Anderson, Journal of Physics A 19, 1605 (1986).
  * Sherrington and Kirkpatrick (1975) D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975).
  * Jacucci  _et al._ (2022) G. Jacucci, L. Delloye, D. Pierangeli, M. Rafayelyan, C. Conti, and S. Gigan, Phys. Rev. A 105, 033502 (2022).
  * Pierangeli  _et al._ (2019) D. Pierangeli, G. Marcucci, and C. Conti, Phys. Rev. Lett. 122, 213902 (2019).
  * Pierangeli  _et al._ (2021) D. Pierangeli, M. Rafayelyan, C. Conti, and S. Gigan, Phys. Rev. Applied 15, 034087 (2021).
  * Pierangeli  _et al._ (2020) D. Pierangeli, G. Marcucci, and C. Conti, Optica 7, 1535 (2020).
  * Tommasi  _et al._ (2016) F. Tommasi, E. Ignesti, S. Lepri, and S. Cavalieri, Scientific Reports 6, 37113 (2016).
  * Ghofraniha  _et al._ (2015) N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, L. Leuzzi, and C. Conti, Nature Communications 6, 6058 (2015).
  * Johnson  _et al._ (2011) M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose, Nature 473, 194 (2011).
  * Boixo  _et al._ (2014) S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Nature Physics 10, 218 (2014).
  * Berloff  _et al._ (2017) N. G. Berloff, M. Silva, K. Kalinin, A. Askitopoulos, J. D. Töpfer, P. Cilibrizzi, W. Langbein, and P. G. Lagoudakis, Nature Materials 16, 1120 (2017).
  * Kalinin  _et al._ (2020) K. P. Kalinin, A. Amo, J. Bloch, and N. G. Berloff, Nanophotonics 9, 4127 (2020).
  * Harris  _et al._ (2018) R. Harris, Y. Sato, A. J. Berkley, M. Reis, F. Altomare, M. H. Amin, K. Boothby, P. Bunyk, C. Deng, C. Enderud, S. Huang, E. Hoskinson, M. W. Johnson, E. Ladizinsky, N. Ladizinsky, T. Lanting, R. Li, T. Medina, R. Molavi, R. Neufeld, T. Oh, I. Pavlov, I. Perminov, G. Poulin-Lamarre, C. Rich, A. Smirnov, L. Swenson, N. Tsai, M. Volkmann, J. Whittaker, and J. Yao, Science 361, 162 (2018).
  * Hamerly  _et al._ (2019) R. Hamerly, T. Inagaki, P. L. McMahon, D. Venturelli, A. Marandi, T. Onodera, E. Ng, C. Langrock, K. Inaba, T. Honjo, K. Enbutsu, T. Umeki, R. Kasahara, S. Utsunomiya, S. Kako, K. ichi Kawarabayashi, R. L. Byer, M. M. Fejer, H. Mabuchi, D. Englund, E. Rieffel, H. Takesue, and Y. Yamamoto, Science Advances 5, 0823 (2019).
  * Pierangeli  _et al._ (2017) D. Pierangeli, A. Tavani, F. Di Mei, A. J. Agranat, C. Conti, and E. DelRe, Nature Communications 8, 1501 (2017).
  * Waidyasooriya  _et al._ (2022) H. M. Waidyasooriya, Y. Ohma, and M. Hariyama, in  _2022 IEEE 15th International Symposium on Embedded Multicore/Many-core Systems-on-Chip (MCSoC)_ (2022) pp. 195–199.
  * Marandi  _et al._ (2014) A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, Nature Photonics 8, 937 (2014).
  * Haribara  _et al._ (2016) Y. Haribara, S. Utsunomiya, and Y. Yamamoto, Lecture Notes in Physics , 251–262 (2016).
  * McMahon  _et al._ (2016) P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Utsunomiya, K. Aihara, R. L. Byer, M. M. Fejer, H. Mabuchi, and Y. Yamamoto, Science 354, 614 (2016).
  * Inagaki  _et al._ (2016a) T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu, O. Tadanaga, H. Takenouchi, K. Aihara, K. ichi Kawarabayashi, K. Inoue, S. Utsunomiya, and H. Takesue, Science 354, 603 (2016a).
  * Böhm  _et al._ (2019) F. Böhm, G. Verschaffelt, and G. Van der Sande, Nature Communications 10, 3538 (2019).
  * Takesue  _et al._ (2020) H. Takesue, K. Inaba, T. Inagaki, T. Ikuta, Y. Yamada, T. Honjo, T. Kazama, K. Enbutsu, T. Umeki, and R. Kasahara, Phys. Rev. Applied 13, 054059 (2020).
  * Honjo  _et al._ (2021) T. Honjo, T. Sonobe, K. Inaba, T. Inagaki, T. Ikuta, Y. Yamada, T. Kazama, K. Enbutsu, T. Umeki, R. Kasahara, K. ichi Kawarabayashi, and H. Takesue, Science Advances 7, 0952 (2021).
  * Nixon  _et al._ (2013) M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, Phys. Rev. Lett. 110, 184102 (2013).
  * Hamerly  _et al._ (2020) R. Hamerly, T. Inagaki, P. L. McMahon, D. Venturelli, A. Marandi, D. R. Englund, and Y. Yamamoto, in  _AI and Optical Data Sciences_, Vol. 11299, edited by B. Jalali and K. ichi Kitayama, International Society for Optics and Photonics (SPIE, 2020) pp. 42 – 48\.
  * Calvanese Strinati  _et al._ (2021) M. Calvanese Strinati, L. Bello, E. G. Dalla Torre, and A. Pe’er, Phys. Rev. Lett. 126, 143901 (2021).
  * Gershenzon  _et al._ (2020) I. Gershenzon, G. Arwas, S. Gadasi, C. Tradonsky, A. Friesem, O. Raz, and N. Davidson, Nanophotonics 9, 4117 (2020).
  * Edri  _et al._ (2021) H. Edri, B. Raz, G. Fleurov, R. Ozeri, and N. Davidson, New Journal of Physics 23, 053005 (2021).
  * Inagaki  _et al._ (2016b) T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, Nature Photonics 10, 415 (2016b).
  * Honjo  _et al._ (2022) T. Honjo, K. Inaba, T. Inagaki, T. Ikuta, Y. Yamada, and H. Takesue, in  _2022 IEEE 22nd International Conference on Nanotechnology (NANO)_ (2022) pp. 405–408.
  * Cen  _et al._ (2022) Q. Cen, H. Ding, T. Hao, S. Guan, Z. Qin, J. Lyu, W. Li, N. Zhu, K. Xu, Y. Dai, and M. Li, Light: Science & Applications 11, 333 (2022).
  * (43) W. Sun, W. Zhang, Y. Liu, Q. Liu, and Z. He,  47, 1498.
  * Kumar  _et al._ (a) S. Kumar, H. Zhang, and Y.-P. Huang,  3, 1 (a).
  * Kumar  _et al._ (b) S. Kumar, Z. Li, T. Bu, C. Qu, and Y. Huang,  6, 1 (b).
  * (46) J. Huang, Y. Fang, and Z. Ruan,  4, 1.
  * (47) Y. Fang, J. Huang, and Z. Ruan,  127, 043902, 2011.02771 .
  * Mézard  _et al._ (1984) M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, Journal de Physique 45, 843 (1984).
  * (49) M. Mezard, G. Parisi, and M. Virasoro,  _Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications_, World Scientific Lecture Notes in Physics, Vol. 9 (WORLD SCIENTIFIC).
  * Popoff  _et al._ (2010) S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
  * Leonetti  _et al._ (2021) M. Leonetti, E. Hörmann, L. Leuzzi, G. Parisi, and G. Ruocco, Proceedings of the National Academy of Sciences 118 (2021).
  * Hobbs (2022) P. C. Hobbs,  _Building electro-optical systems: making it all work_ (John Wiley & Sons, 2022).
  * Sebbah (2001) P. Sebbah,  _Waves and imaging through complex media_ (Springer Science & Business Media, 2001).
  * Mézard and Young (1992) M. Mézard and A. P. Young, Europhysics Letters 18, 653 (1992).
  * Mézard  _et al._ (1984) M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, Phys. Rev. Lett. 52, 1156 (1984).
  * Parisi (1983) G. Parisi, Phys. Rev. Lett. 50, 1946 (1983).


# Images of Fractional Brownian motion with deterministic drift: Positive
Lebesgue measure and non-empty interior

###### Abstract

Let $B^{H}$ be a fractional Brownian motion in $\mathbb{R}^{d}$ of Hurst index
$H\in\left(0,1\right)$, $f:\left[0,1\right]\longrightarrow\mathbb{R}^{d}$ a
Borel function and $A\subset\left[0,1\right]$ a Borel set. We provide
sufficient conditions for the image $(B^{H}+f)(A)$ to have a positive Lebesgue
measure or to have a non-empty interior. This is done through the study of the
properties of the density of the occupation measure of $(B^{H}+f)$. Precisely,
we prove that if the parabolic Hausdorff dimension of the graph of $f$ is
greater than $Hd$, then the density is a square integrable function. If, on
the other hand, the Hausdorff dimension of $A$ is greater than $Hd$, then it
even admits a continuous version. This allows us to establish the result
already cited.

MOHAMED ERRAOUI

Department of mathematics, Faculty of science El jadida,

Chouaïb Doukkali University, Morocco

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

Department of mathematics, Faculty of science Semlalia,

Cadi Ayyad University, 2390 Marrakesh, Morocco

e-mail<EMAIL_ADDRESS>
Keywords: Fractional Brownian motion, Hausdorff dimension, Parabolic
dimension.

Mathematics Subject Classification Primary 60J65

##  1 Introduction

Many sets from the trajectories of the fractional Brownian motion are somehow
random fractals. These sets have an interesting geometric structure at all
scales. Among the most studied in the literature, it is worth mentioning
images and graphs of a fractional Brownian motion. For standard reference we
refer to Adler [1], Kahane [10]. So, we start by recalling the exact Hausdorff
dimension of the image $B^{H}(A)$ where $B^{H}$ is a fractional Brownian
motion in $\mathbb{R}^{d}$ of Hurst index $H\in\left(0,1\right)$ and
$A\subset\mathbb{R}$ is a Borel set. It is visibly that $B^{H}(A)$ is a random
set in $\mathbb{R}^{d}$. There has been considerable interest in the study of
its geometric and arithmetic properties. It is proved that almost surely

$\dim B^{H}(A)=\min\left\\{\dfrac{\dim(A)}{H},d\right\\}.$

We write $\dim(\centerdot)$ for the Hausdorff dimension, and refer to [6] and
[10] for its definition and basic properties. Several works have focused
strongly on determining whether $B^{H}(A)$ is a.s. a set of positive Lebesgue
measure or a Salem set by comparing $\dim(A)$ and $Hd$, see [10] and [18]. Two
distinct cases are clearly apparent : $\dim(A)>Hd$ or $\dim(A)<Hd$.

In the first case, a proof based on Fourier transform due to Kahane (see [10])
provided a preliminary result, namely $B^{H}(A)$ a.s. has positive
$d$-dimensional Lebesgue measure. Pitt [17] and Kahane [10] strengthened this
result by proving that $B^{H}(A)$ a.s. has non-empty interior. However, it
should be noted that, for the one dimensional Brownian motion the fact that
$B^{H}(A)$ has non-empty interior when $\dim(A)>1/2$ is due to Kaufman in
[11]. In the case $\dim(A)<Hd$, Kahane [10] established that $B^{H}(A)$ is
a.s. a Salem set. The reader might refer to the book of Kahane [10] for the
precise definition and the basic properties of Salem set since it will not be
treated here.

Recently Peres and Sousi [16] studied fractal properties of images and graphs
of $B^{H}+f$ where $f:[0,1]\rightarrow\mathbb{R}^{d}$ is a Borel measurable
function. They expressed, for a Borel set $A\subset[0,1]$, the dimension of
the image set $\left(B^{H}+f\right)(A)$ in terms of the so-called parabolic
Hausdorff dimension of the graph of $f$ restricted to $A$ denoted by
$\dim_{\Psi,H}(Gr_{A}(f))$, see Definition 2.1. Precisely, the authors stated
that almost surely

$\dim\left((B^{H}+f)(A)\right)=\min\left(\dfrac{\dim_{\Psi,H}(Gr_{A}(f))}{H},d\right).$

It is important to note that the parabolic Hausdorff dimension has first been
considered by Taylor and Watson [19] for the study of polar sets of the heat
equation. It has also been shown to be useful for analyzing the geometry of
the images of Brownian motion, see Khoshnevisan and Xiao [13]. Once again when
$B^{H}$ is the multifractional Brownian motion (mBm), the parabolic Hausdorff
dimension again happens to be the relevant tool to describe the geometry of
its graph and the images of sets, cf [3]. For all these reasons, it is
therefore quite natural to ask the following question: what are the properties
of the set $\left(B^{H}+f\right)(A)$ that can be explored deeply from the
informations provided by the parabolic dimension of the graph of $f$? This
issue will be the main goal of this work, which we consider as a continuation
of [16].

In Section 2, first we give a comparison result for parabolic Hausdorff
dimensions for different parameters. We look also seek to determine the exact
value of $\dim_{\Psi,H}(Gr_{A}(f))$. It is already becoming apparent that
calculation of parabolic Hausdorff dimensions can be a little involved, even
for simple sets. For this, as a first step we establish lower and upper bounds
of $\dim_{\Psi,H}(Gr_{A}(f))$, that take into account the Hölder continuity
and $\dim(A)$. We note that the lower bound of $\dim_{\Psi,H}(Gr_{A}(f))$ can
be seen as a kind of the Mastrand projection theorem. Secondly, in order to
give sharp bounds, we looked for functions with known graphs, such as the
paths of Hölder’s continuous stochastic process. Precisely, we have considered
the paths of the fractional Brownian motion with a Hurst parameter less than
$H$. For this class we were able to calculate the exact value of the desired
parabolic Hausdorff dimension.

In this section 3, our investigation focuses on the fractional Brownian motion
with drift $\left(B^{H}+f\right)$ and turned to the question of whether the
image $\left(B^{H}+f\right)(A)$ admit a positive Lebesgue measure or have a
non-empty interior. The question we are examining arises from the subject of
the polar functions of the planar Brownian motion initiated by Graversen [9]
and explored in a more general context, including both the spatial dimension
and the Hölder property, by Le Gall [14] and Antunović et al [2]. Here the
idea is to construct, under the condition $\dim_{\Psi,H}(Gr_{A}(f))>Hd$, a
probability $\nu$ carried by $A$ such that its image measure $\mu$ under the
mapping $A\ni t\longrightarrow\left(B^{H}+f\right)(t)$, commonly known as
occupation measure, which is supported by $\left(B^{H}+f\right)(A)$ has a
density with respect to the Lebesgue measure. This latter density is known as
occupation density and interpreted as local time over the set $A$. The idea of
introducing local times as densities of occupation measure has been fruitful
in a variety of contexts. Important papers in this direction are Geman and
Horowitz [7] and Geman, Horowitz and Rosen [8]. Our first result indicates
that image $\left(B^{H}+f\right)(A)$ admit a positive Lebesgue measure since
the occupation density is a square integrable function. In terms of the second
part of our question, we will follow the approach used in Kahane [10] and Pitt
[17]. The keystone of this approach is to reduce the problem of covering an
open set to the existence of a continuous occupation density function. To do
this, we’ll draw upon on the condition used in the case without drift, which
is a little stronger than that already assumed, namely $\dim(A)>Hd$, see
Propositon 2.7. It should be noted that the results of this section are indeed
extensions of those established by Pitt and Kahane for fractional Brownian
motion to the case of fractional Brownian motion with variable drift through
the comparison between $\dim_{\Psi,H}(Gr_{A}(f))$ and $Hd$.

Our motivation for section 4 is as follows: in his note [14] on Graversen’s
conjecture on the polar functions for the planar Brownian motion, Le Gall
asked if for each $\gamma<1/d$ there exist $\gamma$-Hölder continuous
functions which are non-polar for $d$-dimensional Brownian motion
$B^{\nicefrac{{1}}{{2}}}$. An answer is given by Antunović et al [2] in terms
of modifications of the standard Hilbert curve perturbed by Brownian motion.
Precisely, they showed that for any $\gamma<1/d$, there exists a
$\gamma$-Hölder continuous function $f:[0,1]\longrightarrow\mathbb{R}^{d}$ for
which the set
$\left(B^{\nicefrac{{1}}{{2}}}-f\right)\left(\left[0,1\right]\right)$ covers
an open set almost surely. On the other hand, as mentioned before when
$\dim(A)<Hd$, $B^{H}(A)$ is a.s. a Salem set. It is natural to ask whether for
each $\alpha\in\left(0,\dim(A)/d\right)$ there exist $\alpha$-Hölder
continuous function $f:[0,1]\rightarrow\mathbb{R}^{d}$ for which the range
$(B^{H}+f)(A)$ has a non-empty interior a.s.?

Since $\dim(A)>\alpha d$ then the previous case allows us to assert that the
set $\left(B^{\alpha}+f\right)(A)$ have a non-empty interior where
$B^{\alpha}$ is another fractional Brownian motion with Hurst index possibly
defined on different probability space. Our idea is to consider $B^{H}$ as a
drift of $B^{\alpha}$. This induces us to bring them together on the same
space while preserving their distributions. The best way to do this is to work
on the product space and to consider processes on this latter.

Here are some notations that we will use throughout this work.
$\lVert\cdot\rVert_{\infty}$ denotes the maximum norm on $\mathbb{R}^{d}$ .
$\left\langle\cdot,\cdot\right\rangle$ and $\lVert\cdot\rVert$ are the
ordinary scalar product and the Euclidean norm in $\mathbb{R}^{d}$
respectively. If $(E,\rho)$ is a metric space, then the Borel $\sigma$-algebra
over $E$ will be denoted by $\mathcal{B}(E)$. We denote by $\dim A$ the
Hausdorff dimension of a set $A\subset\mathbb{R}$. For a function
$f:\mathbb{R}_{+}\rightarrow\mathbb{R}^{d}$, $Gr_{A}(f)=\\{(t,f(t)):t\in A\\}$
is the graph of $f$ over the set $A$. We will use $C,C_{1},\ldots,C_{4}$ to
denote unspecified positive finite constants which may not necessarily be the
same in each occurrence.

##  2 Preliminaries on Parabolic Hausdorff dimension

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

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

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

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

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

$\displaystyle\Psi_{H}^{\beta}(F)=\inf\left\\{\sum_{j}\delta_{j}^{\beta}:F\subseteq\underset{j}{\cup}\left[a_{j},a_{j}+\delta_{j}\right]\times\left[b_{j,1},b_{j,1}+\delta_{j}^{H}\right]\times\ldots\times\left[b_{j,d},b_{j,d}+\delta_{j}^{H}\right]\right\\}$
(2.1)

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

$\operatorname{dim}_{\Psi,H}(F)=\inf\left\\{\beta:\Psi_{H}^{\beta}(F)=0\right\\}.$

######  Remark 2.2.

Let $\rho_{H}$ be the metric defined on $\mathbb{R}_{+}\times\mathbb{R}^{d}$
by

$\rho_{H}((s,x),(t,y))=\max\\{|s-t|^{H},\lVert
x-y\rVert_{\infty}\\}\quad\forall(s,x),(t,y)\in\mathbb{R}_{+}\times\mathbb{R}^{d}.$
(2.2)

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

$\displaystyle\mathcal{H}_{\rho_{H}}^{\beta}(F)=\inf\left\\{\sum_{j}diam(U_{j})^{\beta}:F\subseteq\underset{j}{\cup}\,U_{j}\right\\}$
(2.3)

where $\left\\{U_{j}\right\\}$ 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 diameter of the set
$\left[a_{j},a_{j}+\delta_{j}\right]\times\left[b_{j,1},b_{j,1}+\delta_{j}^{H}\right]\times\ldots\times\left[b_{j,d},b_{j,d}+\delta_{j}^{H}\right]$
is $\delta_{j}^{H}$ then it can be seen without difficulty that for any
$\beta>0$ and $F\subseteq\mathbb{R}_{+}\times\mathbb{R}^{d}$ we have

$\displaystyle\Psi_{H}^{\beta}(F)=\mathcal{H}_{\rho_{H}}^{\beta/H}(F).$ (2.4)

Hence we obtain

$\dim_{\Psi,H}(F)=H\times\dim_{\rho_{H}}(F).$ (2.5)

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

$\displaystyle\dim_{\Psi,H}(F)=\sup\\{\beta:\mathcal{C}_{\rho_{H},\beta/H}(F)>0\\}=\inf\\{\beta:\mathcal{C}_{\rho_{H},\beta/H}(F)=0\\},$
(2.6)

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

$\mathcal{C}_{\rho_{H},\beta}(F)=\left[\inf_{\mu\in\mathcal{P}(F)}\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\frac{\mu(du)\mu(dv)}{(\rho_{H}(u,v))^{\beta}}\right]^{-1}.$
(2.7)

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 Theorem 3.4.2
in [4] or Theorem 4.32 in [15] . ∎

The next theorem is the analogue of Frostman’s theorem for parabolic Hausdorff
dimension. The statement can be found in Taylor and Watson in [19, 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

$\mu\left([a,a+\delta]\times\prod_{j=1}^{d}\left[b_{j},b_{j}+\delta^{H}\right]\right)\leq
C\delta^{\kappa},$ (2.8)

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

Now we give a comparison result for the Hausdorff parabolic dimensions with
different parameters. We note that it is a generalization of Lemma 5.4 in [3]
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 $H,H^{\prime}\in(0,1)$
such that $H<H^{\prime}$. Then we have

$\dim_{\Psi,H}(F)\vee\left(\dfrac{H^{\prime}}{H}\dim_{\Psi,H}(F)+1-\dfrac{H^{\prime}}{H}\right)\leq\dim_{\Psi,H^{\prime}}(F)\leq\left(\frac{H^{\prime}}{H}\dim_{\Psi,H}(F)\right)\wedge\left(\dim_{\Psi,H}(F)+(H^{\prime}-H)d\right).$
(2.9)

An equivalent reformulation is

$\dfrac{H}{H^{\prime}}\dim_{\rho_{H}}(F)\vee\left(\dim_{\rho_{H}}(F)+\dfrac{1}{H^{\prime}}-\dfrac{1}{H}\right)\leq\dim_{\rho_{H^{\prime}}}(F)\leq\left(\dim_{\rho_{H}}(F)\wedge\left(d+\dfrac{H}{H^{\prime}}\left(\dim_{\rho_{H}}(F)-d\right)\right)\right).$

###### 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

$\dim_{\Psi,H}(F)\leq\dim_{\Psi,H^{\prime}}(F).$ (2.10)

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

${\displaystyle\underset{n\geq 1}{\sum}\delta_{n}^{\gamma
H^{\prime}+1}\leq\varepsilon.}$ (2.11)

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

${\displaystyle\underset{l\geq
1}{\sum}\,\left(\delta_{l}^{H^{\prime}/H}\right)^{\gamma H+1}\leq
2\underset{n\geq
1}{\sum}\,\delta_{n}^{1-\frac{H^{\prime}}{H}}\,\left(\delta_{n}^{H^{\prime}/H}\right)^{\gamma
H+1}\leq 2\varepsilon.}$ (2.12)

From $\eqref{eq:Haus-H}$ we deduce that $\dim_{\Psi,H}(F)\leq\gamma H+1$,
which implies $\dfrac{\dim_{\Psi,H}(F)-1}{H}\leq\gamma$. Therefore letting
$\gamma$ go to $\dfrac{\dim_{\Psi,H^{\prime}}(F)-1}{H^{\prime}}$ we conclude

$\dfrac{H^{\prime}}{H}\dim_{\Psi,H}(F)+1-\dfrac{H^{\prime}}{H}\leq\dim_{\Psi,H^{\prime}}(F).$
(2.13)

Combining (2.10) and (2.13) , we obtain the lower bound

$\dim_{\Psi,H}(F)\vee\left(\dfrac{H^{\prime}}{H}\dim_{\Psi,H}(F)+1-\dfrac{H^{\prime}}{H}\right)\leq\dim_{\Psi,H^{\prime}}(F).$

For the second inequality let $\kappa<\dim_{\Psi,H^{\prime}}(F)$. Then by
Frostman’s theorem 2.4 there exists a probability measure $\mu$ supported on
$F$ such that (2.8) is satisfied. Our aim is to show that

$\displaystyle\mu\left(\left[a,a+\delta\right]\times\prod_{j=1}^{d}\left[b_{j},b_{j}+\delta^{H}\right]\right)\leq
C\left(\delta^{\kappa+d(H-H^{\prime})}\wedge\delta^{\kappa
H/H^{\prime}}\right),$ (2.14)

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

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

Using (2.8) we obtain

$\displaystyle\mu\left(\left[a,a+\delta\right]\times\prod_{j=1}^{d}\left[b_{j},b_{j}+\delta^{H}\right]\right)\leq
C\delta^{\kappa\,H/H^{\prime}}.$ (2.15)

Now, for any $j=1,...,d$, the interval $[b_{j},b_{j}+\delta^{H}]$ can be
covered by at most $\delta^{(H-H^{\prime})}$ interval of length
$\delta^{H^{\prime}}$. Using once again (2.8) we deduce that

$\displaystyle\mu\left(\left[a,a+\delta\right]\times\prod_{j=1}^{d}\left[b_{j},b_{j}+\delta^{H}\right]\right)\leq
C\delta^{\kappa+d(H-H^{\prime})}.$ (2.16)

Thus the inequality (2.14) 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 Theorem $4.19$ in
[15]), implies that

$\dim_{\rho_{H}}(F)\geq\dfrac{\kappa}{H^{\prime}}\vee\dfrac{\kappa+d(H-H^{\prime})}{H}.$

From (2.5) it follows that

$\kappa\leq\left(\frac{H^{\prime}}{H}\dim_{\Psi,H}(F)\right)\wedge\left(\dim_{\Psi,H}(F)+d(H^{\prime}-H)\right).$

Therefore letting $\kappa\uparrow\dim_{\psi,H^{\prime}}(F)$ the assertion
(2.9) follows. ∎

######  Remark 2.6.

It is easy to see that

$\dim_{\Psi,H}(F)\vee\left(\dfrac{H^{\prime}}{H}\dim_{\Psi,H}(F)+1-\dfrac{H^{\prime}}{H}\right)=\left\\{\begin{array}[]{llc}\dim_{\Psi,H}(F)&\text{if}&\dim_{\Psi,H}(F)\leq
1\\\ \\\
\dfrac{H^{\prime}}{H}\dim_{\Psi,H}(F)+1-\dfrac{H^{\prime}}{H}&\text{if}&\dim_{\Psi,H}(F)>1.\end{array}\right.$

Hence the lower bound in (2.9) is sharp as long as $\dim_{\Psi,H}(F)>1$.

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

######  Proposition 2.7.

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

$\dim(A)\leq\dim_{\Psi,H}(Gr_{A}(f)).$ (2.17)

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

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

then we have

$\dim_{\Psi,H}(Gr_{A}(f))\leq\left(\frac{H}{\alpha}\dim(A)\right)\wedge\left(\dim(A)+(H-\alpha)d\right).$
(2.18)

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

$\dim(A)=\dim_{\Psi,H}(Gr_{A}(f))$ (2.19)

###### Proof.

We begin by proving (2.17). Let $\gamma>\dim_{\Psi,H}(Gr_{A}(f))$. Since
$\Psi_{H}^{\gamma}(Gr_{A}(f))=0$ 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 $Gr_{A}(f)$ 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

$\sum_{l\geq 1}\left|[a_{l},a_{l}+\delta_{l}]\right|^{\gamma}\leq\varepsilon.$

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

$\displaystyle\mathcal{H}^{\gamma}(A):=\inf\left\\{\sum_{j}\lvert
U_{j}\rvert^{\gamma}:A\subseteq\underset{j}{\cup}\,U_{j}\right\\}\leq\varepsilon.$
(2.20)

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

$\mu\left([a,a+\delta]\times\prod_{j=1}^{d}\left[b_{j},b_{j}+\delta^{H}\right]\right)\leq
C\delta^{\kappa},$ (2.21)

for any $\left(a,b_{1},\cdots,b_{d}\right)\in A\times\mathbb{R}^{d}$ 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

$\nu([a,a+\delta])=\mu(P_{1}^{-1}([a,a+\delta]))\leq
C_{1}\left(\delta^{\kappa+d(\alpha-H)}\,\wedge\,\delta^{\kappa\,\alpha/H}\right),$
(2.22)

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
$f_{j}(s)\in[f_{j}(a)-K\delta^{\alpha},f_{j}(a)+K\delta^{\alpha}]$. There are
two ways to cover

$P_{1}^{-1}\left([a,a+\delta]\right)=\left\\{(s,f(s))\in
Gr_{A}(f);s\in[a,a+\delta]\right\\}.$

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 $\eqref{ineq Frostman parabolic-1}$ we conclude that there exists a
constant $C^{\prime}>0$ which depends only on $K$ and $d$, such that

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

for the first cover. For the second one we get

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

where $C^{\prime\prime}>0$ is a constant depending only on $K$ and $d$. Thus
inequality (2.22) is proved. Thanks to the mass distribution principle, see
Theorem $4.19$ in [15], 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
(2.18) follows.

Finally it is easy to see from (2.17) and (2.18) that when $f$ is
$\left(H-\varepsilon\right)$-Hölder continuous for all $\varepsilon>0$ we have
$\dim(A)=\dim_{\Psi,H}(Gr_{A}(f))$. ∎

######  Remark 2.8.

It is worth noting that the inequality (2.18) is also obtained by assuming
only that the function $f$ is $(\alpha-\varepsilon)$-Hölder continuous for all
$\varepsilon>0$.

A natural question that arises from Proposition 2.7 whether (2.18) is a sharp
estimate? The main idea is to use the paths of Hölder continuous stochastic
process. Precisely, we have the following result

######  Theorem 2.9.

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

$\displaystyle\dim_{\Psi,H}(Gr_{A}(B^{\alpha}))=\left(\left(\frac{H}{\alpha}\dim(A)\right)\wedge(\dim(A)+d(H-\alpha))\right)\text{
a.s.}$ (2.23)

Moreover, if $\alpha<H$ we have

$\dim_{\Psi,H}(Gr_{A}(B^{\alpha}))>\dim(A).$ (2.24)

For the proof we need the following lemma.

######  Lemma 2.10.

There exists a constants $C$ such that for all $t\in(0,1]$ we have

$\mathbb{E}\left[\frac{1}{(\max\\{t^{H},\lVert
B^{\alpha}(t)\rVert_{\infty}\\})^{\gamma/H}}\right]\leq\left\\{\begin{array}[]{lll}C\,t^{-\gamma\alpha/H}\mbox{
}&\text{if}&\gamma<Hd,\\\ \\\ C\,t^{d(H-\alpha)-\gamma}\mbox{
}&\text{if}&\gamma>Hd.\end{array}\right.$ (2.25)

###### Proof.

We first note that the denominator on the left hand side of (2.25) 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

$\begin{array}[]{lll}\mathbb{E}\left[\dfrac{1}{\left(\max\left(t^{H},t^{\alpha}\lVert
N\rVert_{\infty}\right)\right)^{\gamma/H}}\right]&=&\dfrac{1}{t^{\gamma}}\mathbb{P}\left[\lVert
N\rVert_{\infty}\leq
t^{H-\alpha}\right]+\dfrac{1}{t^{\gamma\alpha/H}}\mathbb{E}\left[\dfrac{1}{\lVert
N\rVert_{\infty}^{\gamma/H}}1\\!\\!1_{\\{\lVert
N\rVert_{\infty}>t^{H-\alpha}\\}}\right]\\\ \\\
&\leq&C\left(\dfrac{1}{t^{\gamma}}\displaystyle\int_{\lVert
y\rVert_{\infty}\leq t^{H-\alpha}}\,e^{-\lVert
y\rVert^{2}/2}dy+\dfrac{1}{t^{\gamma\alpha/H}}\displaystyle\int_{\lVert
y\rVert_{\infty}>t^{H-\alpha}}\,\frac{e^{-\lVert y\rVert^{2}/2}}{\lVert
y\rVert_{\infty}^{\gamma/H}}dy\right)\\\ \\\
&\leq&C_{1}\left(t^{d(H-\alpha)-\gamma}+t^{-\gamma\alpha/H}\displaystyle\int_{t^{H-\alpha}}^{1}r^{d-1-\gamma/H}dr+t^{-\gamma\alpha/H}\right)\\\
\\\ &\leq&C_{2}\left(t^{d(H-\alpha)-\gamma}+t^{-\gamma\alpha/H}\right)\\\ \\\
&\leq&C_{3}\,t^{d(H-\alpha)-\gamma},\end{array}$

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

###### Proof of Theorem 2.9.

The upper bound of $\dim_{\Psi,H}(Gr_{A}(B^{\alpha}))$ follows directly from
Proposition 2.7.

Now let us prove the lower bound. First, we consider the case
$\dim(A)\leq\alpha d$. Let
$\gamma<\frac{H}{\alpha}\dim(A)\leq\dim(A)+d(H-\alpha)$. By Theorem 4.32 in
[15], there exists a probability measure $\nu$ on $A$ such that

$\displaystyle\mathcal{E}_{\gamma\alpha/H}(\nu):=\int_{A}\int_{A}\frac{1}{|t-s|^{\gamma\alpha/H}}\nu(ds)\nu(dt)<\infty.$
(2.26)

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

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

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 $\left(\max\left(|s-t|^{H},\lVert
B^{\alpha}(|s-t|)\rVert_{\infty}\right)\right){}^{\gamma/H}$. It follows that

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

Since $\gamma<Hd$ we deduce from Lemma 2.10 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 (2.26). 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
$Hd<\gamma<\dim(A)+d(H-\alpha)<\frac{H}{\alpha}\dim(A)$. 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 $\gamma\uparrow(\frac{H}{\alpha}\dim(A))\wedge(\dim(A)+d(H-\alpha))$
finishes the proof. ∎

As a consequence of (2.5) and Theorem 2.9, we have the following result

######  Corollary 2.11.

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

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

##  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 $[0,1]\ni t\longrightarrow Y(t)(w)\in\mathbb{R}^{d}$ is defined by

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

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 Theorem 21.15 in
[7], 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:

$\displaystyle 1.$
$\displaystyle\nu_{Y}<<\lambda_{d}\,\,\,\text{with}\,\,\,\frac{d\nu_{Y}}{d\lambda_{d}}(.)\in
L^{2}(\lambda_{d}\otimes\mathbb{P}).$ $\displaystyle 2.$
$\displaystyle\liminf_{r\downarrow
0}\,r^{-d}\int_{A}\int_{A}\mathbb{P}\left\\{\lVert
Y(s)-Y(t)\rVert<r\right\\}\,d\nu(s)\,d\nu(t)<\infty.$

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

$\mu\left([a,a+\delta]\times\prod_{j=1}^{d}\left[b_{j},b_{j}+\delta^{H}\right]\right)\leq
C\delta^{\kappa},$ (3.1)

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
$\nu=\mu\circ P_{1}^{-1}$. We are now ready to state

######  Theorem 3.2.

Let $\\{B^{H}(t):t\in[0,1]\\}$ be a $d$-dimensional fractional Brownian motion
of Hurst index $H\in(0,1)$. Let $f:[0,1]\rightarrow\mathbb{R}^{d}$ be a Borel
measurable function and let $A\subset[0,1]$ be a Borel set. If
$\eta:=\dim_{\Psi,H}(Gr_{A}(f))>Hd$ then $\lambda_{d}((B^{H}+f)(A))>0$ almost
surely.

###### 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

$\displaystyle\mathbb{P}\left\\{\lVert(B^{H}+f)(s)-(B^{H}+f)(t)\rVert<r\right\\}=\dfrac{1}{(2\pi)^{d/2}|t-s|^{Hd}}\int_{\lVert
y\rVert<r}\exp\left(-\frac{\lVert
y-f(t)+f(s)\rVert^{2}}{2|t-s|^{2H}}\right)dy.$

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

$\displaystyle\displaystyle\int_{A}\mathbb{P}\left\\{\lVert(B^{H}+f)(s)-(B^{H}+f)(t)\rVert<r\right\\}\,d\nu(s)$
$\displaystyle=\dfrac{1}{(2\pi)^{d/2}}\int_{\lVert
y\rVert<r}\int_{A}\dfrac{1}{|t-s|^{Hd}}\exp\left(-\frac{\lVert
y-f(t)+f(s)\rVert^{2}}{2|t-s|^{2H}}\right)\,d\nu(s)\,dy$ $\displaystyle\leq
C\,r^{d}\sup_{\lVert
y\rVert<r}\underset{=I(y,\left(t,f(t))\right)}{\underbrace{\int_{A}\,\dfrac{1}{|t-s|^{Hd}}\exp\left(-\frac{\lVert
y-f(t)+f(s)\rVert^{2}}{2|t-s|^{2H}}\right)\,d\nu(s)}}.$ (3.2)

where $C$ is a positive constant depending only on $d$.

For any fixed $y$ in $\left\\{\lVert y\rVert<r\right\\}$ 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

$\begin{array}[]{l}I_{1}\left(y,\left(t,f(t)\right)\right)=\displaystyle\int_{\left\\{s\in
A:\lVert y-f(t)+f(s)\rVert\leq
C_{1}\,|s-t|^{H}\sqrt{|\log|s-t||}\right\\}}\dfrac{1}{|s-t|^{Hd}}\exp\left(-\frac{\lVert
y-f(t)+f(s)\rVert^{2}}{2|s-t|^{2H}}\right)\,d\nu(s)\\\ \\\
=\displaystyle\int_{\left\\{(s,f(s))\in Gr_{A}(f):\lVert y-f(t)+f(s)\rVert\leq
C_{1}\,|s-t|^{H}\sqrt{|\log|s-t||}\right\\}}\dfrac{1}{|s-t|^{Hd}}\exp\left(-\frac{\lVert
y-f(t)+f(s)\rVert^{2}}{2|s-t|^{2H}}\right)\,d\mu(s,f(s)).\end{array}$

and

$\begin{array}[]{l}I_{2}\left(y,\left(t,f(t)\right)\right)=\displaystyle\int_{\left\\{s\in
A:\lVert
y-f(t)+f(s)\rVert>C_{1}\,|s-t|^{H}\sqrt{|\log|s-t||}\right\\}}\dfrac{1}{|s-t|^{Hd}}\exp\left(-\frac{\lVert
y-f(t)+f(s)\rVert^{2}}{2|s-t|^{2H}}\right)\,d\nu(s)\\\ \\\
=\displaystyle\int_{\left\\{(s,f(s))\in Gr_{A}(f):\lVert
y-f(t)+f(s)\rVert>C_{1}\,|s-t|^{H}\sqrt{|\log|s-t||}\right\\}}\dfrac{1}{|s-t|^{Hd}}\exp\left(-\frac{\lVert
y-f(t)+f(s)\rVert^{2}}{2|s-t|^{2H}}\right)\,d\mu(s,f(s)),\end{array}$

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

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

By the inequality (3.1) we can verify that the measure $\nu$ is no atomic.
Then we have

$\displaystyle I_{1}(y,\left(t,f(t))\right)$
$\displaystyle\leq\sum_{n=1}^{\infty}\,2^{nHd}\,\mu\left\\{\lVert
y-f(t)+f(s)\rVert\leq C_{1}\,|s-t|^{H}\sqrt{|\log|s-t||},\,\,2^{-n}<|s-t|\leq
2^{1-n}\right\\}$
$\displaystyle\leq\sum_{n=1}^{\infty}\,2^{nHd}\,\mu\left\\{\lVert
y-f(t)+f(s)\rVert\leq C_{2}\,2^{-nH}\sqrt{n},\,\,2^{-n}<|s-t|\leq
2^{1-n}\right\\},$

where $C_{2}=C_{1}\,2^{H}\sqrt{\log(2)}$. Now, for $n\geq 1$, we set

$S_{n}\left(y,\left(t,f(t)\right)\right)=\left\\{\left(s,f(s)\right)\in
Gr_{A}(f):\lVert y-f(t)+f(s)\rVert\leq
C_{2}\,2^{-nH}\sqrt{n},\,\,2^{-n}<|s-t|\leq 2^{1-n}\right\\}.$

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

$\underset{\left(y,\left(t,f(t)\right)\right)\in\left\\{\lVert
y\rVert<r\right\\}\times
Gr_{A}(f)}{\sup}\,\mu(S_{n}\left(y,\left(t,f(t)\right)\right)\leq
C_{3}\,n^{d/2}\,2^{-\kappa n}.$

Therefore we get

$\underset{\left(y,\left(t,f(t)\right)\right)\in\left\\{\lVert
y\rVert<r\right\\}\times
Gr_{A}(f)}{\sup}I_{1}\left(y,\left(t,f(t)\right)\right)\leq
C_{4}\sum_{n=1}^{\infty}\,2^{-(\kappa-Hd)n}\,n^{d/2}<\infty,$ (3.3)

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

$\displaystyle I_{2}\left(y,\left(t,f(t)\right)\right)\leq$
$\displaystyle\sum_{n=1}^{\infty}2^{nHd}\exp\left(-\dfrac{C_{1}^{2}}{2}(n-1)\ln
2\right)\times$ $\displaystyle\,\mu\left\\{\lVert
f(s)-f(t)+y\rVert>C_{1}\,|s-t|^{H}\sqrt{|\log|s-t||},\,\,2^{-n}<|s-t|\leq
2^{1-n}\right\\}$

Thus, for $C_{1}>\sqrt{2Hd}$ we obtain

$\underset{\left(y,\left(t,f(t)\right)\right)\in\left\\{\lVert
y\rVert<r\right\\}\times
Gr_{A}(f)}{\sup}I_{2}\left(y,\left(t,f(t)\right)\right)\leq e^{C_{1}^{2}\ln
2/2}\sum_{n=1}^{\infty}2^{-n\left(C_{1}^{2}/2-Hd\right)}<+\infty.$ (3.4)

Now putting all this together yields

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

Thus we get from (3.2) that

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

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

$\int_{\mathbb{R}^{d}}\int_{A}\int_{A}\,\mathbb{E}\left(e^{i\langle\theta,(Y(s)-Y(t)\rangle}\right)\,d\nu(s)\,d\nu(t)\,d\theta<\infty.$
(3.5)

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

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

However we are unable to ensure finiteness of the integral,

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

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 (Theorems 1 and 2
p. 267 in [10]) 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’s what prompted us to
consider the case $Hd<\dim(A)$. Let us note that in the light of (2.17), 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 Lemma 7.1 of [17] 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 $n\geq 1$ 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 $t_{0}=0$. The following
elementary formula allows to estimate the determinant of the covariance matrix
of the Gaussian random vector $(Z_{1},\ldots,Z_{n})$

$\text{det}\text{Cov}(Z_{1},\ldots,Z_{n})=\text{Var}(Z_{1})\underset{k=2}{\overset{n}{\prod}}\text{Var}\left(Z_{k}/Z_{1},\ldots,Z_{k-1}\right).$

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

$\text{det}\text{Cov}\left(B_{0}^{H}(t_{j}),1\leq j\leq
p\right)\geq\underset{j=1}{\overset{p}{{\displaystyle\prod}}}\left(\min\,\left\\{\lvert
t_{j}-t_{i}\rvert^{2H},0\leq i\leq j-1\right\\}\right).$ (3.6)

We need also the following lemma which is due to Cuzick and DuPreez [5]

######  Lemma 3.4.

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

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

for all $\varepsilon>0$. Then

$\displaystyle\int_{\mathbb{R}^{n}}g(v_{1})$
$\displaystyle\exp\left(-\frac{1}{2}\text{Var}\left(\sum_{j=1}^{n}v_{j}Z_{j}\right)\right)dv_{1}...dv_{n}$
$\displaystyle=\frac{(2\pi)^{(n-1)/2}}{(\text{det}\text{Cov}(Z_{1},...,Z_{n}))^{1/2}}\int_{-\infty}^{\infty}g\left(\frac{v}{\sigma_{1}}\right)e^{-v^{2}}dv,$

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.

######  Theorem 3.5.

Let $\\{B^{H}(t):t\in[0,1]\\}$ be a $d$-dimensional fractional Brownian motion
of Hurst index $H\in(0,1)$. Let $f:[0,1]\rightarrow\mathbb{R}^{d}$ be a
continuous function and let $A\subset[0,1]$ be a closed set. Assume that
$\dim(A)>Hd$. Then we have $(B^{H}+f)(A)$ has a non-empty interior almost
surely.

###### 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

$\tau([a,a+\delta])\leq C\,\delta^{\kappa},$ (3.7)

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 $(25.7)$ in [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

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

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

$\underset{j=1}{\overset{p}{\prod}}|\exp\left(i\langle
x,\xi_{j}\rangle\right)-\exp\left(i\langle y,\xi_{j}\rangle\right)|\leq
2^{(1-\gamma)p}\,\,\lVert x-y\rVert^{\gamma
p}\,\sum\prod_{j=1}^{p}|\xi_{j,k_{j}}|^{\gamma}$

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

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

Now the generalized Hölder’s inequality leads to

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

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

$\displaystyle J\leq$
$\displaystyle\,\,\dfrac{(2\pi)^{(pd-1)/2}}{\left[\det\text{Cov}\left(B^{H}_{l}(t_{j}),0\leq
l\leq d,1\leq j\leq
p\right)\right]^{1/2}}\,\,\displaystyle\int_{\mathbb{R}}|r|^{p\gamma}e^{-r^{2}/2}dr\,\prod_{j=1}^{p}\,\,\dfrac{1}{\sigma_{j}^{\gamma}}$
$\displaystyle\leq$
$\displaystyle\,\,\dfrac{(dK)^{p}(p!)^{\gamma}}{\left[\det\text{Cov}\left(B^{H}_{0}(t_{j}),1\leq
j\leq
p\right)\right]^{d/2}}\,\,\underset{j=1}{\overset{p}{\displaystyle\prod}}\,\,\dfrac{1}{\sigma_{j}^{\gamma}}$
(3.8)

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 Lemma $2.5$ in [20], that

$\underset{j=1}{\overset{p}{\displaystyle\prod}}\dfrac{1}{\sigma_{j}^{\gamma}}\leq\frac{K^{p}}{\left[\det\text{Cov}\left(B^{H}_{0}(t_{j}),1\leq
j\leq
p\right)\right]^{\gamma}}\leq\dfrac{K^{p}}{\underset{j=1}{\overset{p}{{\displaystyle\prod}}}\left(\min\,\left\\{\lvert
t_{j}-t_{i}\rvert^{2H\gamma},0\leq i\leq j-1\right\\}\right)},$ (3.9)

we conclude

$J\leq\dfrac{\left(dK^{2}\right)^{p}(p!)^{\delta}}{\underset{j=1}{\overset{p}{{\displaystyle\prod}}}\left(\min\,\left\\{\lvert
t_{j}-t_{i}\rvert^{H(d+2\gamma)},0\leq i\leq j-1\right\\}\right)}.$ (3.10)

Now, we can prove that

$\displaystyle\int_{A^{p}}\,\dfrac{d\tau(t_{1})\ldots
d\tau(t_{p})}{\underset{j=1}{\overset{p}{{\displaystyle\prod}}}\left(\min\,\left\\{|t_{j}-t_{i}|,0\leq
i\leq j-1\right\\}\right)^{H(d+2\gamma)}}\leq
p!\left(\sup_{s\in[0,1]}\displaystyle\int_{A}\frac{d\tau(t)}{|t-s|^{H(d+2\gamma)}}\right)^{p}.$
(3.11)

First, for all $j\in\\{1,...,p\\}$ we splite $A$ into the regions
$A_{i,j}:=\left\\{t\in A\cap I,|t-t_{i}|=\min\\{|t-t_{l}|,0\leq l\leq
j-1\\}\right\\}$ for all $i\in\\{0,...,j-1\\}$, and $t_{0}=0$. Then, for all
$j\in\\{1,...,p\\}$ we have

$\displaystyle\int_{A}\frac{d\nu(t)}{(\min\\{|t-t_{i}|,0\leq i\leq
j-1\\})^{H(d+2\gamma)}}=\sum_{i=0}^{j-1}\int_{A_{i,j}}\frac{d\nu(t)}{|t-t_{i}|^{H(d+2\gamma)}}\leq
j\sup_{s\in[0,1]}\int_{A}\frac{d\nu(t)}{|t-s|^{H(d+2\gamma)}},$

Hence (3.11) follows by iterating the above estimation, and by using (3.7), if
we take

$0<\gamma<\dfrac{1}{2}(\kappa/H-d),$

the right hand side in (3.11) is finite. Putting all the above facts together,
we arrive at

$\mathbb{E}(\vartheta(x)-\vartheta(y))^{p}\leq C_{p,\gamma}\,\lVert
x-y\rVert^{\gamma p},$ (3.12)

where $C_{p,\gamma}$ is a constant depends only on $p$ and $\gamma$. Hence we
can apply the Kolmogorov’s continuity theorem (Theorem 2.3.1 p.158 in [12]) 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 [17, p. 324] or
Geman and Horowitz [7, 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
ends 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
$\dim(A)=\dfrac{\ln\left(2\right)}{\ln\left(3\right)}$. For
$0<\alpha\,d<\dfrac{\ln\left(2\right)}{\ln\left(3\right)}<H\,d$, we have from
$\eqref{parabolic dimension of FBM}$ 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 drifts 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.

######  Theorem 4.1.

Let $\\{B^{H}(t):t\in[0,1]\\}$ be a $d$-dimensional fractional Brownian motion
of Hurst index $H\in(0,1)$. Let $A\subset[0,1]$ a closed set such that
$0<\dim(A)\leq Hd$. Then for all $\alpha\in\left(0,\dim(A)/d\right)$ there
exists a $\alpha$-Hölder continuous function
$f:[0,1]\rightarrow\mathbb{R}^{d}$ for which the range $(B^{H}+f)(A)$ has a
non-empty interior a.s.

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 $\left\\{B^{\alpha^{\prime}}(t):t\in[0,1]\right\\}$ 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

$\displaystyle
Z(t,(\omega,\omega^{\prime}))=B^{H}(t,\omega)+B^{\alpha^{\prime}}(t,\omega^{\prime})\text{
for all }t\in[0,1]\text{ and
}(\omega,\omega^{\prime})\in\Omega\times\Omega^{\prime}.$ (4.1)

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

$\widetilde{\mathbb{E}}(Z_{0}(s)Z_{0}(t))=\mathbb{E}(B^{H}_{0}(s)B^{H}_{0}(t))+\mathbb{E}^{\prime}(B^{\alpha^{\prime}}_{0}(s)B^{\alpha^{\prime}}_{0}(t))=\dfrac{1}{2}(s^{2H}+t^{2H}-|t-s|^{2H})+\dfrac{1}{2}(s^{2\alpha^{\prime}}+t^{2\alpha^{\prime}}-|t-s|^{2\alpha^{\prime}}),$

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

######  Proposition 4.2.

The real-valued process $\\{Z_{0};t\geq 0\\}$ satisfy the following

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

$|s-t|^{2\alpha^{\prime}}\leq\widetilde{\mathbb{E}}\left(Z_{0}(t)-Z_{0}(s)\right)^{2}\leq
2\,|s-t|^{2\alpha^{\prime}}.$ (4.2)

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

$Var\left(Z_{0}(u)|Z_{0}(t_{1}),...,Z_{0}(t_{n})\right)\geq C\left[\min_{0\leq
k\leq n}|u-t_{k}|^{2\alpha^{\prime}}+\min_{0\leq k\leq
n}|u-t_{k}|^{2H}\right],$ (4.3)

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

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

$Var\left(Z_{0}(t)|Z_{0}(s),|s-t|\geq r\right)\geq C\,r^{2\alpha^{\prime}}.$
(4.4)

###### Proof.

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

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

Since $\alpha\leq H$ it follows that

$|t-s|^{2\alpha^{\prime}}\leq\begin{array}[]{lll}\widetilde{\mathbb{E}}\left(Z_{0}(t)-Z_{0}(s)\right)^{2}&\leq&\,2\,|t-s|^{2\alpha^{\prime}}.\end{array}$

2\. By definition we can write

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

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 $u$, $t_{1},...,t_{n}\in I$,

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

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 $\left\\{Z_{0}(s),|s-t|\geq r\right\\}$ 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 $|s_{j}-t|\geq
r,\,(j=1,\cdots,n)$,

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

What follows from (4.3). Thus the proof is ended. ∎

The following is a direct consequence of Corollary 3.3 in [18].

######  Proposition 4.3.

$Z(A)$ has $\mathbb{P}\otimes\mathbb{P}^{\prime}$-$a.s.$ non-empty interior.

###### 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 $B^{\alpha^{\prime}}$. ∎

## References

  * [1] R. J. Adler. The Geometry of Random Fields. Wiley, New York, 1981.
  * [2] T. Antunović, Y. Peres and B. Vermesi. Brownian motion with variable drift can be space filling. Proc. Amer. Math. Soc. 139, no. 9 (2011), 3359-3373.
  * [3] P. Balança. Some sample path properties of multifractional Brownian motion. Stochastic Process. Appl. 125, no. 10 (2015), 3823-3850.
  * [4] C. J. Bishop and Y. Peres. Fractals in probability and analysis. Cambridge Studies in Advanced Mathematics, 162. Cambridge University Press, Cambridge, 2017.
  * [5] J. Cuzick and J. P. DuPreez. Joint continuity of Gaussian local times. Ann. Probab. 10, no. 3 (1982), 810-817.
  * [6] K. J. Falconer. Fractal geometry-mathematical foundations and applications. John Wiley Sons, Chichester, 1990.
  * [7] D. Geman and J. Horowitz. Occupation densities. Ann. Probab. 8, no. 1 (1980), 1-67.
  * [8] D. Geman, J. Horowitz and J. Rosen. A local time analysis of intersections of Brownian motion in the plane. Ann. Probab. 12 (1984), 86–107 .
  * [9] S. E. Graversen. Polar-functions for Brownian motion. Z. Wahrsch. Verw. Gebiete 61, no. 2 (1982), 261-270.
  * [10] J. P. Kahane. Some random series of functions, Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985.
  * [11] R. Kaufman. Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A727-A728.
  * [12] D. Khoshnevisan. Multiparameter processes. An introduction to random fields. Springer-Verlag, New York, 2002.
  * [13] D. Khoshnevisan and Y. Xiao. Brownian motion and thermal capacity. Ann. Probab. 43, no. 1 (2015), 405-434.
  * [14] J. F. Le Gall. Sur les fonctions polaires pour le mouvement brownien. Séminaire de Probabilités, XXII, 186-189, Lecture Notes in Math., 1321, Springer, Berlin, 1988.
  * [15] P. Mörters and Y. Peres. Brownian motion. Cambridge Series in Statistical and Probabilistic Mathematics, 30. Cambridge University Press, Cambridge, 2010.
  * [16] Y. Peres and P. Sousi. Dimension of fractional Brownian motion with variable drift. Probab. Theory Related Fields 165, no. 3-4 (2016), 771-794.
  * [17] L. D. Pitt. Local times for Gaussian vector fields. Indiana Univ. Math. J. 27, no. 2 (1978), 309-330.
  * [18] R. Shieh. and Y. Xiao. Images of Gaussian random fields: Salem sets and interior points. Studia Math. 176, no. 1 (2006), 37-60.
  * [19] S. J. Taylor and N. A. Watson. A Hausdorff measure classification of polar sets for the heat equation. Math. Proc. Cambridge Philos. Soc. 97, no. 2 (1985), 325–344.
  * [20] Y. Xiao. Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Related Fields 109, no. 1 (1997), 129-157