arXiv subjects

Youssef Hakiki

Publications and source records attributed to Youssef Hakiki.

7 recordsLinked to original sources

arXiv2026

On Itô-Stratonovich formula for rough sheets

In this paper, we explore a new strategy towards an Itô-Stratonovich type formula for rough sheets. Historically, planar integration for irregular paths has been notoriously cumbersome. The emergence of mixed differential terms in 2D leads to overlapping iterated integrals, which previously required the construction of exhaustive combinatorial structures. As an example of this kind of structure, let us mention the massive 36-element planar signature introduced by K. Chouk and M. Gubinelli in their influential paper 'Rough sheets'. In this work we propose a simplified setting for rough calculus in the plane, which relies on elementary Taylor expansions in a more fundamental way. We claim that this simple trick allows us to significantly streamline the complexity of planar algebraic integration. We illustrate this methodology for a specialized Rough-Young framework: we extend the classical planar change-of-variable formula to paths possessing an asymmetric Hölder regularity: $γ_1 > 1/3$ in the first direction and $γ_2 > 1/2$ in the second direction. Relying on a structured controlled path expansions, we rigorously minimize the set of iterated integrals needed in the signature, and express the resulting planar change-of-variable formula as the explicit limit of Riemann sums.

math.PR↗

arXiv2026

Polarity of points for Gaussian random fields in critical dimension

We study the property of hitting points for a class of $\mathbb{R}^d$-valued continuous Gaussian random fields on $\mathbb{R}^N$ with stationary increments, i.i.d. coordinates, and a regularly varying variance function $σ$ of index $0<H<1$. We first prove that if \[ \lim_{r\to 0^+} \frac{r^N}{σ^d\left(r\left( \log\log\frac{1}{r}\right)^{-1/N}\right)} = \infty, \] then every fixed point is polar (i.e., not hit almost surely). In general, this criterion may not be optimal in the critical dimension $d=N/H$. To aim for an optimal condition, we consider the specific case $σ(r) = r^H (\log(1/r))^γ$ and prove that, in the critical dimension $d=N/H$, points are polar if and only if $γ\le 1/d$, or equivalently in this specific case, \[ \int_{0^+} \frac{r^{N-1}}{σ^d(r)} dr = \infty. \] This integral condition is also necessary for points to be polar under general assumptions. Our main contribution lies in the proof of sufficiency of this condition in the specific case, where we extend a covering argument of Talagrand (1998) based on sojourn time estimates to obtain Hausdorff measure bounds and solve polarity of points in the critical dimension.

math.PR↗

arXiv2023

Irregularity scales for Gaussian processes: Hausdorff dimensions and hitting probabilities

Let $X$ be a $d$-dimensional Gaussian process in $[0,1]$, where the component are independent copies of a scalar Gaussian process $X_0$ on $[0,1]$ with a given general variance function $γ^2(r)=\operatorname{Var}\left(X_0(r)\right)$ and a canonical metric $δ(t,s):=(\mathbb{E}\left(X_0(t)-X_0(s)\right)^2)^{1/2}$ which is commensurate with $γ(t-s)$. Under a weak regularity condition on $γ$, referred to below as $\mathbf{(C_{0+})}$, which allows $γ$ to be far from Hölder-continuous, we prove that for any Borel set $E\subset [0,1]$, the Hausdorff dimension of the image $X(E)$ and of the graph $Gr_E(X)$ are constant almost surely. Furthermore, we show that these constants can be explicitly expressed in terms of $\dim_δ(E)$ and $d$. However, when $\mathbf{(C_{0+})}$ is not satisfied, the classical methods may yield different upper and lower bounds for the underlying Hausdorff dimensions. This case is illustrated via a class of highly irregular processes known as logBm. Even in such cases, we employ a new method to establish that the Hausdorff dimensions of $X(E)$ and $Gr_E(X)$ are almost surely constant. The method uses the Karhunen-Loève expansion of $X$ to prove that these Hausdorff dimensions are measurable with respect to the expansion's tail sigma-field. Under similarly mild conditions on $γ$, we derive upper and lower bounds on the probability that the process $X$ can reach the Borel set $F$ in $\mathbb{R}^d$ from the Borel set $E$ in $[0,1]$. These bounds are obtained by considering the Hausdorff measure and the Bessel-Riesz capacity of $E\times F$ in an appropriate metric $ρ_δ$ on the product space, relative to appropriate orders. Moreover, we demonstrate that the dimension $d$ plays a critical role in determining whether $X\lvert_E$ hits $F$ or not.

math.PR↗

arXiv2023

Fractional Brownian motion with deterministic drift: How critical is drift regularity in hitting probabilities

Let $B^{H}$ be a $d$-dimensional fractional Brownian motion with Hurst index $H\in(0,1)$, $f:[0,1]\longrightarrow\mathbb{R}^{d}$ a Borel function, and $E\subset[0,1]$, $F\subset\mathbb{R}^{d}$ are given Borel sets. The focus of this paper is on hitting probabilities of the non-centered Gaussian process $B^{H}+f$. It aims to highlight how each component $f$, $E$ and $F$ is involved in determining the upper and lower bounds of $\mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}$. When $F$ is a singleton and $f$ is a general measurable drift, some new estimates are obtained for the last probability by means of suitables Hausdorff measure and capacity of the graph $Gr_E(f)$. As application we deal with the issue of polarity of points for $(B^H+f)\vert_E$ (the restriction of $B^H+f$ to the subset $E\subset (0,\infty)$).

math.PR↗

arXiv2021

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

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.

math.PR↗

arXiv2021

Hitting probabilities for fractional Brownian motion with deterministic drift

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

math.PR↗

arXiv2021

Hausdorff dimensions and Hitting probabilities for some general Gaussian processes

Let $B$ be a $d$-dimensional Gaussian process on $\mathbb{R}$, where the component are independents copies of a scalar Gaussian process $B_0$ on $\mathbb{R}_+$ with a given general variance function $γ^2(r)=\operatorname{Var}\left(B_0(r)\right)$ and a canonical metric $δ(t,s):=(\mathbb{E}\left(B_0(t)-B_0(s)\right)^2)^{1/2}$ which is commensurate with $γ(t-s)$. We provide some general condition on $γ$ so that for any Borel set $E\subset [0,1]$, the Hausdorff dimension of the image $B(E)$ is constant a.s., and we explicit this constant. Also, we derive under some mild assumptions on $γ\,$ an upper and lower bounds of $\mathbb{P}\left\{B(E)\cap F\neq \emptyset \right\}$ in terms of the corresponding Hausdorff measure and capacity of $E\times F$. Some upper and lower bounds for the essential supremum norm of the Hausdorff dimension of $B(E)\cap F$ and $E\cap B^{-1}(F)$ are also given in terms of $d$ and the corresponding Hausdorff dimensions of $E\times F$, $E$, and $F$.

math.PR↗