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Maher Boudabra

Publications and source records attributed to Maher Boudabra.

At least 19 recordsLinked to original sources
arXiv2026

Space-time transport of Brownian exit laws

The present paper is devoted to a systematic study of the $p$-Brownian convergence introduced in \cite{boudabra2026stability} (in press) to study the stability of the planar Skorokhod embedding problem \cite{gross2019,Boudabra2020}. The first part is an illustration of some geometric aspects of the $p$-Brownian convergence. The second part turns this notion into a metric between domains. More precisely, we place it within the framework of optimal transport theory. Several results are obtained, namely asymptotic behavior in case of homothetic domains. Numerical illustrations are provided as well.

math.PR↗
arXiv2026

Brownian Convergence of Planar Domains and Stability of the Planar Skorokhod Embedding Problem

We present a numerical framework for approximating the $\mu$-domain in the planar Skorokhod embedding problem PSEP, recently introduced in \cite{gross2019}. We show that under weak convergence of a sequence of probability measures $(\mu_{n})_{n}$, the corresponding sequence of $\mu_{n}$-domains converges, in an appropriate sense, to the domain associated with the limit measure $\mu$. In addition, we provide implementation strategies, convergence rate estimates, and a numerical example. The method is robust and versatile, offering a concrete computational approach for the approximation of $\mu$-domains. As part of this analysis, we introduce a novel mode of convergence for planar domains via planar Brownian motion, which we call $p$-Brownian convergence.

math.PR↗
arXiv2026

A seminorm-only characterization of analytic Besov spaces on the disc

We introduce the space $\mathcal{W}^{s,p}(\mathbb{D})$ of analytic functions $u$ on the unit disc such that the radial restrictions $u_{r}(\xi):=u(r\xi)$ satisfy the Gagliardo seminorm-only bound \[ \sup_{0<r<1}[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}<\infty, \] with no $\emph{a priori}$ control of $\sup_{r}\|u_{r}\|_{L^{p}(\mathbb{S}^{1})}$. Our main result shows that this assumption already forces $u\in H^{p}(\mathbb{D})$ and that the radial boundary trace $u^{*}$ belongs to $W^{s,p}(\mathbb{S}^{1})$, with $u_{r}\to u^{*}$ in $W^{s,p}(\mathbb{S}^{1})$ as $r\to1^{-}$. The key mechanism combines the mean-value property (which pins the constant mode at $u(0)$) with a fractional Poincar$\'e$ inequality on $\mathbb{S}^{1}$, recovering $L^{p}$ control from oscillation alone. As a consequence, the trace map $u\mapsto u^{*}$ is a surjective isomorphism $\mathcal{W}^{s,p}(\mathbb{D})\xrightarrow{\sim}B^{s}_{p,p,+}(\mathbb{S}^{1})$ with explicit norm equivalence.

math.AP↗
arXiv2025

A note on the planar Skorokhod embedding problem

The planar Skorokhod embedding problem was first proposed and solved by R. Gross in 2019 [#gross2019]. Gross worked with probability distributions having finite second moment. In [#boudabra2019remarks, #Boudabra2020], the solutions extended to all distributions with a finite $p^{th}$ moment for $p>1$. The case $p=1$ remained uncovered since then. In this note we show that the planar Skorokhod embedding problem is solvable for $p=1$ when the Hilbert transform of its quantile function is integrable, effectively closing this line of investigation.

math.PR↗
arXiv2025

Symmetrization and the Planar Skorokhod Embedding Problem

This paper continues our earlier work \cite{becher2025skorokhod} on variational questions arising from the planar Skorokhod embedding problem (PSEP). Given a centered probability measure $\mu$ on $\mathbb R$ with finite second moment, PSEP asks for a simply connected domain $U\subset\mathbb C$ containing $0$ such that planar Brownian motion $(Z_t)$ started at $0$ exits $U$ at time $\tau_U$ with real part $\Re(Z_{\tau_U})\sim\mu$. Among all such $\mu$-domains, we study optimal design problems and focus in particular on area minimization and its fractional boundary-energy extensions. We formalize and define the Brownian symmetrization of planar domains, and we clarify the relation between Brownian (Gross) symmetrization and the Baernstein-Pruss symmetrization theory, and how Brownian symmetrization applies to a wider category of domains. Within the simply connected class, Gross' $\mu$-domain $U_\mu^G$ minimizes a whole fractional scale of boundary energies $\mathcal E_s$, $0<s<1$. The proof is formulated in a nonlocal Hardy--Sobolev language: it relies only on the exit law $\mu$ and on fractional Sobolev (Gagliardo) seminorms, rather than on an explicit uniformizer or star-function techniques. We introduce deficiency ratios $\rho_s$ that quantify how far a given $\mu$-domain is from the Gross optimizer; we state several related open problems.

math.PR↗
arXiv2025

A Numerical scheme to approximate the solution of the planar Skorokhod embedding problem

We present a numerical framework to approximate the $\mu$-domain in the planar Skorokhod embedding problem (PSEP), recently appeared in \cite{gross2019}. Our approach investigates the continuity and convergence properties of the solutions with respect to the underlying distribution $\mu$. We establish that, under weak convergence of a sequence of probability measures $(\mu_n)$ with bounded support, the corresponding sequence of $\mu_n$-domains converges to the domain associated with $\mu$, limit of $(\mu_n)$. We derive explicit convergence results in the $L^1$ norm, supported by a generalization using the concept of $\alpha_p$-convergence. Furthermore, we provide practical implementation techniques, convergence rate estimates, and numerical simulations using various distributions. The method proves robust and adaptable, offering a concrete computational pathway for approximating $\mu$-domains in the PSEP.

math.PR↗
arXiv2024

Skorokhod energy of planar domains

In this work, we introduce the Skorokhod energy of a simply connected domain. We show that among all domains solving the planar Skorokhod embedding problem, Gross solution generates the domain with the minimal Skorokhod energy.

math.PR↗
arXiv2024

Brownian symmetrization of planar domains

One of aims of this note is to capture the interest of the mathematical community to a novel transformation, which we shall call Brownian symmetrization. This transformation arises from the solution of the planar Skorokhod embedding problem. Brownian symmetrization shares some properties with the famous Steiner symmetrization. However, we show that these two transformations are not the same and they do not affect each other.

math.PR↗
arXiv2023

A note on a deterministic property to obtain the long run behavior of the range of a stochastic process

A Brownian motion with drift is simply a process $V^η_t$ of the form $V^η_t=B_{t}+ηt$ where $B_{t}$ is a standard Brownian motion and $η>0$ \footnote{The case $η<0$ is deducible by remarking $V^{-η}(t)=-V^η(t)$.} In \cite{tanre2006range}, the authors considered the drifted Brownian motion and studied the statistics of some related sequences defined by certain stopping times. In particular, they provided the law of the range $R_{t}(V^η)$ of $V^η$ as well as its first range process $θ_{V^η}(a)$. In particular, they investigated the asymptotic comportment of $R_{t}(V^η)$ and $θ_{V^η}(a)$. They proved that if $V_{t}^η$ is a Brownian motion with a positive drift $η$ then its range $R_{t}(V^η)=\sup_{0\leq s\leq t}V_{t}^η-\inf_{0\leq s\leq t}V_{t}^η$ is asymptotically equivalent to $ηt$. In other words \begin{equation} \frac{R_{t}(V^η)}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}η.\label{range} \end{equation} In this paper, we show that (\ref{range}) follows from a striking deterministic property. More precisely, we show that the long run behavior of the range of a deterministic continuous function is obtainable straightaway from that of the function itself. Our result can be deemed as the continuous version of a similar one appeared in \cite{mgrw}.

math.PR↗
arXiv2021

A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion

We prove a number of results relating exit times of planar Brownian with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the Cauchy distribution using the optional stopping theorem.

math.PR↗
arXiv2021

On the duration of stays of Brownian motion in domains in Euclidean space

Let $T_D$ denote the first exit time of a Brownian motion from a domain $D$ in ${\mathbb R}^n$. Given domains $U,W \subseteq {\mathbb R}^n$ containing the origin, we investigate the cases in which we are more likely to have fast exits from $U$ than $W$, meaning ${\bf P}(T_U {\bf P}(T_W t) > {\bf P}(T_W>t)$ for $t$ large. This result, which applies only in two dimensions, shows that the unit disk has the lowest probability of long stays amongst all Schlicht domains.

math.PR↗
arXiv2021

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

A comb domain is defined to be the entire complex plain with a collection of vertical slits, symmetric over the real axis, removed. In this paper, we consider the question of determining whether the exit time of planar Brownian motion from such a domain has finite $p$-th moment. This question has been addressed before in relation to starlike domains, but these previous results do not apply to comb domains. Our main result is a sufficient condition on the location of the slits which ensures that the $p$-th moment of the exit time is finite. Several auxiliary results are also presented, including a construction of a comb domain whose exit time has infinite $p$-th moment for all $p \geq 1/2$.

math.PR↗
arXiv2020

A note on the moments of sequences of complex numbers

We give a short proof that the limsup of the p-th root of the modulus of the p-th moment of a sequence of complex numbers is equal to the modulus of the maximum of the sequence.This strengthens known results, and provides an analog to a recent result concerning moments of complex polynomials.

math.CV↗
arXiv2020

On the Dirichlet eigenvalue problem and the conformal Skorokhod embedding problem

In a recent work by Gross, the following problem was stated and solved: given a measure $μ$ with finite second moment, find a simply connected domain $U$ in $\CC$ such that the real part of a Brownian motion stopped when it leaves $U$ is distributed as $μ$. The construction developed by Gross yields a domain which is symmetric with respect to the real axis, but it has been noted by other authors that other domains are also possible, in particular there are a number of examples which have the property that a vertical ray starting at a point in the domain lies entirely within the domain. In this paper we give a new solution to the problem posed by Gross, and show that these other cases noted before are special cases of this method. We further show that the domain generated by this method has the property that it always has the minimal rate (as defined in terms of the spectrum of the Laplacian operator) among all possible domains corresponding to a fixed distribution $μ$, which gives a partial solution to a question posed by Mariano and Panzo. We show that the domain is unique, provided certain conditions are imposed, and use this to give several examples. We also describe a method for identifying the boundary curve of the domain, and discuss several other related topics.

math.PR↗
arXiv2020

Remarks on the speeds of a class of random walks on the integers

In recent years, there has been an interest in deriving certain important probabilistic results as consequences of deterministic ones; see for instance \cite{beig} and \cite{acc}. In this work, we continue on this path by deducing a well known equivalence between the speed of random walks on the integers and the growth of the size of their ranges. This result is an immediate consequence of the Kesten-Spitzer-Whitman theorem, and by appearances is probabilistic in nature, but we will show that it follows easily from an elementary deterministic result. We also investigate the common property of recurrent random walks of having speed zero, and show by example that this property need not be shared by deterministic sequences. However, if we consider the inter-arrival times (times at which the sequence is equal to 0) then we find a sufficient deterministic condition for a sequence to have zero speed, and show that this can be used to derive several probabilistic results.

math.PR↗
arXiv2020

Maximizing the $p$-th moment of exit time of planar Brownian motion from a given domain

In this paper we address the question of finding the point which maximizes the $p$-th moment of the exit time of planar Brownian motion from a given domain. We present a geometrical method of excluding parts of the domain from consideration which makes use of a coupling argument and the conformal invariance of Brownian motion. In many cases the maximizing point can be localized to a relatively small region. Several illustrative examples are presented.

math.PR↗