arXiv subjects

Fathi Haggui

Publications and source records attributed to Fathi Haggui.

6 recordsLinked to original sources

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↗

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↗

arXiv2021

Khavinson conjecture for hyperbolic harmonic functions on the unit ball

In this paper, we prove the Khavinson conjecture for hyperbolic harmonic functions on the unit ball. This conjecture was partially solved in \cite{JKM2020}.

math.CV↗

arXiv2020

Schwarz Lemma for mappings satisfying Biharmonic Equations

In this paper, we establish some Schwarz type lemmas for mappings $\Phi$ satisfying the inhomogeneous biharmonic Dirichlet problem $ \Delta (\Delta(\Phi)) = g$ in $\mathbb{D}$, $\Phi=f$ on $\mathbb{T}$ and $\partial_n \Phi=h$ on $\mathbb{T}$, where $g$ is a continuous function on $\overline{\mathbb{D}}$, $f,h$ are continuous functions on $\mathbb{T}$, where $\mathbb{D}$ is the unit disc of the complex plane $\mathbb{C}$ and $\mathbb{T}=\partial \mathbb{D}$ is the unit circle. To reach our aim, we start by investigating some properties of $T_2$-harmonic functions. Finally, we prove a Landau-type theorem.

math.CV↗

arXiv2019

Study of some holomorphic curves in $\C^3$ and their projection into the complex projectve space $\C P^2$

We study holomorphic curves $f:\C\longrightarrow \C^3$ avoiding four complex hyperplanes and a real subspace of real dimension four or five in $\C^3$. We show that the projection of $f$ into the complex projective space $\C P^2$ is not necessarily constant.

math.CV↗