Paul Bekima | Searcharxiv

arXiv2025

Hartman-Grobman Theorem for Stochastic Dynamical Systems

In this paper, we extend the Hartman-Grobman theorem to systems perturbed with white noises. Let's recall that, in deterministic systems, the Hartman-Grobman theorem establishes the "topological equivalence" of the local phase portrait between a system and its linearization around hyperbolic fixed points; hence, simplifying the study of the stability at those points. However, it should be pointed out that, the conditions for a "useful" linear approximation of a non-linear system do not solely involved hyperbolic points; indeed, Nils Berglund and Barbara Gentz for example conspicuously used it in their book [3]; particularly during their study of white noise perturbed slow-fast dynamical systems. Yet, since our focus is on behavior of critical points of a system, we need to make sure that the linear approximation of our perturbed system is equivalent in some sense to the perturbed system of the linear approximation of the corresponding deterministic system. The paper is organized as follow: we first establish the theorem when the "diffusion" matrix is invertible. We continue by examining the case of non-invertible matrices, and non square matrices. We then apply the results to the study of Multi-dimensional slow-fast systems done by Berglund and Gentz [3] by weakening regularity conditions.

math.DS↗

arXiv2024

Well-Posedness and Stability of the Stochastic OGTT Model

Oral Glucose Tolerance Test (OGTT) is one of many way to produce data in the study of the diabetes dynamic. In a recent paper [1.]:\textit{ Estimating insulin sensitivity and $ \beta $-cell function from the oral glucose tolerance test: validation of a new insulin sensitivity and secretion (ISS) model, \textit{J. American Physiological Society },(2024)},Ha J., Chung S.T., and al. proposed a comprehensive OGTT model under the form of a dynamic system. But their model was fully deterministic. Yet, our natural environment interacts with noise; thus taking into account that inherent perturbation could potentially improve the model, which in turn could lead to a better estimation of the parameters of the system. The current paper endeavors to explore the OGTT model proposed by Ha and al. but this time with the addition of white noise perturbations to the system.\ The work is organized as follow: we first establish the existence and uniqueness of a global positive solution to the proposed stochastic model. Then follows the study the statistical stability of the model by examining the existence of an invariant measure to the perturbed system. Once done with the well-posedness and the stability of the stochastic model, we then examine a Maximum Likelihood Estimation (MLE) scheme to estimate the parameters involved in the model. The paper ends with a brief discussion on potential future developments follows by an appendix whose goal is to make the paper as self contained as possible.

math.PR↗