Hartman-Grobman Theorem for Stochastic Dynamical Systems
Paul Bekima
TL;DR
The paper tackles the problem of extending the Hartman-Grobman local topological equivalence to stochastic dynamical systems perturbed by white noise. It analyzes the SDE $dX_t = f(X_t) dt + \epsilon \sigma(X_t) dB_t$ near hyperbolic fixed points, treating both invertible and non-invertible diffusion matrices via a Girsanov change of measure and appropriate exponential martingale constructions. It further extends the stochastic Hartman-Grobman framework to multi-dimensional slow-fast systems by weakening regularity assumptions to Sobolev spaces and imposing a Fundamental Condition, using mollification and the Ascoli-Arzelà theorem to preserve the local equivalence under noise. These contributions provide a rigorous basis for stability analysis of nonlinear systems under stochastic perturbations, with implications for slow-manifold dynamics and potential transport-PDE interpretations. The work integrates stochastic analysis with dynamical systems to broaden the applicability of local linearization in noisy, high-dimensional settings.
Abstract
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.