Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2410.02382

Continuity of Lyapunov exponents for stochastic differential equations

Zhenxin Liu·Lixin Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

For non-autonomous linear stochastic differential equations (SDEs), we establish that the top Lyapunov exponent is continuous if the coefficients "almost" uniformly converge. For autonomous SDEs, assuming the existence of invariant measures and the convergence of coefficients and their derivatives in pointwise sense, we get the continuity of all Lyapunov exponents. Furthermore, we demonstrate that for autonomous SDEs with strict monotonicity condition, all Lyapunov exponents are Lipschitz continuous with respect to the coefficients under the $L^{p,p}$ norm ($p>2$). Similarly, the Hölder continuity of Lyapunov exponents holds under weaker regularity conditions. It seems that the continuity of Lyapunov exponents has not been studied for SDEs so far, in spite that there are many results in this direction for discrete-time dynamical systems.

Explore related subjects

math.DSmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhenxin Liu, Lixin Zhang. 2024-10-03. Continuity of Lyapunov exponents for stochastic differential equations. https://arxiv.org/abs/2410.02382

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Equidistribution of saddle periodic points for Hénon-like maps

We prove that under a natural assumption on the dynamical degrees, the saddle periodic points of a Hénon-like map in any dimension equidistribute with respect to the equilibrium measure. Our work is a generalization of the results of Bedford-Lyubich-Smillie, Dujardin, and Dinh-Sibony along with improvements of their techniques. We also investigate some fine properties of Green currents associated with the map.

math.DS↗
arXiv2026

On dissonance and orthogonal projections of self-conformal measures

Let $μ$ be a self-conformal measure on $\mathbb{R}^d$. We establish conditions for $μ$ under which $\dim(μ*ν) = \min\lbrace d,\dimμ+\dimν\rbrace$ holds when $ν$ is any Ahlfors-regular or self-conformal measure on $\mathbb{R}^d$. Our main result states the following sufficient condition: $μ$ is totally non-linear and not supported on a smooth hypersurface. We also establish sufficient (likely non-sharp) algebraic conditions for self-conformal measures which are not totally non-linear. In addition, we show that $\dim μ\circπ^{-1} = \min\{ k, \dim μ\}$ for every ortohogonal projection $π:\mathbb{R}^d\to\mathbb{R}^k$, $0<k<d$, when either $d=2$ and $μ$ is not self-similar and not supported on a line, or $d\geq 3$ and $μ$ is totally non-linear and not supported on a smooth hypersurface.

math.DS↗
arXiv2026

Equation-Free Screening of Mittag-Leffler-Compatible Dynamics from Scalar Time Series via kNN Multi-Horizon Profiles

Fractional models provide a natural description of systems with memory, but a noninteger derivative should not be introduced solely because a time series is curved or slowly relaxing. We develop an equation-free preliminary screening framework that asks whether a scalar time series produces a multi-horizon k-nearest-neighbor (kNN) profile more compatible with Mittag-Leffler-type behavior than with selected conventional alternatives. In an ideal matched Caputo-relaxation benchmark, the complete generation-kNN-profile-model-comparison pipeline reproduces the expected Mittag-Leffler geometry and recovers the generating order to within approximately $10^{-3}$; this is interpreted as controlled calibration rather than as general fractional-order identification. Under 3% trajectory-specific observational noise, the held-out Mittag-Leffler preference is most consistent when the generating dynamics are well separated from the integer-order limit and becomes progressively less decisive as $α\rightarrow1$. The fitted order $α_{\mathrm{fit}}$, however, shows substantially larger realization-to-realization variability. Thus, relative model compatibility is more robust than single-realization order estimation in the present noisy benchmark. Noise-free nonfractional controls show a separate limitation of specificity: a stretched exponential can generate a strongly Mittag-Leffler-compatible profile, whereas inclusion of the generating rational/Hill family recovers that family and its parameters to numerical precision in the matched setting. A positive Mittag-Leffler-versus-exponential screen therefore does not uniquely establish fractional origin. A fractional chaotic system is treated only as an exploratory extension: the Mittag-Leffler growth family gives lower finite-window RMSE than exponential and logistic/saturating alternatives over the detected pre-transition interval.

math.DS↗