Curvature-Dependent Path Concentration in Stochastic Fast-Slow Systems with Noise on the Slow Variable
Year: 2026
Type: preprint
Abstract: We study stochastic fast-slow systems with noise acting exclusively on the slow variable near uniformly attracting, normally hyperbolic critical manifolds $x=X^\ast(y)$. The deviation equation contains transmitted diffusion and an Itô curvature forcing of order $O(σ^2\lVert D^2X^\ast\rVert)$. Using a parameter-dependent Lyapunov function and stopped martingale estimates, we obtain geometry-dependent concentration bounds under the sufficient noise restriction $σ\le C_0\min(\sqrt{\varepsilon},\varepsilon^{1/4}L_{\mathrm{geom}}/\sqrt{λ_0})$, where $λ_0$ is the uniform spectral gap and $L_{\mathrm{geom}}=\sqrt{λ_0/(\lVert D^2X^\ast\rVert\lVert h\rVert^2)}$. For fixed geometry and any fixed $p>0$, the confidence radius on $[0,T/\varepsilon]$ is $O(\varepsilon+σ^2+σ\sqrt{\log(1/\varepsilon)})$, with stopped fast-variable failure probability at most $\varepsilon^p$; slow-variable exit is controlled separately. We also establish an adiabatic error bound of order $O(\varepsilon+σ\sqrt{\varepsilon}+σ^2\lVert D^2X^\ast\rVert)$, with explicit failure probabilities. These estimates quantify the curvature dependence of fast-variable concentration and its $O(\varepsilon)$ contribution to the slow-error upper bound for fixed geometry.
Source: arXiv (Cornell University)
Author: Wu, Yefan
Institution: The University of Sydney
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7169868388