Curvature-Dependent Path Concentration in Stochastic Fast-Slow Systems with Noise on the Slow Variable
Abstract
We study stochastic fast-slow systems with noise acting exclusively on the slow variable near uniformly attracting, normally hyperbolic critical manifolds x=X∗(y). The deviation equation contains transmitted diffusion and an Itô curvature forcing of order O(σ2lVert D2X∗rVert). Using a parameter-dependent Lyapunov function and stopped martingale estimates, we obtain geometry-dependent concentration bounds under the sufficient noise restriction σ≤ C0min(√ε,ε1/4Lgeom/√(λ0)), where λ0 is the uniform spectral gap and Lgeom=√(λ0/(lVert D2X∗rVertlVert hrVert2)). For fixed geometry and any fixed p>0, the confidence radius on [0,T/ε] is O(ε+σ2+σ√(log(1/ε))), with stopped fast-variable failure probability at most εp; slow-variable exit is controlled separately. We also establish an adiabatic error bound of order O(ε+σ√ε+σ2lVert D2X∗rVert), with explicit failure probabilities. These estimates quantify the curvature dependence of fast-variable concentration and its O(ε) contribution to the slow-error upper bound for fixed geometry.
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BibTeX
@misc{wu2026curvaturedependent,
title = {{Curvature-Dependent Path Concentration in Stochastic Fast-Slow Systems with Noise on the Slow Variable}},
author = {Yefan Wu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.16217},
eprint = {2607.16217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.16217},
}
RIS
TY - UNPB
TI - Curvature-Dependent Path Concentration in Stochastic Fast-Slow Systems with Noise on the Slow Variable
AU - Yefan Wu
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2607.16217
UR - https://arxiv.org/abs/2607.16217
AB - We study stochastic fast-slow systems with noise acting exclusively on the slow variable near uniformly attracting, normally hyperbolic critical manifolds x=X^∗(y). The deviation equation contains transmitted diffusion and an Itô curvature forcing of order O(σ^2lVert D^2X^∗rVert). Using a parameter-dependent Lyapunov function and stopped martingale estimates, we obtain geometry-dependent concentration bounds under the sufficient noise restriction σ≤ C_0min(√ε,ε^1/4L_geom/√(λ_0)), where λ_0 is the uniform spectral gap and L_geom=√(λ_0/(lVert D^2X^∗rVertlVert hrVert^2)). For fixed geometry and any fixed p>0, the confidence radius on [0,T/ε] is O(ε+σ^2+σ√(log(1/ε))), with stopped fast-variable failure probability at most ε^p; slow-variable exit is controlled separately. We also establish an adiabatic error bound of order O(ε+σ√ε+σ^2lVert D^2X^∗rVert), with explicit failure probabilities. These estimates quantify the curvature dependence of fast-variable concentration and its O(ε) contribution to the slow-error upper bound for fixed geometry.
ER -