Curvature-Dependent Path Concentration in Stochastic Fast-Slow Systems with Noise on the Slow Variable

Abstract

We study stochastic fast-slow systems in which the noise acts exclusively on the slow variable: $dx=f(x,y)dt$, $dy=\epsilon g(x,y)dt+\ddot{I}ƒh(y)d{W}_t$. While the path-concentration theory for noise on the fast variable is well developed, the complementary case of noise only on the slow variable has remained largely unexplored, with a recent exception treating the fold bifurcation. For general, uniformly normally-hyperbolic deterministic slow manifolds $x=X^{\ast }(y)$, we derive rigorous pathwise estimates showing that the deviation $z=x-X^{\ast }(y)$ concentrates with exponential tail bounds over the slow timescale $[0,T/\epsilon ]$. A central finding is that the Itô correction arising from the curvature $D^2{X}^{\ast }$ of the slow manifold introduces a systematic $O(\ddot{I}{ƒ}^2\Vert D^2{X}^{\ast }\Vert )$ bias that tightens the concentration bound beyond the classical $\ddot{I}ƒ/\sqrt{\hat{I}{»}_0}$ tube width. We identify a geometric critical noise scale $\ddot{I}{ƒ}_c(\epsilon )=C_0\min (\sqrt{\epsilon },{\epsilon }^{1/4}{L}_{\mathrm{geom}}/\sqrt{\hat{I}{»}_0})$, where $L_{\mathrm{geom}}=\sqrt{\hat{I}{»}_0/(\Vert D^2{X}^{\ast }\Vert \Vert h{\Vert }^2)}$ is a local geometric scale of the manifold. For $\ddot{I}ƒ\le \ddot{I}{ƒ}_c$, the fast variable tracks the manifold to within $C(\epsilon /\hat{I}{»}_0+\ddot{I}{ƒ}^2\Vert D^2{X}^{\ast }\Vert \Vert h{\Vert }^2/(2\hat{I}{»}_0)+\ddot{I}ƒ/\sqrt{\hat{I}{»}_0})$ with probability at least $1-e^{-\hat{I}º/\epsilon }-e^{-C_K}$, where $C_K>0$ depends on the confinement of the slow dynamics. We also prove that the slow-variable adiabatic error is $O(\epsilon +\ddot{I}ƒ\sqrt{\epsilon }+\ddot{I}{ƒ}^2\Vert D^2{X}^{\ast }\Vert )$, which is dominated by classical terms when $\ddot{I}ƒ\le \ddot{I}{ƒ}_c$; hence curvature governs fast-variable path concentration but not adiabatic validity.

34 pages, 1 figure. Submitted to Journal of Differential Equations