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: $dx = f(x,y)\,dt$, $dy = \varepsilon\,g(x,y)\,dt + \sigma\,h(y)\,dW_t$. 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 $x = X^*(y)$, we derive rigorous pathwise estimates showing that the deviation $z = x - X^*(y)$ concentrates with exponential tail bounds over the slow timescale $[0,T/\varepsilon]$. A central finding is that the It\^o correction arising from the curvature $D^2X^*$ of the slow manifold introduces a systematic $O(\sigma^2\|D^2X^*\|)$ bias that tightens the concentration bound beyond the classical $\sigma/\sqrt{\lambda_0}$ tube width. We identify a geometric critical noise scale $\sigma_c(\varepsilon) = C_0\min\!\bigl(\sqrt{\varepsilon},\, \varepsilon^{1/4} L_{\mathrm{geom}}/\sqrt{\lambda_0}\bigr)$, where $L_{\mathrm{geom}} = \sqrt{\lambda_0/(\|D^2X^*\|\|h\|^2)}$ is a local geometric scale of the manifold. For $\sigma \le \sigma_c$, the fast variable tracks the manifold to within $C\bigl(\varepsilon/\lambda_0 + \sigma^2\|D^2X^*\|\|h\|^2/(2\lambda_0) + \sigma/\sqrt{\lambda_0}\bigr)$ with probability at least $1 - e^{-\kappa/\varepsilon} - e^{-C_K}$, where $C_K > 0$ depends on the confinement of the slow dynamics. We also prove that the slow-variable adiabatic error is $O(\varepsilon + \sigma\sqrt{\varepsilon} + \sigma^2\|D^2X^*\|)$, which is dominated by classical terms when $\sigma \le \sigma_c$; hence curvature governs fast-variable path concentration but not adiabatic validity.