AI 中文总结
研究慢变量带噪声的随机快-慢系统，推导路径估计，发现慢流形曲率产生的伊藤修正引入偏差并确定临界噪声尺度，证明快变量跟踪情况及慢变量绝热误差，表明曲率控制快变量路径集中而非绝热有效性。
AI 中文摘要
我们研究噪声仅作用于慢变量的随机快-慢系统：$dx = f(x,y)\,dt$，$dy = \varepsilon\,g(x,y)\,dt + \sigma\,h(y)\,dW_t$。虽然快变量噪声的路径集中理论已充分发展，但慢变量仅带噪声的情况大多未被探索，近期一个例外是处理折叠分岔。对于一般的、一致法向双曲确定性慢流形$x = X^*(y)$，我们推导了严格的路径估计，表明偏差$z = x - X^*(y)$在慢时间尺度$[0,T/\varepsilon]$上以指数尾部边界集中。一个核心发现是慢流形曲率$D^2X^*$产生的伊藤修正引入了一个系统的$O(\sigma^2\|D^2X^*\|)$偏差，该偏差收紧了集中边界，超出了经典$\sigma/\sqrt{\lambda_0}$管宽度。我们确定了一个几何临界噪声尺度$\sigma_c(\varepsilon) = C_0\min\!\bigl(\sqrt{\varepsilon},\, \varepsilon^{1/4} L_{\mathrm{geom}}/\sqrt{\lambda_0}\bigr)$，其中$L_{\mathrm{geom}} = \sqrt{\lambda_0/(\|D^2X^*\|\|h\|^2)}$是流形的局部几何尺度。对于$\sigma \le \sigma_c$，快变量在概率至少为$1 - e^{-\kappa/\varepsilon} - e^{-C_K}$的情况下跟踪流形至$C\bigl(\varepsilon/\lambda_0 + \sigma^2\|D^2X^*\|\|h\|^2/(2\lambda_0) + \sigma/\sqrt{\lambda_0}\bigr)$，其中$C_K > 0$取决于慢动力学的限制。我们还证明慢变量绝热误差为$O(\varepsilon + \sigma\sqrt{\varepsilon} + \sigma^2\|D^2X^*\|)$，当$\sigma \le \sigma_c$时由经典项主导；因此曲率控制快变量路径集中但不控制绝热有效性。
英文摘要
We study stochastic fast-slow systems in which the noise acts exclusively on the slow variable: $dx = f(x,y)\,dt$, $dy = \varepsilon\,g(x,y)\,dt + σ\,h(y)\,dW_t$. 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 Itô correction arising from the curvature $D^2X^*$ of the slow manifold introduces a systematic $O(σ^2\|D^2X^*\|)$ bias that tightens the concentration bound beyond the classical $σ/\sqrt{λ_0}$ tube width. We identify a geometric critical noise scale $σ_c(\varepsilon) = C_0\min\!\bigl(\sqrt{\varepsilon},\, \varepsilon^{1/4} L_{\mathrm{geom}}/\sqrt{λ_0}\bigr)$, where $L_{\mathrm{geom}} = \sqrt{λ_0/(\|D^2X^*\|\|h\|^2)}$ is a local geometric scale of the manifold. For $σ\le σ_c$, the fast variable tracks the manifold to within $C\bigl(\varepsilon/λ_0 + σ^2\|D^2X^*\|\|h\|^2/(2λ_0) + σ/\sqrt{λ_0}\bigr)$ with probability at least $1 - e^{-κ/\varepsilon} - e^{-C_K}$, where $C_K > 0$ depends on the confinement of the slow dynamics. We also prove that the slow-variable adiabatic error is $O(\varepsilon + σ\sqrt{\varepsilon} + σ^2\|D^2X^*\|)$, which is dominated by classical terms when $σ\le σ_c$; hence curvature governs fast-variable path concentration but not adiabatic validity.
Comments34 pages, 1 figure. Submitted to Journal of Differential Equations
