2026/06/06 by Yefan Wu
Mathematics · #math.PR #math.DS
We study stochastic fast-slow systems in which the noise acts exclusively on the slow variable: dx = f(x,y) dt, dy = ε g(x,y) dt + σ h(y) dWt. 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/ε]. A central finding is that the Itô correction arising from the curvature D2X^* of the slow manifold introduces a systematic O(σ2‖D2X^*‖) bias that tightens the concentration bound beyond the classical σ/√(λ0) tube width. We identify a geometric critical noise scale σc(ε) = C0min (√(ε), ε1/4 Lgeom/√(λ0)), where Lgeom = √(λ0/(‖D2X^*‖‖h‖2)) is a local geometric scale of the manifold. For σ≤ σc, the fast variable tracks the manifold to within C(ε/λ0 + σ2‖D2X^*‖‖h‖2/(2λ0) + σ/√(λ0)) with probability at least 1 - e-κ/ε - e-CK, where CK > 0 depends on the confinement of the slow dynamics. We also prove that the slow-variable adiabatic error is O(ε + σ√(ε) + σ2‖D2X^*‖), which is dominated by classical terms when σ≤ σc; hence curvature governs fast-variable path concentration but not adiabatic validity.