THE EFFECT OF A SMALL BOUNDED NOISE ON THE HYPERBOLICITY FOR AUTONOMOUS SEMILINEAR DIFFERENTIAL EQUATIONS

The paper proves Theorem 3.12 by (i) a Lyapunov–Perron fixed-point construction that yields a bounded global solution near the hyperbolic equilibrium under the smallness conditions (3.10)–(3.11) (Theorem 3.9), and (ii) robustness of exponential dichotomy for the linearized cocycle along that solution (Theorem 3.1), thereby establishing random hyperbolicity (Definition 3.11) . The candidate solution reproduces the same architecture: a Lyapunov–Perron map built from the Green operator to get a unique bounded entire solution staying within ε of y0*, followed by roughness of exponential dichotomies to ensure hyperbolicity. Differences are cosmetic: constants and notational choices (K/α vs. M/β), and the roughness step cites a classical source rather than the paper’s Theorem 3.1. One minor omission in the model sketch is not explicitly checking measurability of the dichotomy projections; the paper addresses measurability downstream when treating stochastic examples (Theorem 3.16) via measurability of the fixed point construction . Overall, both proofs align closely in substance.