Uniform Exponential Contraction for Viscous Hamilton-Jacobi Equations

The paper proves uniform exponential contraction for kicked viscous Hamilton–Jacobi equations under Assumption 1 (unique nondegenerate minimum), by Hopf–Cole, a conjugated kernel L̃_ν, a Lyapunov-function/Markov-chain contraction theorem, and a bootstrap that yields constants independent of ν; see Theorem A and the outlined Steps 1–4 in the paper (Hopf–Cole and Lν, Theorem 4.1, Lyapunov approach, and bootstrap) . By contrast, the candidate solution relies on a spectral/Hilbert-metric route: it claims projective-metric contraction at a rate (λ1(ν)/ρν)^n with a prefactor C0 “depending only on kernel bounds,” and then a ν-uniform spectral gap via a Mehler-kernel limit. This outline misses crucial uniformity: as ν→0 the kernel degenerates and the projective-metric/L∞–L2 comparison constants (and related Harnack/Birkhoff constants) typically blow up with ν, so the asserted ν-uniform C0 is not justified; the argument never supplies bounds ensuring that this prefactor remains ν-independent. The paper circumvents precisely this difficulty via Markov minorization/Lyapunov techniques and a bootstrap that restores uniform constants (e.g., Theorem 6.1 and its use of Corollary 5.3) , whereas the model’s spectral approach leaves these steps unproven.