A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations

The candidate solution proves exactly the Fisher-information identities and the difference formula stated as Proposition 1.4 and equations (1.6)–(1.7) of the paper, including FI(ρ) = −∫Q dµ = −λΣ and FI(f) = −∫ Q̃ dν = nλ1 − 2λΣ, and FI(f) − FI(ρ) = (1/2)∑k∫M∫SxM |(Xk − V∗∇Xk(x))hx|2/hx dv dµ = nλ1 − λΣ . The route (stationarity + integration-by-parts identity + Liouville/Oseledets for the Jacobian rates) aligns closely with the paper’s Section 2 derivation via the Kolmogorov equation and Furstenberg–Khasminskii/Liouville formulas . One technical flaw in the model’s write-up is a sign error in the stated adjoint relation used in the one-vector-field identity, although the final identity employed is correct; with this minor fix, the argument matches the paper’s proof.