Rigorously enclosing Lyapunov exponents of stochastic flows is a long-standing and notoriously difficult problem: even determining the sign of the top exponent is often out of reach outside special structures or perturbative regimes. In this paper, we develop a computer-assisted method to obtain rigorous upper and lower bounds on Lyapunov exponents of stochastic flows under mild hypoellipticity assumptions.

Our approach starts from the Furstenberg-Khasminskii representation of the top Lyapunov exponent as an ergodic average over the invariant measure of the associated projective process, but crucially avoids any rigorous computation of that invariant measure. Instead, we introduce a numerical adjoint method, amenable to rigorous numerics, and show that an approximate solution of the associated Poisson equation yields a certified enclosure of the Lyapunov exponent via an explicit bound on the residual. This converts a non-rigorous numerical approximation into a rigorous quantitative estimate.

The method applies to systems on both compact and non-compact state spaces, does not require special geometric structure, and is not restricted to small-noise or other perturbative settings. We use it to prove positivity of the top Lyapunov exponent for several stochastic systems, including examples exhibiting noise-induced chaos and parameter-dependent sign changes, and we combine it with continuation methods to obtain rigorous bounds over large parameter regions.