We develop a powerful and general method to provide arbitrarily accurate rigorous upper and lower bounds for Lyapunov exponents of stochastic flows. Our approach is based on computer-assisted tools, the adjoint method and established results on the ergodicity of diffusion processes. We do not require any structural assumptions on the stochastic system and work under mild hypoellipticity conditions outside of perturbative regimes. Therefore, our method allows for the treatment of systems that were so far inaccessible from existing mathematical tools. We demonstrate our method to exhibit the chaotic nature of three non-Hamiltonian systems. Finally, we show that our approach is robust to continuation methods to produce bounds on Lyapunov exponents for large parameter regions.

翻译：我们提出了一种强大且通用的方法，可为随机流的李雅普诺夫指数提供任意精度的严格上下界。我们的方法基于计算机辅助工具、伴随方法以及关于扩散过程遍历性的既定结果。我们无需对随机系统施加任何结构性假设，且在非微扰区域下仅需满足温和的亚椭圆性条件。因此，本方法能够处理现有数学工具至今无法处理的系统。我们通过三个非哈密顿系统展示了本方法在揭示混沌特性方面的应用。最后，我们证明该方法对延拓方法具有鲁棒性，能够为大参数区域内的李雅普诺夫指数生成界。