We develop a powerful and general method to provide rigorous and accurate 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, work under mild hypoellipticity conditions and outside of perturbative regimes. Therefore, our method allows for the treatment of systems that were so far out of reach from existing mathematical tools. We demonstrate our method to exhibit the chaotic nature of three different systems. Finally, we show the robustness of our approach by combining it with continuation methods to produce bounds on Lyapunov exponents over large parameter regions.

翻译：我们提出了一种强大而通用的方法，为随机流的李雅普诺夫指数提供严格且精确的上下界。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已有结果。我们不对随机系统施加任何结构性假设，在温和的亚椭圆性条件下工作，且不局限于微扰体系。因此，我们的方法能够处理以往数学工具无法触及的系统。我们通过三个不同系统展示了该方法以揭示其混沌特性。最后，我们结合延拓方法证明了本方法的鲁棒性，可在广阔参数区域内生成李雅普诺夫指数的界。