Explicit results for ergodic properties of SDEs driven by cylindrical symmetric stable noise

We consider the exponentially ergodic properties of systems of SDEs in ℝⁿ driven by cylindrical stable processes, potentially with different indices across different coordinates. Our approach is based on the well-known Foster-Lyapunov criteria and a careful selection of Lyapunov functions, alongside recent advances in regularity and transition density estimates for solutions to SDEs driven by Lévy processes with independent coordinates. These results are novel, even in the one-dimensional case. Notably, our findings suggest that multiplicative cylindrical stable processes can enhance the ergodicity of the system when the stable noise indices in all directions fall within [1, 2).