For switching diffusion, the long-time position cumulants equal free cumulants of the diffusion-coefficient distribution, and the rate function shows dynamical transitions depending on the tail of that distribution.
autonomous ratcheting by stochastic resetti ng
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Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions
For switching diffusion, the long-time position cumulants equal free cumulants of the diffusion-coefficient distribution, and the rate function shows dynamical transitions depending on the tail of that distribution.