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Information Anatomy of Stochastic Equilibria

A stochastic nonlinear dynamical system generates information, as measured by its entropy rate. Some---the ephemeral information---is dissipated and some---the bound information---is actively stored and so affects future behavior. We derive analytic expressions for the ephemeral and bound informations in the limit of small-time discretization for two classical systems that exhibit dynamical equilibria: first-order Langevin equations (i) where the drift is the gradient of a potential function and the diffusion matrix is invertible and (ii) with a linear drift term (Ornstein-Uhlenbeck) but a noninvertible diffusion matrix. In both cases, the bound information is sensitive only to the drift, while the ephemeral information is sensitive only to the diffusion matrix and not to the drift. Notably, this information anatomy changes discontinuously as any of the diffusion coefficients vanishes, indicating that it is very sensitive to the noise structure. We then calculate the information anatomy of the stochastic cusp catastrophe and of particles diffusing in a heat bath in the overdamped limit, both examples of stochastic gradient descent on a potential landscape. Finally, we use our methods to calculate and compare approximations for the so-called time-local predictive information for adaptive agents.