Dissipation-recurrence inequalities at the steady state

For Markov jump processes in out-of-equilibrium steady state, we present inequalities which link the average rate of entropy production with the timing of the site-to-site recurrences. Such inequalities are upper bounds on the average rate of entropy production. The combination with the finite-time thermodynamic uncertainty relation (a lower bound) yields inequalities of the pure kinetic kind for the relative precision of a dynamical output. After having derived the main relations for the discrete case, we sketch the possible extension to overdamped Markov dynamics on continuous degrees of freedom, treating explicitly the case of one-dimensional diffusion in tilted periodic potentials; an upper bound on the average velocity is derived, in terms of the average rate of entropy production and the microscopic diffusion coefficient, which corresponds to the finite-time thermodynamic uncertainty relation in the limit of vanishingly small observation time.