### Abstract

1. Long range space correlations as a signature of violation of time reversal invariance (TRI)

We introduce the notions of strong and weak violation of time reversal invariance (TRI) according to whether the violation of TRI in the microscopic dynamics shows up or not at the hydrodynamical level. We then argue that long range space correlations seem to be a generic feature of dynamics strongly violating TRI. In particular, on the basis af a recently established Hamilton-Jacobi equation for the free energy, we show that equilibrium states of Glauber-Kawasaki type dynamics under strong violation of TRI have space correlations over a macroscopic scale. This result indicates that long range correlations are not specific to non equilibrium stationary states.

2. Dynamical phase transitions in large current fluctuations of stochastic lattice gases

In works in collaboration with L. Bertini, A. De Sole, D. Gabrielli and C. Landim we have shown that in large fluctuations of the current, averaged over long intervals of time, transitions to different dynamical regimes can take place. These are revealed by the time dependence of the thermodynamic variables associated to the fluctuations. In this case time shift invariance is spontaneously broken. So far two examples are known, the weakly asymmetric simple exclusion process with periodic boundary conditions discussed by Bodineau and Derrida and the Kipnis-Marchioro-Presutti model for which we have provided a rigorous proof of the transition.