Statistical properties of a biased out-of-equilibrium Brownian gyrator

We study the two-dimensional stochastic model called Brownian gyrator, known to reach a non equilibrium steady state rather peculiar characterized by persistent rotation. We generalize the original model introducing constant forces doing work on the gyrator, for which we derive exact asymmetry relations, that are reminiscent of the standard fluctuation relations. Unlike the latter, our relations concern instantaneous and not time averaged values of the observables of interest. We investigate the full two-dimensional dynamics as well as the dynamics projected on the x- and y-axes, and define in both cases "effective temperatures". Remarkably, the effective temperatures appearing in full dynamics are distinctly different from the ones emerging in its projections, confirming that they are not thermodynamic quantities, although they precisely characterize the state of the system. We then compute the power spectral density for the single com- ponents of this process, based on individual realization. We derive exact expression for the probability distribution of this quantity and compare it to the standard ensemble averaged power spectral density. We find that the frequency dependent behavior of the latter is already inferable at single trajectory level. Finally, we highlight differences in correlations behavior in and out equilibrium by looking at the joint distribution of the single component spectra, confirmed by numerical simulation.