  The thermodynamic uncertainty relation is a universal trade-off relation connecting the precision of a current with the average dissipation at large times. For continuous time Markov chains (also called Markov jump processes) this relation is valid in the time-homogeneous case, while it fails in the time-periodic case. The latter is relevant for the study of several small thermodynamic systems. We consider here a time-periodic Markov chain with continuous time and a broad class of functionals of stochastic trajectories, which are general linear combinations of the empirical flow and the empirical density. Inspired by the analysis done in our previous work [1], we provide general methods to get local quadratic bounds for large deviations, which lead to universal lower bounds on the ratio of the diffusion coefficient to the squared average value in terms of suitable universal rates, independent of the empirical functional. These bounds are called "generalized thermodynamic uncertainty relations" (GTUR's), being generalized versions of the thermodynamic uncertainty relation to the time-periodic case and to functionals which are more general than currents. Previously, GTUR's in the time-periodic case have been obtained in [1, 27, 42]. Here we recover the GTUR's in [1, 27] and produce new ones, leading to even stronger bounds and also to new trade-off relations for time-homogeneous systems. Moreover, we generalize to arbitrary protocols the GTUR obtained in [42] for time-symmetric protocols. We also generalize to the time-periodic case the GTUR obtained in [19] for the so called dynamical activity, and provide a new GTUR which, in the time-homogeneous case, is stronger than the one in [19]. The unifying picture is completed with a comprehensive comparison between the different GTUR's. 
A unifying picture of generalized thermodynamic uncertainty relations