  Periodically driven Kitaev chains are known to exhibit novel Floquet Majorana
fermions and anomalous Floquet end modes. The fact that `quasienergy in Floquet
systems is periodic' poses a difficulty in defining the ground state in
periodically driven systems. To overcome this problem, we start with the ground
state of the undriven Kitaev chain and gradually switch on the periodic driving
in chemical potential over a timescale $\tau$. The extent to which the single
particle eigenstates of the undriven system get distributed among the Floquet
states upon driving can be characterized by inverse participation ratio. In a
Josephson junction between two periodically driven Kitaev chains not differing
in phase of the pair potentials but differing in phases of the driving
potentials, a net average current flows from one superconductor to the other.
We term such a current anomalous current. Further, we study current phase
relation in junctions between two periodically driven superconductors and find
that the system exhibits nonequilibrium Josephson diode effect. The Floquet
Majorana end modes and anomalous Floquet end modes whenever present contribute
significantly to the anomalous current and the diode effect. Further, the
anomalous current and the nonequilibrium Josephson diode effect survive when
the periodic driving is adiabatically switched on.
