  Periodic measures are the time-periodic counterpart to invariant measures for
dynamical systems and can be used to characterise the long-term periodic
behaviour of stochastic systems. This paper gives sufficient conditions for the
existence, uniqueness and geometric convergence of a periodic measure for
time-periodic Markovian processes on a locally compact metric space in great
generality. In particular, we apply these results in the context of
time-periodic weakly dissipative stochastic differential equations, gradient
stochastic differential equations as well as Langevin equations. We will
establish the Fokker-Planck equation that the density of the periodic measure
sufficiently and necessarily satisfies. Applications to physical problems shall
be discussed with specific examples.
