Periodic measures are the time-periodic counterpart to invariant measures for dynamical systems that can characterise the long-term periodic behaviour of stochastic dynamical systems. In this thesis, sufficient conditions are given for the existence, uniqueness and geometric convergence of periodic measures of time-periodic Markovian systems on locally compact metric spaces. The results will be applied specifically to time-periodic weakly dissipative stochastic differential equations (SDEs), gradient SDEs and Langevin equations. We show that the periodic measure density sufficiently and necessarily satisfies a time-periodic Fokker-Planck equation. We will also rigorously derive that the expected exit duration of time-periodic SDEs is the time-periodic solution of a second-order linear parabolic partial differential equation (PDE). Collectively, this rigorously establishes two novel Feynman-Kac dualities for time-periodic SDEs. Casting the time-periodic solution of the PDE as a fixed point problem and a convex optimisation problem, we give sufficient conditions in which the PDE is well-posed in a weak and classical sense. With no known closed formulae, we show that these approaches can be readily implemented to compute the expected exit time numerically. Periodic measures and expected exit times are novel tools to understand physical phenomena exhibiting periodicity. Particular application towards stochastic resonance will be discussed
