In this paper, we derive a parabolic partial differential equation for the expected exit time of non-autonomous time-periodic non-degenerate stochastic differential equations. This establishes a Feynman–Kac duality between expected exit time of time-periodic stochastic differential equations and time-periodic solutions of parabolic partial differential equations. Casting the time-periodic solution of the parabolic partial differential equation as a fixed point problem and a convex optimisation problem, we give sufficient conditions in which the partial differential equation is well-posed in a weak and classical sense. With no known closed formulae for the expected exit time, we show our method can be readily implemented by standard numerical schemes. With relatively weak conditions (e.g. locally Lipschitz coefficients), the method in this paper is applicable to wide range of physical systems including weakly dissipative systems. Particular applications towards stochastic resonance will be discussed.

在本文中，我们针对非自治时间周期非退化随机微分方程的期望退出时间推导了一个抛物型偏微分方程。这在时间周期随机微分方程的预期退出时间与抛物线偏微分方程的时间周期解之间建立了Feynman-Kac对偶。将抛物线偏微分方程的时间周期解转换为不动点问题和凸优化问题，我们给出了充分的条件，使偏微分方程在弱经典意义上具有良好的位置。在没有已知的预期退出时间的封闭公式的情况下，我们证明了我们的方法可以很容易地通过标准数值方案实现。在相对较弱的条件下（例如局部的Lipschitz系数），本文中的方法适用于各种物理系统，包括弱耗散系统。将讨论对随机共振的特殊应用。