The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. Here we present an accurate and efficient numerical method for computing this probability for systems described by non-autonomous (time-dependent) stochastic differential equations (SDEs) or their equivalent Fokker-Planck partial differential equations. The method is based on the direct approximation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the forward SDE. The Feynman-Kac formula is approximated using the Gauss-Hermite quadrature rules and piecewise cubic Hermite interpolating polynomials, and a GPU accelerated matrix representation is used to compute the entire time evolution of the exit time probability using a single pass of the algorithm. The method is unconditionally stable, exhibits second-order convergence in space, first-order convergence in time, and is straightforward to parallelize. Applications are presented to the advection-diffusion of a passive tracer in a fluid flow exhibiting chaotic advection, and to the runaway acceleration of electrons in a plasma in the presence of an electric field, collisions, and radiation damping. Benchmarks against analytical solutions as well as comparisons with explicit and implicit finite difference standard methods for the adjoint Fokker-Planck equation are presented.

翻译：退出时间概率使初始条件有可能在特定时间或之前离开动态系统阶段空间的指定区域,这可以说是最为自然和重要的运输问题之一。在这里,我们提出了一个准确有效的数字方法,用于计算非自主(依赖时间的)随机差异方程式(SDEs)或其等效的Fokker-Planck部分差异方程所描述的系统的这一概率。这种方法基于Feynman-Kac公式的直接接近,该公式在Fokker-Planc 联合方程式和前方SDE之间建立了联系。Feynman-Kac公式使用高斯-赫米特二次方程式规则以及小巧的Hermite间对聚氨基体间对等组合规则来估计这一概率。GPU加速矩阵表用于用算算出退出时间概率的全时间演变。这种方法是无条件稳定的,显示空间的次等级对等比对等方方方方方平方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方