New theorems for the moments of the first passage time of one dimensional nonlinear stochastic processes with an entrance boundary are formulated. This important class of one dimensional stochastic processes results among others from approximations of the energy or amplitude of second order nonlinear stochastic differential equations. Since the diffusion of a stochastic process vanishes at an entrance boundary, the entrance boundary is called a singular point of the stochastic process. The theorems for the moments of the first passage times are validated based on existing analytical results. In addition, the first passage times of a nonlinear stochastic differential equation, which is important for the determination of dangerous ship roll dynamics, are calculated. The proposed analytical expressions for the moments of the first passage times can be calculated very fast using standard quadrature formulas.

中文翻译：

---

具有奇异边界行为的非线性扩散过程的首次通过时间

公式化了具有入口边界的一维非线性随机过程的第一次通过时间的矩的新定理。这一类重要的一维随机过程是二阶非线性随机微分方程的能量或振幅的近似值。由于随机过程的扩散在入口边界处消失，因此入口边界被称为随机过程的奇异点。基于现有分析结果验证了第一次通过时间的矩的定理。此外，计算了非线性随机微分方程的第一次通过时间，这对于确定危险的船舶横摇动力学很重要。