Path integral solutions for n-dimensional stochastic differential equations underα-stable Lévy excitation

In this paper,the path integral solutions for a general n-dimensional stochastic differential equa-tions(SDEs)withα-stable Lévy noise are derived and verified.Firstly,the governing equations for the solutions of n-dimensional SDEs under the excitation ofα-stable Lévy noise are obtained through the characteristic function of stochastic processes.Then,the short-time transition probability density func-tion of the path integral solution is derived based on the Chapman-Kolmogorov-Smoluchowski(CKS)equation and the characteristic function,and its correctness is demonstrated by proving that it satis-fies the governing equation of the solution of the SDE,which is also called the Fokker-Planck-Kolmogorov equation.Besides,illustrative examples are numerically considered for highlighting the feasibility of the proposed path integral method,and the pertinent Monte Carlo solution is also calculated to show its correctness and effectiveness.

Reliability evaluation of the roll motion under the wind and irregular beam waves

This paper intends to study the stochastic response and reliability of the roll motion under the action of wind and wave excitation.The roll motion in random beam seas is described by a four-dimensional(4D)Markov dynamic system whose probabilistic properties are governed by the Fokker-Planck(FP)equation.The 4D path integration(PI)method,an efficient numerical technique based on the Markov property of the 4D system,is applied in order to solve the high dimensional FP equation and then the stochastic statistics of the roll motion are derived.Based on the obtained response statistics,the reliability evaluation of the ship stability is performed and the effect of wind action is studied.The accuracy of the 4D PI method and the reliability evaluation is assessed by the versatile Monte Carlo simulation(MCS)method.