Path integration method for stochastic responses of differential equations under Lévy white noise

A path integration (PI) approach that is progressive for studying the stochastic response driven by Lévy white noise is presented. First, a probability mapping is constructed, which decouples the domain of interest for the system state and the probability space derived from the randomness of Lévy white noise within a short time interval. Then, solving the probability mapping yields the short-time response of the system. Finally, the stochastic evolution of the system can be grasped in a stepwise manner based on the fundamental concept of the PI method. The applicability and effectiveness of our approach in addressing the transient and stationary responses under Lévy white noises are verified by Monte Carlo simulation results. Moreover, the advances in utilization of this method are that it eliminates the restriction of the previous PI method on the controlling parameter of Lévy white noises, and it is highly efficient for solving responses of systems under Lévy white noises.