Statistics of residence time for Lévy flights in unstable parabolic potentials

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.

Interplay of noise induced stability and stochastic resetting

Stochastic resetting and noise-enhanced stability are two phenomena that can affect the lifetime and relaxation of nonequilibrium states. They can be considered measures of controlling the efficiency of the completion process when a stochastic system has to reach the desired state. Here, we study the interaction of random (Poissonian) resetting and stochastic dynamics in unstable potentials. Unlike noise-induced stability that increases the relaxation time, the stochastic resetting may eliminate winding trajectories contributing to the lifetime and accelerate the escape kinetics from unstable states. In this paper, we present a framework to analyze compromises between the two contrasting phenomena in noise-driven kinetics subject to random restarts.