Crossover between Lévy and Gaussian regimes in first-passage processes

Area of Science:

• Statistical Physics
• Stochastic Processes
• Probability Theory

Background:

• First-passage time problems are fundamental in analyzing stochastic processes.
• Lévy flights exhibit heavy-tailed distributions, posing challenges for traditional analysis.
• Truncated Lévy flights (KoBoL processes) modify Lévy distributions by cutting off extreme values.

Purpose of the Study:

• To propose a novel approach for solving first-passage time problems.
• To investigate the behavior of first-passage time distributions in truncated Lévy flights.
• To analyze the crossover phenomena between non-Gaussian and Gaussian regimes.

Main Methods:

• Development of a general method applicable to Wiener processes, Lévy flights, and stable distributions.
• Focus on truncated Lévy flights (KoBoL processes) to demonstrate the method's utility.
• Analysis of asymptotic scaling laws for first-passage time distributions.

Main Results:

• Identified a crossover in the asymptotic scaling law of the first-passage time distribution.
• Observed a transition from a t(-(alpha+1)/alpha)-law (non-Gaussian Lévy regime) to a t(-3/2)-law (Gaussian regime).
• Demonstrated ultraslow convergence from the non-Gaussian to the Gaussian regime in truncated Lévy flights.

Conclusions:

• The proposed method effectively analyzes first-passage time problems in complex stochastic systems.
• Truncated Lévy flights exhibit a distinct crossover in scaling laws, bridging non-Gaussian and Gaussian behaviors.
• The findings have implications for understanding anomalous diffusion and related phenomena.