Lévy Diffusion Under Power-Law Stochastic Resetting

We investigated the diffusive dynamics of a Lévy walk subject to stochastic resetting through combined numerical and theoretical approaches. Under exponential resetting, the process mean squared displacement (MSD) undergoes a sharp transition from free superdiffusive behavior with exponent γ0 to a steady-state saturation regime. In contrast, power-law resetting with exponent β exhibits three asymptotic MSD regimes: free superdiffusion for β<1, superdiffusive scaling with a linearly β-decreasing exponent for 1<β<γ0+1, and localization characterized by finite steady-state plateaus for β>γ0+1. MSD scaling laws derived via renewal theory-based analysis demonstrate excellent agreement with numerical simulations. These findings offer new insights for optimizing search strategies and controlling transport processes in non-equilibrium environments.