Lévy Diffusion Under Power-Law Stochastic Resetting

We investigated the diffusive dynamics of a L\'evy 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 \( γ_0 \) 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 linearly \( β \)-decreasing exponent for \( 1 < β < γ_0 + 1 \), and localization characterized by finite steady-state plateaus for \( β > γ_0 + 1 \). 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.