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Active velocity processes with suprathermal stationary distributions and long-time tails

When a particle moves through a spatially-random force field, its momentum may change at a rate which grows with its speed. Suppose moreover that a thermal bath provides friction which gets weaker for large speeds, enabling high-energy localization. The result is a unifying framework for the emergence of heavy tails in the velocity distribution, relevant for understanding the power-law decay in the electron velocity distribution of space plasma or more generally for explaining non-Maxwellian behavior of driven gases. We also find long-time tails in the velocity autocorrelation, indicating persistence at large speeds for a wide range of parameters and implying superdiffusion of the position variable.