Descending from infinity: Convergence of tailed distributions
  We investigate the relaxation of long-tailed distributions under stochastic dynamics that do not support such tails. Linear relaxation is found to be a borderline case in which long tails are exponentially suppressed in time but not eliminated. Relaxation stronger than linear suppresses long tails immediately, but may lead to strong transient peaks in the probability distribution. A delta function initial distribution under stronger than linear decay displays not one but two different regimes of diffusive spreading. 
Christian Van den Broeck, Upendra Harbola, Raul Toral and Katja   Lindenberg
cond-mat.stat-mech