Functional limit theorems for the Bouchaud trap model with slowly   varying traps
  We consider the Bouchaud trap model on the integers in the case that the trap distribution has a slowly varying tail at infinity. Our main result is a functional limit theorem for the model under the annealed law, analogous to the functional limit theorems previously established in the literature in the case of integrable or regularly varying trap distribution. Reflecting the fact that the clock process is dominated in the limit by the contribution from the deepest-visited trap, the limit process for the model is a spatially-subordinated Brownian motion whose associated clock process is an extremal process. 
David Croydon and Stephen Muirhead
math.PR