Sub-Gaussian bound for the one-dimensional Bouchaud trap model

We establish a sub-Gaussian lower bound for the transition kernel of the one-dimensional, symmetric Bouchaud trap model, which provides a positive answer to the behavior predicted by Bertin and Bouchaud in (Phys. Rev. E (3) 67 (2013) 026128). The proof rests on the Ray Knight description of the local time of a one-dimensional Brownian motion. Using the same ideas, we also prove the corresponding result for the FIN singular diffusion.