Almost sure asymptotics for the maximum local time in Brownian environment

Roland Diel (1)

1 LJAD - Laboratoire Jean Alexandre Dieudonné

We study the asymptotic behaviour of the maximum local time L*(t) of the Brox's process, the diffusion in Brownian environment. Shi proved that the maximum speed of L*(t) is surprisingly, at least t log(log(log t)) whereas in the discrete case it is t. We show here that t log(log(log t)) is the proper rate and we prove that for the minimum speed the rate is the same as in the discrete case namely t/log(log(log t)).

Mots clés: Local time; Brox's process; Diffusion process in Brownian environment

Domaines: Probabilités [math.PR]

Roland Diel. Almost sure asymptotics for the maximum local time in Brownian environment. Stochastic Processes and their Applications, 2011, 121 (10), pp.2303-2330. ⟨10.1016/j.spa.2011.06.002⟩. ⟨hal-00460810v2⟩