Besov-Orlicz regularity of local Broownian time

Let (B-t, t epsilon [0,1]) be a linear Brownian motion starting from 0 and denote by (L-t(x), t greater than or equal to 0, x epsilon R) its local tim e. We prove that the spatial trajectories of the Brownian local time have t he same Besov-Orlicz regularity as the Brownian motion itself (i.e. for all t > 0, a.s. the function x bar right arrow L-t(x) belongs to the Besov-Orl icz space B-M2,infinity(1/2), with M-2(x) = e \ x \(2) - 1). Our result is optimal.