Régularité Besov-Orlicz du temps local Brownien

Let ${\displaystyle (B_t,t\in [0,1])}$ be a linear Brownian motion starting from 0 and denote by ${\displaystyle (L_t\left(x\right),t\ge 0,x\in ℝ)}$ its local time. We prove that the spatial trajectories of the Brownian local time have the same Besov-Orlicz regularity as the Brownian motion itself (i.e. for all t>0, a.s. the function ${\displaystyle x\to L_t\left(x\right)}$ belongs to the Besov-Orlicz space ${\displaystyle B^{\frac{1}{2}}\_\{M_2,\infty \}}$ with ${\displaystyle M_2\left(x\right)=e^{{\left|x\right|}^2}-1}$). Our result is optimal.