Quelques approximations du temps local brownien

We give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. We prove that $\frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon)\wedge t}^+ \indi_{\{X_u \leqslant 0\}} du + \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon) \wedge t}^- \indi_{\{X_u>0\}} du$ and $\frac{4}{\epsilon}\int_0^{t} X_u^- \indi_{\{X_{(u+\epsilon) \wedge t} > 0\}} du$ converge in the ucp sense to $L_t^0$, as $\epsilon \to 0$. We show that $\frac{1}{\epsilon}\int_0^t (\indi_{\{x<X_{s+\epsilon}\}} - \indi_{\{x<X_{s}\}}) (X_{s+\epsilon}-X_{s})ds$ goes to $L_t^x$ in $L^2(\Omega)$ as $\epsilon \to 0$, and that the rate of convergence is of order $\epsilon^\alpha$, for any $\alpha < {1/4}$.