Hi

Given a Brownian Motion $B_t$ is it possible to reconstruct it from the knowledge of the local times $L^x_t$ ?

Using occupation time formula this would mean solving for some $f$ the following equation :

$$B_t=\int_{-\infty}^{+\infty}f(x)L^x_t.dx=\int_0^t f(B_s)ds$$

This seems achievable but I couldn't find out the solution or prove that there is none.

By the way, if someone can achieve this reconstruction of $B_t$ from $L^x_t$ using some other device than the occupation formula I would be equally interested.

Best regards