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



Getting $B_t$ from its local times $L^x_t$
Knowing local times you can derive if the path $\gamma=\{(t,B\_t): t\in[0,T]\}$ passes through any rectangle of the following form: $[k/2^n,(k+1)/2^n]\times[j/2^n,(j+1)/2^n]$. For fixed $n$, denote by $G\_n$ the union of all these visited rectangles.


Since $B\_t$ is uniformly continuous on $[0,T]$, we have $\gamma=\bigcap\_n G\_n$.


