Getting $B_t$ from its local times $L^x_t$ - MathOverflow

Getting $B_t$ from its local times $L^x_t$

2011-11-18T16:55:56Z

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

Answer by Yuri Bakhtin for Getting $B_t$ from its local times $L^x_t$

2011-11-19T04:06:01Z

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 $\gamma=\bigcap_n G_n$.

Answer by pgassiat for Getting $B_t$ from its local times $L^x_t$

2012-01-21T09:40:48Z

This paper seems to answer (something very close to) your question :

Warren, J. and Yor, M. (1998), The Brownian burglar: conditioning Brownian motion by its local time process. Seminaire de Probabilites XXXII, pages 328-342. (pdf link)

Abstract :

Imagine a Brownian crook who spent a month in a large metropolis. The number of nights he spent in hotels A,B,C...etc. is known; but not the order, nor his itinerary. So the only information the police has is total hotel bills.....