Getting $B_t$ from its local times $L^x_t$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:10:42Z http://mathoverflow.net/feeds/question/81265 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81265/getting-b-t-from-its-local-times-lx-t Getting $B_t$ from its local times $L^x_t$ The Bridge 2011-11-18T16:55:56Z 2012-01-21T09:40:48Z <p>Hi </p> <p>Given a Brownian Motion $B_t$ is it possible to reconstruct it from the knowledge of the local times $L^x_t$ ?</p> <p>Using occupation time formula this would mean solving for some $f$ the following equation :</p> <p>$$B_t=\int_{-\infty}^{+\infty}f(x)L^x_t.dx=\int_0^t f(B_s)ds$$</p> <p>This seems achievable but I couldn't find out the solution or prove that there is none.</p> <p>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. </p> <p>Best regards</p> http://mathoverflow.net/questions/81265/getting-b-t-from-its-local-times-lx-t/81305#81305 Answer by Yuri Bakhtin for Getting $B_t$ from its local times $L^x_t$ Yuri Bakhtin 2011-11-19T04:06:01Z 2011-11-19T04:06:01Z <p>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.</p> <p>Since $B_t$ is uniformly continuous on $[0,T]$, we have $\gamma=\bigcap_n G_n$.</p> http://mathoverflow.net/questions/81265/getting-b-t-from-its-local-times-lx-t/86294#86294 Answer by pgassiat for Getting $B_t$ from its local times $L^x_t$ pgassiat 2012-01-21T09:40:48Z 2012-01-21T09:40:48Z <p>This paper seems to answer (something very close to) your question :</p> <p>Warren, J. and Yor, M. (1998), The Brownian burglar: conditioning Brownian motion by its local time process. Seminaire de Probabilites XXXII, pages 328-342. (<a href="http://archive.numdam.org/ARCHIVE/SPS/SPS_1998__32_/SPS_1998__32__328_0/SPS_1998__32__328_0.pdf" rel="nofollow">pdf link</a>)</p> <p>Abstract :</p> <blockquote> <p>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.....</p> </blockquote>