Random paths with bounded local time

Abstract

We consider one-dimensional Brownian motion conditioned (in a suitable sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.58. . . as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). We also study the random walk case and show that the process is asymptotically ballistic but with an unknown speed.