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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 fixedconstant. Our main result shows that a phenomenon of entropic repulsion occurs:that is, this process is ballistic and has an asymptotic velocity approximately4.58... as high as required by the conditioning (the exact value of thisconstant involves the first zero of a Bessel function). We also study therandom walk case and show that the process is asymptotically ballistic but withan unknown speed.

Random paths with bounded local time