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.
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Itai Benjamini, Nathanaël Berestycki, Random paths with bounded local time. J. Eur. Math. Soc. 12 (2010), no. 4, pp. 819–854DOI 10.4171/JEMS/216