Marginal densities of the “true” self-repelling motion

Let X ( t ) be the true self-repelling motion (TSRM) constructed by Tَth and Werner (1998) [22], L ( t , x ) its occupation time density (local time) and H ( t ) : = L ( t , X ( t ) ) the height of the local time profile at the actual position of the motion. The joint distribution of ( X ( t ) , H ( t ) ) was identified by Tَth (1995) [20] in somewhat implicit terms. Now we give explicit formulas for the densities of the marginal distributions of X ( t ) and H ( t ) . The distribution of X ( t ) has a particularly surprising shape: its density has a sharp local minimum with discontinuous derivative at 0 . As a consequence we also obtain a precise version of the large deviation estimate of Dumaz (2011) [5].