Marginal densities of the “true” self-repelling motion

Let X(t) be the true self-repelling motion (TSRM) constructed by Tóth 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 Tóth (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].