Marginal densities of the "true" self-repelling motion

Let X(t) be the true self-repelling motion (TSRM) constructed by B.T. and Wendelin Werner in 1998, 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 B.T. in 1995 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: It 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 arXiv:1105.2948v3.