Marginal densities of the "true" self-repelling motion

TL;DR

This paper derives explicit formulas for the marginal densities of the true self-repelling motion and its local time height, revealing surprising distribution shapes and refining large deviation estimates.

Contribution

It provides explicit formulas for the marginal densities of the TSRM and its local time height, advancing understanding of their distributional properties.

Findings

Distribution of X(t) has a sharp local minimum at 0

Explicit formulas for marginal densities are obtained

Refines large deviation estimates for TSRM

Abstract

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.