The Third and Fourth Moment of the Renormalized Intersection Local Time

TL;DR

This paper computes the third and fourth moments of the renormalized intersection local time for planar Brownian motion, providing analytical and numerical results, and also examines moments for planar random walks.

Contribution

It presents the first analytical calculation of the third moment and numerical estimation of the fourth moment of the renormalized intersection local time.

Findings

Third moment calculated analytically.

Fourth moment estimated numerically.

Third moment for planar random walk derived in leading order.

Abstract

In this article we calculate the third and fourth moment of the renormalized intersection local time of a planar Brownian motion. The third moment is calculated anlaytically, the fourth moment numerically. For the closed planar random walk the third moment of the distribution of the multiple point range is also calculated in leading order.