Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion

TL;DR

This paper revisits the higher order derivatives of fractional Brownian motion's self-intersection local time, offering new proofs and extending convergence ranges through renormalization techniques.

Contribution

It provides alternative proofs using Wiener chaos expansion and extends the convergence range for derivatives via Varadhan-type renormalization.

Findings

New proofs of existence for derivatives over certain Hurst parameter regions

Extension of convergence range for even derivatives through renormalization

Enhanced understanding of regularity properties of self-intersection local time

Abstract

In a recent paper by Yu (arXiv:2008.05633, 2020), higher order derivatives of self-intersection local time of fractional Brownian motion were defined, and existence over certain regions of the Hurst parameter $H$ was proved. Utilizing the Wiener chaos expansion, we provide new proofs of Yu's results, and show how a Varadhan-type renormalization can be used to extend the range of convergence for the even derivatives.