Derivative of self-intersection local time for multidimensional fractional Brownian motion

TL;DR

This paper investigates the behavior of the derivative of self-intersection local time for multidimensional fractional Brownian motion at the critical Hurst parameter value, extending previous existence results.

Contribution

It establishes a limit theorem for the derivative of self-intersection local time at the critical case where H equals 1 divided by the dimension, which was previously unexplored.

Findings

Limit theorem proved at critical H=1/d

Extends understanding of local time derivatives at boundary conditions

Provides new insights into fractional Brownian motion properties

Abstract

The existence condition $H<1/d$ for first-order derivative of self-intersection local time for $d\geq3$ dimensional fractional Brownian motion can be obtained in Yu (2021). In this paper, we show a limit theorem under the non-existence critical condition $H=1/d$.