Intermediate dimension of images of sequences under fractional Brownian motion

TL;DR

This paper calculates the almost sure intermediate dimension of images of specific sets under fractional Brownian motion, revealing a precise formula that is smaller than the H"older bound, and establishing their box-counting dimension.

Contribution

It provides a new exact formula for the intermediate dimension of images of certain sets under fractional Brownian motion, refining previous bounds.

Findings

Derived the exact intermediate dimension formula $rac{ heta}{ph+ heta}$

Established the box-counting dimension of the images

Showed the dimension is smaller than the H"older bound

Abstract

We show that the almost sure $\theta$-intermediate dimension of the image of the set $F_p =\{0, 1,\frac{1}{2^p},\frac{1}{3^p},\ldots\}$ under index-$h$ fractional Brownian motion is $\frac{\theta}{ph+\theta}$, a value that is smaller than that given by directly applying the H\"{o}lder bound for fractional Brownian motion. In particular this establishes the box-counting dimension of these images.