Brownian dimension

One way to determine an objects' dimension is to answer the following question

How does the number $\mathbf{N}$ of circles of radius $r$ that are required to cover the object change with $r$ ?

For instance, a line of length $L$ can be covered by $L/r$ circles of radius $r$ , and $\mathbf{N}\sim r^{-1}$ . A square of side length $L$ can be covered by $\sim L^2/r^2$ circles of radius $r$ , and $\mathbf{N}\sim r^{-2}$ . It is for this reason that we call a line 1 dimensional, and we call the square 2 dimensional. Here we'd like to ask, what is the dimensionality of a particle undergoing Brownian motion (diffusion)?

A particle undergoes Brownian motion through the action of a random force so that its displacement in a time interval $\Delta t$ behaves as $\sim\sqrt{2D\Delta t}$ . Because the particle is 0d, we might expect that its trajectory is 1d, as for particles in projectile motion.

Suppose we have a perfectly sampled trajectory of a particle, $\gamma_B$ . We want to lay down a set of circles, $C$ , of radius $r$ to cover this path, i.e. so that every section of the particle's path in the 2d plane is overlaid by a portion of a circle $C_i \in C$ . $\mathbf{N}_\textrm{Brownian}$ changes with $r$ according to $\mathbf{N}_\textrm{Brownian} \sim r^d$ . What is $\lvert d \rvert$ , i.e. what is the dimensionality of Brownian motion?