Our result in "Quasi-sure non-self-intersection for rough differential equations
driven by fractional Brownian motion" says that for any rough differential equation driven by fractional Brownian motion
there is 0 $(r,q)$-capacity for the set of paths with self-intersection when

$\frac{2}{H} +rq < d$.

Note that $H$ is the Hurst
parameter of the Brownian motion and that $(0,1)$-capacity is equal to probability.
This visualization illustrates a fractional Brownian motion in $\mathbb{R}^3$.
Play with the slider and see the self-intersections appear and disappear at the critical value for $H=2/3$ for 0 probability of self-intersection on $\mathbb{R}^3$.