Offset returns => Normal distribution => Brownian motion
Linear returns => Log-Normal distribution => Geometric Brownian motion
Affine returns => Log-Normal distribution + other => Geometric Brownian motion + other
Where other appears to be a logit-normal distribution stretched out. Over time you get change in the shape of the distribution which may mirror what Paul Wilmott described in his FAQ.
Pretty pics and link to a pre-print on Github here