  This paper concerns the intermediate dimensions, a spectrum of dimensions
that interpolate between the Hausdorff and box dimensions. Potential theoretic
methods are used to produce dimension bounds for images of sets under H\"older
maps and certain stochastic processes. We apply this to compute the almost-sure
value of the dimension of Borel sets under index-$\alpha$ fractional Brownian
motion in terms of dimension profiles defined using capacities. As a corollary,
this establishes continuity of the profiles for Borel sets and allows us to
obtain an explicit condition showing how the Hausdorff dimension of a set may
influence the typical box dimension of H\"older images such as projections. The
methods used propose a general strategy for related problems; dimensional
information about a set may be learned from analysing particular fractional
Brownian images of that set. To conclude, we obtain bounds on the Hausdorff
dimension of exceptional sets, with respect to intermediate dimensions, in the
setting of projections.
