This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Capacity theoretic methods are used to produce dimension bounds for images of sets under Holder 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 capacity theoretic dimension profiles. As a corollary, this establishes continuity of the profiles for all Borel sets, allowing us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Holder 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 in the setting of projections.
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