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We study the small-time asymptotics for hypoelliptic diffusion processesconditioned by their initial and final positions, in a model class ofdiffusions satisfying a weak H\"ormander condition where the diffusivity isconstant and the drift is linear. We show that, while the diffusion bridge canexhibit a blow-up behaviour in the small time limit, we can still make sense ofsuitably rescaled fluctuations which converge weakly. We explicitly describethe limit fluctuation process in terms of quantities associated to theunconditioned diffusion. In the discussion of examples, we also find anexpression for the bridge from 0 to 0 in time 1 of an iterated Kolmogorovdiffusion.

Small-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak H\"ormander type