## Electronic Journal of Probability

### Small-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak Hörmander type

Karen Habermann

#### Abstract

We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak Hörmander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expression for the bridge from $0$ to $0$ in time $1$ of an iterated Kolmogorov diffusion.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 11, 19 pp.

Dates
Accepted: 4 February 2019
First available in Project Euclid: 18 February 2019

https://projecteuclid.org/euclid.ejp/1550480425

Digital Object Identifier
doi:10.1214/19-EJP274

#### Citation

Habermann, Karen. Small-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak Hörmander type. Electron. J. Probab. 24 (2019), paper no. 11, 19 pp. doi:10.1214/19-EJP274. https://projecteuclid.org/euclid.ejp/1550480425

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