dc.contributor.author Habermann, Karen dc.date.accessioned 2018-02-09T14:20:33Z dc.date.available 2018-02-09T14:20:33Z dc.date.issued 2018-05-19 dc.date.submitted 2017-08-30 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/271855 dc.description.abstract Sub-Riemannian geometry is the natural setting for studying dynamical systems, as noise often has a lower dimension than the dynamics it enters. This makes sub-Riemannian geometry an important field of study. In this thesis, we analysis some of the aspects of sub-Riemannian diffusion processes on manifolds. We first focus on studying the small-time asymptotics of sub-Riemannian diffusion bridges. After giving an overview of recent work by Bailleul, Mesnager and Norris on small-time fluctuations for the bridge of a sub-Riemannian diffusion, we show, by providing a specific example, that, unlike in the Riemannian case, small-time fluctuations for sub-Riemannian diffusion bridges can exhibit exotic behaviours, that is, qualitatively different behaviours compared to Brownian bridges. We further extend the analysis by Bailleul, Mesnager and Norris of small-time fluctuations for sub-Riemannian diffusion bridges, which assumes the initial and final positions to lie outside the sub-Riemannian cut locus, to the diagonal and describe the asymptotics of sub-Riemannian diffusion loops. We show that, in a suitable chart and after a suitable rescaling, the small-time diffusion loop measures have a non-degenerate limit, which we identify explicitly in terms of a certain local limit operator. Our analysis also allows us to determine the loop asymptotics under the scaling used to obtain a small-time Gaussian limit for the sub-Riemannian diffusion bridge measures by Bailleul, Mesnager and Norris. In general, these asymptotics are now degenerate and need no longer be Gaussian. We close by reporting on work in progress which aims to understand the behaviour of Brownian motion conditioned to have vanishing $N$th truncated signature in the limit as $N$ tends to infinity. So far, it has led to an analytic proof of the stand-alone result that a Brownian bridge in $\mathbb{R}^d$ from $0$ to $0$ in time $1$ is more likely to stay inside a box centred at the origin than any other Brownian bridge in time $1$. dc.description.sponsorship Research supported by EPSRC Grant EP/H023348/1 for the Cambridge Centre for Analysis. dc.language.iso en dc.rights No Creative Commons licence (All rights reserved) dc.subject sub-Riemannian diffusions dc.subject small-time asymptotics dc.subject Malliavin calculus dc.subject signature of a path dc.title Geometry of sub-Riemannian diffusion processes dc.type Thesis dc.type.qualificationlevel Doctoral dc.type.qualificationname Doctor of Philosophy (PhD) dc.publisher.institution University of Cambridge dc.publisher.department Cambridge Centre for Analysis dc.date.updated 2018-02-09T13:39:36Z dc.identifier.doi 10.17863/CAM.18862 dc.contributor.orcid Habermann, Karen [0000-0002-3533-900X] dc.publisher.college Newnham College dc.type.qualificationtitle PhD in Mathematics cam.supervisor Norris, James rioxxterms.freetoread.startdate 2018-02-09
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