On the rough diffusive limits of deterministic geometric mechanics
File(s)
Author(s)
Diamantakis, Theo
Type
Thesis
Abstract
Many branches of modern science must deal with randomness: observation or measurement noise, approximation errors or even quantum fluctuations. This thesis studies a generalised class of (variational) models in a framework known as geometric mechanics. Geometric mechanics allows formulating models for hydrodynamics obeying conservation laws derived from continuous symmetry (Lie groups). The thesis connects two perspectives of geometric mechanics using homogenisation theory; that of deterministic and stochastic (random) geometric mechanics. Our goal is to construct homogenised limits of flows that are incorporated in a variational principle agreeing with the Stochastic Advection by Lie Transport (SALT) model. Reaching this goal will enable this randomness in the SALT model to be interpreted as the diffusive limit of scale separated fast and slow terms in the velocity field and flow map. Our flow map approach also implies a novel random-coefficient "mean" partial differential equation that can be equivalently solved for the SALT dynamics. To derive these models, the subtle convergence properties of these chaotic limits can be handled by the use of rough path theory; in doing so, we deduce that the transport is modified by noise-induced drift. We find that this drift term preserves Lie group structures, but modifies qualitative phenomena such as stability and long time behaviour.
Version
Open Access
Date Issued
2024-10-03
Date Awarded
01/06/2025
Advisor
Holm, Darryl
Pavliotis, Grigorios
Sponsor
Engineering and Physical Sciences Research Council (Great Britain)
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)