EPSRC Reference: 
EP/Y001478/1 
Title: 
LargeN limit of horizontal Brownian motions on Lie groups 
Principal Investigator: 
Habermann, Dr K 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Statistics 
Organisation: 
University of Warwick 
Scheme: 
Standard Research  NR1 
Starts: 
01 December 2023 
Ends: 
30 November 2024 
Value (£): 
56,725

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
17 May 2023

ECR International Collaboration Grants Panel 1

Announced


Summary on Grant Application Form 
This collaborative project aims to explore largeN limits of natural horizontal Brownian motions on Lie groups of N x N matrices. It lies at the intersection of probability theory, differential geometry and group theory, and particularly combines the study of stochastic processes on Lie groups and the study of subRiemannian geometries in a novel way.
SubRiemannian geometries model systems with constraints but set up such that the system moves over all parts of the phase space, that is, the constraints are flexible enough that any two points in the space can be connected by a curve satisfying the constraints. These type of geometries naturally appear in all sciences, ranging from constrained physical systems over motion planning in robotics to modelling the first layer of the visual cortex of the brain. For instance, the position of a vehicle in a field can be described by specifying the coordinates of its centre and the angle of rotation with respect to a reference line. In this threedimensional parameter space, it is not possible to perform motions which correspond to a movement perpendicular to the direction of the wheels of the vehicle. However, by choosing suitable maneuvers it is still possible to reach any target position.
LargeN limits of Brownian motions on Lie groups of N x N matrices have been actively studied. The analysis employs tools from free probability and the results have implications in random matrix theory. In these works, the Lie groups in considerations are equipped with a canonical Riemannian structure.
We plan to now tackle the natural question of what can be said about the largeN limits of horizontal Brownian motions on Lie groups of N x N matrices where, instead of using a Riemannian structure, the Lie groups are equipped with canonical subRiemannian structures.

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


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Project URL: 

Further Information: 

Organisation Website: 
http://www.warwick.ac.uk 