Imperial College London
Portrait Picture

ProfessorThomasCass

Faculty of Natural Sciences, Department of Mathematics

Professor of Mathematics
 
 
 
//

Contact

 

thomas.cass

 
 
//

Location

 

808Weeks BuildingSouth Kensington Campus

//

Summary

 

Publications

Citation

BibTex format

@article{Cass:2015:plms/pdu040,
author = {Cass, T and Lyons, T},
doi = {plms/pdu040},
journal = {Proceedings of the London Mathematical Society},
pages = {83--107},
title = {Evolving communities with individual preferences},
url = {http://dx.doi.org/10.1112/plms/pdu040},
volume = {110},
year = {2015}
}
Download

RIS format (EndNote, RefMan)

TY  - JOUR
AB  - The goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals. We model the population by a measure space (,,) where  determines the abundance of individual preferences. The preferences of an individual ∈ are described by a measurable choice ()of a rough path.We aim to identify, for each individual, a choice for the forward evolution ()for an individual in the community. These choices () must be consistent so that ()correctly accounts for the individual's preference and correctly models their interaction with the aggregate behaviour of the community.In general, solutions are continuum of interacting threads analogous to the huge number of individual atomic trajectories that together make up the motion of a fluid. The evolution of the population need not be governed by any overarching partial differential equation (PDE). Although one can match the standard nonlinear parabolic PDEs of McKean–Vlasov type with specific examples of communities in this case. The bulk behaviour of the evolving population provides a solution to the PDE.We focus on the case of weakly interacting systems, where we are able to exhibit the existence and uniqueness of consistent solutions.An important technical result is continuity of the behaviour of the system with respect to changes in the measure assigning weight to individuals. Replacing the deterministic  with the empirical distribution of an independent and identically distributed sample from leads to many standard models, and applying the continuity result allows easy proofs for propagation of chaos.The rigorous underpinning presented here leads to uncomplicated models which have wide applicability in both the physical and social sciences. We make no presumption that the macroscopic dynamics are modelled by a PDE.This work builds on the fine probability literature considering the limit behaviour for systems where
AU  - Cass,T
AU  - Lyons,T
DO  - plms/pdu040
EP  - 107
PY  - 2015///
SN  - 0024-6115
SP  - 83
TI  - Evolving communities with individual preferences
T2  - Proceedings of the London Mathematical Society
UR  - http://dx.doi.org/10.1112/plms/pdu040
UR  - http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000350125200004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
VL  - 110
ER  -
Download