  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 the measure determines the abundance of
individual preferences. The preferences of an individual are described by a
measurable choice of a rough path. We focus on the case of weakly interacting
systems, where we are able to exhibit the existence and uniqueness of
consistent solutions. 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 over-arching 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. 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 measure with the empirical
distribution of an i.i.d. sample from it 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 a large no of particles are interacting with independent
preferences; there is also work on continuum models with preferences described
by a semi-martingale measure. We mention some of the key papers.
