The Net Charge Process for Interacting, Signed Diffusions

Abstract

We consider a system of $N (\rightarrow \infty)$ interacting one-dimensional diffusions, in which each diffusion is assigned a random charge $(\pm 1)$, and study the behavior of the net charge distribution through space and time. The diffusion equations are a slight variation of those considered in the initial studies of "the propagation of chaos," but the interaction involves the signs of the diffusions and triplet rather than pairwise interactions. This has the effect of leading to a non-Gaussian fluctuation theory, which turns out to be close to the $P(:\Phi^4:)$ models of Euclidean quantum field theory. The main tools of the proofs involve the Stroock-Varadhan martingale theory and a general theory of $U$-statistics.