  We consider a stochastic partial differential equation (SPDE) model for
chemorepulsion, with non-linear sensitivity on the one-dimensional torus. We
show that for any suitable initial data there exists a pathwise unique, global
solution to the SPDE. Furthermore we show that the associated semi-group is
Markov and possesses a unique invariant measure, supported on a H\"older-Besov
space of positive regularity, which the solution law converges to exponentially
fast. We also establish tail bounds on the invariant measure that are heavier
than Gaussian when measured using any $L^p$ norm.
