  Many physical, chemical and biological systems have an inherent discrete
spatial structure that strongly influences their dynamical behaviour. Similar
remarks apply to internal or external noise, as well as to nonlocal coupling.
In this paper we study the combined effect of nonlocal spatial discretization
and stochastic perturbations on travelling waves in the Nagumo equation, which
is a prototypical model for bistable reaction-diffusion partial differential
equations (PDEs). We prove that under suitable parameter conditions, various
discrete-stochastic variants of the Nagumo equation have solutions, which stay
close on long time scales to the classical monotone Nagumo front with high
probability if the noise level and spatial discretization are sufficiently
small.
