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The main contribution of this paper is the development of a novel approach tomulti-scale analysis that we believe can be used to analyse processes withnon-equilibrium dynamics. Our approach will be referred to as \emph{multi-scaleanalysis with non-equilibrium feedback} and will be used to analyse a naturalrandom growth process with competition on $\mathbb{Z}^d$ called \emph{firstpassage percolation in a hostile environment} that consists of two firstpassage percolation processes $FPP_1$ and $FPP_{\lambda}$ that compete for theoccupancy of sites. Initially, $FPP_1$ occupies the origin and spreads throughthe edges of $\mathbb{Z}^d$ at rate 1, while $FPP_{\lambda}$ is initialised atsites called \emph{seeds} that are distributed according to a product ofBernoulli measures of parameter $p\in(0,1)$, where a seed remains dormant until$FPP_1$ or $FPP_{\lambda}$ attempts to occupy it before then spreading throughthe edges of $\mathbb{Z}^d$ at rate $\lambda>0$. Particularly challengingaspects of FPPHE are its non-equilibrium dynamics and its lack of monotonicity(for instance, adding seeds could be benefitial to $FPP_1$ instead of$FPP_\lambda$); such characteristics, for example, prevent the application of amore standard multi-scale analysis. As a consequence of our main result forFPPHE, we establish a coexistence phase for the model for $d\geq3$, answeringan open question in \cite{sidoravicius2019multi}. This exhibits a raresituation where a natural random competition model on $\mathbb{Z}^d$ observescoexistence for processes with \emph{different} speeds. Moreover, we are ableto establish the stronger result that $FPP_1$ and $FPP_{\lambda}$ can bothoccupy a \emph{positive density} of sites with positive probability, which isin stark contrast with other competition processes.

Non-equilibrium multi-scale analysis and coexistence in competing first passage percolation