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A model is presented relating the evolution of genomic GC content over time to AT$\rightarrow$GC and GC$\rightarrow$AT mutation rates. By employing Itô calculus it is shown that if mutation rates in asexually reproducing organisms are subject to random perturbations that can vary over time several implications follow. For instance, an extra Brownian motion term appears influencing nucleotide variability; the greater the variability of the random perturbations on the mutation rates the stronger the impact of the Brownian motion term. Reducing the influence of the random perturbations, to limit fitness decreasing and deleterious mutations, will likely imply divesting resources to genomic repair systems. The stable mutation rates seen in many organisms could thus be an evolved strategy to reduce the influence of the Brownian motion term. Furthermore, if change to genomic GC content, i.e. the GC content of variable sites or single nucleotide polymorphisms (SNPs), is just as likely to increase as to decrease, something that resembles knockout of repair enzymes and removal of selective pressures seen in evolutionary laboratory experiments, the species genome will likely decay unless infinite resources are available. These implications are solely a consequence of allowing random perturbations affect AT- and GC mutation rates and not obtainable using standard non-stochastic methodology. Finally, a connection between the model for genomic GC content evolution and the classical Luria-Delbrück mutation model is presented in a stochastic setting.