  Consider a mean-reverting equation, generalized in the sense it is driven by
a 1-dimensional centered Gaussian process with H\"older continuous paths on
$[0,T]$ ($T > 0$). Taking that equation in rough paths sense only gives local
existence of the solution because the non-explosion condition is not satisfied
in general. Under natural assumptions, by using specific methods, we show the
global existence and uniqueness of the solution, its integrability, the
continuity and differentiability of the associated It\^o map, and we provide an
$L^p$-converging approximation with a rate of convergence ($p\geq 1$). The
regularity of the It\^o map ensures a large deviation principle, and the
existence of a density with respect to the Lebesgue measure, for the solution
of that generalized mean-reverting equation. Finally, we study a generalized
mean-reverting pharmacokinetic model.
