  Consider a $1$-dimensional centered Gaussian process $W$ with
$\alpha$-H\"older continuous paths on the compact intervals of $\mathbb R_+$
($\alpha\in ]0,1[$) and $W_0 = 0$, and $X$ the local solution in rough paths
sense of Jacobi's equation driven by the signal $W$. The global existence and
the uniqueness of the solution are proved via a change of variable taking into
account the singularities of the vector field, because it doesn't satisfy the
non-explosion condition. The regularity of the associated It\^o map is studied.
By using these deterministic results, Jacobi's equation is studied on
probabilistic side : an ergodic theorem in L. Arnold's random dynamical systems
framework, and the existence of an explicit density with respect to Lebesgue's
measure for each $X_t$, $t > 0$ are proved. The paper concludes on a
generalization of Morris-Lecar's neuron model, where the normalized conductance
of the $\textrm{K}^+$ current is the solution of a generalized Jacobi's
equation.
