Ergodicity of a generalized Jacobi equation and applications

Abstract

Consider a 1-dimensional centered Gaussian process W with α-Hölder continuous paths on the compact intervals of R+(α∈]0,1[) and W0=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 does not satisfy the non-explosion condition. The regularity of the associated Itô 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 Xt, t>0 are proved.The paper concludes on a generalization of Morris–Lecar’s neuron model, where the normalized conductance of the K+ current is the solution of a generalized Jacobi’s equation.