Ergodicity of a Generalized Jacobi's Equation and Applications

Abstract

Consider a $1$-dimensional centered Gaussian process $W$ with $\alpha$-Hölder 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 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 $X_t$, $t>0$ are proved. The paper concludes on a generalization of Morris-Lecar's neuron model, where the normalized conductance of the ${\text{K}}^{+}$ current is the solution of a generalized Jacobi's equation.

32 pages, 4 figures. Stochastic Processes and their Applications, 2015

References in corpus (1)

• A Generalized Mean-Reverting Equation and Applications