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arXiv · 1308.4884

Ergodicity of a Generalized Jacobi's Equation and Applications

Nicolas Marie
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Abstract

Consider a $1$-dimensional centered Gaussian process $W$ with $α$-Hölder continuous paths on the compact intervals of $\mathbb R_+$ ($α\in ]0,1[$) 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 $\textrm{K}^+$ current is the solution of a generalized Jacobi's equation.

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Nicolas Marie. 2015-09-02. Ergodicity of a Generalized Jacobi's Equation and Applications. https://doi.org/10.1016/j.spa.2015.07.015

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