Ergodicity of a Generalized Jacobi's Equation and Applications

TL;DR

This paper investigates the ergodic properties and solution regularity of a generalized Jacobi's equation driven by Gaussian processes, establishing existence, uniqueness, and density results, with applications to neuron modeling.

Contribution

It introduces a novel analysis of a generalized Jacobi's equation, proving global existence, uniqueness, ergodicity, and explicit density, extending classical results to rough path driven stochastic equations.

Findings

Proved global existence and uniqueness of solutions despite singularities.

Established ergodic theorem within a random dynamical systems framework.

Derived explicit density for solutions at positive times.

Abstract

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.