Ergodicity of a generalized Jacobi equation and applications

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Marie, Nicolas^1,2 

[1] Univ Paris Ouest, Lab Modal X, F-92000 Nanterre, France

[2] ESME Sudria, Lab ISTI, F-75015 Paris, France

关键词: Euler scheme;    Fractional Brownian motion;    Jacobi's equation;    Malliavin calculus;    Morris-Lecar's model;    Random dynamical systems;    Rough paths;    Stochastic differential equations;   

DOI  :  10.1016/j.spa.2015.07.015

来源: Elsevier

【 摘 要 】

Consider a 1-dimensional centered Gaussian process W with alpha-Holder continuous paths on the compact intervals of R+ (alpha is an element of ]0, l[) 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 does not satisfy the non-explosion condition. The regularity of the associated Ito 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 K+ current is the solution of a generalized Jacobi's equation. (C) 2015 Elsevier B.V. All rights reserved.

【 授权许可】

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