Strongly degenerate time inhomogeneous SDEs: densities and support properties. Application to a Hodgkin-Huxley system with periodic input

Abstract

In this paper we study the existence of densities for strongly degenerate stochastic differential equations (SDEs) whose coefficients depend on time and are not globally Lipschitz. In these models neither local ellipticity nor the strong Hörmander condition is satisfied. In this general setting we show that continuous transition densities indeed exist in all neighborhoods of points where the weak Hörmander condition is satisfied. We also exhibit regions where these densities remain positive. We then apply these results to stochastic Hodgkin-Huxley models with periodic input as a first step towards the study of ergodicity properties of such systems in the sense of [27]-[28].

arXiv comment35 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1310.7373

References in corpus (3)

• Demographic noise and piecewise deterministic Markov processes
• Qualitative properties of certain piecewise deterministic Markov processes
• Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input

Cited by in corpus (3)

• Expected Exit Time for Time-Periodic Stochastic Differential Equations and Applications to Stochastic Resonance
• Ergodic Numerical Approximation to Periodic Measures of Stochastic Differential Equations
• Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Applications to a stochastic Hodgkin-Huxley model