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

2014/10/01 by Höpfner, Reinhard, Löcherbach, E., Thieullen, M.
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1410.0341

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].

Related

• Transition densities for strongly degenerate time inhomogeneous random models
• Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Applications to a stochastic Hodgkin-Huxley model
• Entrance measures for semigroups of time-inhomogeneous SDEs: possibly degenerate and expanding
• Transition densities for stochastic Hodgkin-Huxley models
• Probability density for a hyperbolic SPDE with time dependent coefficients