Transition densities for strongly degenerate time inhomogeneous random models

TL;DR

This paper investigates the existence and positivity of transition densities in strongly degenerate, time-inhomogeneous stochastic differential equations without global Lipschitz conditions, with applications to Hodgkin-Huxley models.

Contribution

It establishes the existence of transition densities near points satisfying the weak Hörmander condition in a degenerate, non-Lipschitz setting, extending previous results.

Findings

Transition densities exist near points satisfying the weak Hörmander condition.

Regions where these densities are positive are identified.

Applications to Hodgkin-Huxley models demonstrate practical relevance.

Abstract

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