A practical criterion for positivity of transition densities

TL;DR

This paper presents a simple, practical criterion to determine where the transition density of a degenerate diffusion process is strictly positive, based on local flow information, which is surprising given the global nature of positivity.

Contribution

It introduces a new criterion linking local Lie algebra data of the SDE to the positivity of transition densities, simplifying analysis in degenerate diffusions.

Findings

Local flow information often determines positivity regions

Positivity is more global than previously thought

The criterion builds on classical Lie algebra and diffusion theory

Abstract

We establish a simple criterion for locating points where the transition density of a degenerate diffusion is strictly positive. Throughout, we assume that the diffusion satisfies a stochastic differential equation (SDE) on $\mathbf{R}^d$ with additive noise and polynomial drift. In this setting, we will see that it is often that case that local information of the flow, e.g. the Lie algebra generated by the vector fields defining the SDE at a point $x\in \mathbf{R}^d$, determines where the transition density is strictly positive. This is surprising in that positivity is a more global property of the diffusion. This work primarily builds on and combines the ideas of Ben Arous and L\'eandre (1991) and Jurdjevic and Kupka (1981, 1985).