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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 Rd with additive noise and polynomial drift. In this setting, we will see that it is often the case that local information of the flow, e.g. the Lie algebra generated by the vector fields defining the SDE at a point x ∈ Rd, 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 Arous and Léandre (1991 Décroissance exponentielle du noyau de la chaleur sur la diagonale. II Probab. Theory Relat. Fields 90 377–402 ) and Jurdjevic and Kupka (1985 Polynomial control systems Math. Ann. 272 361–8 ).
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