First hitting times for general non-homogeneous 1d diffusion processes: density estimates in small time

Motivated by some applications in neurosciences, we here collect several estimates for the density of the first hitting time of a threshold by a non-homogeneous one-dimensional diffusion process and for the density of the associated process stopped at the threshold. We first remind the reader of the connection between both. We then provide some Gaussian type bounds for the density of the stopped process. We also discuss the stability of the density with respect to the drift. Proofs mainly rely on the parametrix expansion.