A New Proof of the Engelbert-Schmidt Zero-One Law for Time-Homogeneous Diffusions

In this paper we give a new proof to the Engelbert-Schmidt zero-one law for time-homogeneous diffusions, which provides deterministic criteria for the convergence of integral functional of diffusions. Our proof is based on stochastic time change and Feller's test of explosions, and does not rely on advanced methods such as the first Ray-Knight theorem, Wiliam's theorem, Shepp's dichotomy result for Gaussian processes or Jeulin's lemma as in the previous literature (see Mijatovi c and Urusov (2012) for a pointer to the literature). The new proof has an intuitive interpretation as we link the integral functional to the explosion time of an associated diffusion process