Convergence of integral functionals of one-dimensional diffusions

In this paper we describe the pathwise behaviour of the integral functional integral(t)(0) f(Y-u) du for any t is an element of [0, zeta], where zeta is (a possibly infinite) exit time of a one-dimensional diffusion process Y from its state space, f is a nonnegative Borel measurable function and the coefficients of the SDE solved by Y are only required to satisfy weak local integrability conditions. Two proofs of the deterministic characterisation of the convergence of such functionals are given: the problem is reduced in two different ways to certain path properties of Brownian motion where either the Williams theorem and the theory of Bessel processes or the first Ray-Knight theorem can be applied to prove the characterisation. As a simple application of the main results we give a short proof of Feller's test for explosion.