Convergence of Integral Functionals of One-Dimensional Diffusions

In this expository paper we describe the pathwise behaviour of the integral functional ${\int }_0^{t}f(Y_u)\text{dd}u$ for any $t\in [0,\zeta ]$, 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.