Mijatović, Aleksander; Urusov, Mikhail Convergence of integral functionals of one-dimensional diffusions. (English) Zbl 1372.60044 Electron. Commun. Probab. 17, Paper No. 61, 13 p. (2012). Summary: In this paper, we describe the pathwise behaviour of the integral functional \(\int^t_0 f(Y_u)du\) 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. As a simple application of the main results we give a short proof of Feller’s test for explosion. Cited in 1 ReviewCited in 25 Documents MSC: 60F17 Functional limit theorems; invariance principles 60J60 Diffusion processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J55 Local time and additive functionals Keywords:integral functionals; diffusions; local time; Bessel process; Ray-Knight theorem; Williams theorem × Cite Format Result Cite Review PDF Full Text: DOI arXiv