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Convergence of integral functionals of one-dimensional diffusions. (English) Zbl 1372.60044

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.

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
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