2023/04/18 by Berger, Quentin, Béthencourt, Loïc, Tardif, Camille
#60G51 #60J25 #60J55 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2304.09034
In this article, we consider additive functionals ζt = ∫0t f(Xs)d s of a càdlàg Markov process (Xt)t≥ 0 on ℝ. Under some general conditions on the process (Xt)t≥ 0 and on the function f, we show that the persistence probabilities verify ℙ(ζs < z for all s≤ t ) ∼ V(z) ς(t) t-θ as t→∞, for some (explicit) V(⋅), some slowly varying function ς(⋅) and some θ∈ (0,1). This extends results in the literature, which mostly focused on the case of a self-similar process (Xt)t≥ 0 (such as Brownian motion or skew-Bessel process) with a homogeneous functional f (namely a pure power, possibly asymmetric). In a nutshell, we are able to deal with processes which are only asymptotically self-similar and functionals which are only asymptotically homogeneous. Our results rely on an excursion decomposition of (Xt)t≥ 0, together with a Wiener--Hopf decomposition of an auxiliary (bivariate) Lévy process, with a probabilistic point of view. This provides an interpretation for the asymptotic behavior of the persistence probabilities, and in particular for the exponent θ, which we write as θ= ρβ, with β the scaling exponent of the local time of (Xt)t≥ 0 at level 0 and ρ the (asymptotic) positivity parameter of the auxiliary Lévy process.