  In this article, we consider additive functionals $\zeta_t = \int_0^t
f(X_s)\mathrm{d} s$ of a c\`adl\`ag Markov process $(X_t)_{t\geq 0}$ on
$\mathbb{R}$. Under some general conditions on the process $(X_t)_{t\geq 0}$
and on the function $f$, we show that the persistence probabilities verify
$\mathbb{P}(\zeta_s < z \text{ for all } s\leq t ) \sim \mathcal{V}(z)
\varsigma(t) t^{-\theta}$ as $t\to\infty$, for some (explicit)
$\mathcal{V}(\cdot)$, some slowly varying function $\varsigma(\cdot)$ and some
$\theta\in (0,1)$. This extends results in the literature, which mostly focused
on the case of a self-similar process $(X_t)_{t\geq 0}$ (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
$(X_t)_{t\geq 0}$, together with a Wiener--Hopf decomposition of an auxiliary
(bivariate) L\'evy 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 $\theta$, which we write as $\theta = \rho
\beta$, with $\beta$ the scaling exponent of the local time of $(X_{t})_{t\geq
0}$ at level $0$ and $\rho$ the (asymptotic) positivity parameter of the
auxiliary L\'evy process.
