## 1. Introduction

[2] Many geophysical systems can be modeled as deterministic dynamics perturbed by stochastic impulses. One type of perturbation is the marked Poisson process with exponentially distributed inter-arrival times and random (exponential) i.i.d. jumps. Such a process is often used in geophysics to model, for example, stochastic rainfall [*Rodriguez*-*Iturbe et al.*, 1999], river flow variability [*Lefebvre and Guilbault*, 2008], fire occurrence [*D*'*Odorico et al.*, 2007], landslides [*D*'*Odorico and Fagherazzi*, 2003], earthquakes [*Kagan and Jackson*, 1991], etc.

[3] An important aspect of stochastic dynamics lies in the properties of threshold crossings. These properties are especially relevant to the study of geophysical processes having, or being influences by, threshold behaviors as discussed by *Zehe and Sivapalan* [2009]. Crossing properties are often characterized by the probability distribution of the time required to cross a threshold starting from a generic initial point. Noteworthy results concerned with the moments of such distribution include, e.g., the mean first passage time for non-Markovian processes [*Masoliver*, 1987] and for Poisson processes with exponential jumps [*Laio et al.*, 2001b]. *McGrath et al.* [2007] derived all the moments of the probability distribution of first upcrossing times, with specific reference to the case of linear systems driven by Poisson noise. All these works are related to dynamical systems with additive disturbances.

[4] We study here the excursion times, intended as the time intervals spent above/below a threshold, of dynamical systems forced by a marked Poisson noise, either additive or multiplicative. We derive exact expressions for the probability of excursions of infinitesimal duration; such expressions, together with mean first passage times, enable the approximation of the probability density function of excursion times with parametric distributions having two parameters. These functions, although not providing a conclusive solution to the complex problem of crossing properties of Poisson-driven systems, are sound approximations based on the dynamical systems characteristics and represent a significant improvement compared to the standard assumption of exponentially distributed excursion times [see, e.g., *Botter et al.*, 2007]. Examples of ecohydrological processes that can benefit for the solutions proposed here include surface runoff generation, water soil infiltration and leaching, vegetation water stress and anoxic stress, and hydroperiods (or time intervals spent above the soil surface by a fluctuating water table).