Related papers: Excursions away from a regular point for one-dimen…

We prove that the spatial Wiener-Hopf factorisation of a L\'evy process or random walk without killing is unique.

Trawl processes belong to the class of continuous-time, strictly stationary, infinitely divisible processes; they are defined as Levy bases evaluated over deterministic trawl sets. This article presents the first nonparametric estimator of…

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on $\mathbb R^2$ under small spatial-invariant random perturbations. An expansion formula for mean curvatures is derived when the magnitude of the perturbation…

What is the analogue of L\'evy processes for random surfaces? Motivated by scaling limits of random planar maps in random geometry, we introduce and study L\'evy looptrees and L\'evy maps. They are defined using excursions of general L\'evy…

In this work we derive limit theorems for trawl processes. First,we study the asymptotic behaviour of the partial sums of the discretized trawl process $(X_{i\Delta_{n}})_{i=0}^{\lfloor nt\rfloor-1}$, under the assumption that as…

We study the homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can…

We determine the rate of decrease of the right tail distribution of the exponential functional of a Levy process with a convolution equivalent Levy measure. Our main result establishes that it decreases as the right tail of the image under…

We characterise the convergence of a certain class of discrete time Markov processes toward locally Feller processes in terms of convergence of associated operators. The theory of locally Feller processes is applied to L\'evy-type processes…

Let $X$ be a real L\'evy process and let $\Xpos $ be the process conditioned to stay positive. We assume that $ 0 $ is regular for $(-\infty, 0)$ and $(0, +\infty) $ with respect to $X$. Using elementary excursion theory arguments, we…

This paper is concerned with the limit theory of the extreme order statistics derived from random walks. We establish the joint convergence of the order statistics near the minimum of a random walk in terms of the Feller chains. Detailed…

In [16], under mild conditions, a Wiener-Hopf type factorization is derived for the exponential functional of proper L\'evy processes. In this paper, we extend this factorization by relaxing a finite moment assumption as well as by…

Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the Revuz measure used in the time change. In…

We study L\'{e}vy-like and truncated L\'{e}vy-like flights with step probability distribution of the form $r^{-1+\nu}$ for negative, positive, and zero $\nu$, focusing on the appearance of fractal geometry characteristics in the generated…

We prove that a general (not necessarily symmetric) L\'evy process killed on exiting a bounded open set (without regular condition on the boundary) is intrinsically ultracontractive, provided that $B(0,R_0)\subseteq \rm{supp}(\nu)$ for some…

We consider a continuous-time random walk which is defined as an interpolation of a random walk on a point process on the real line. The distances between neighboring points of the point process are i.i.d. random variables in the normal…

Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive L\'evy process with non-zero L\'evy measure. In this paper we study the asymptotic behavior of the local time process,…

We prove asymptotic behaviour of transition density for a large class of spectrally one-sided L\'evy processes of unbounded variation satisfying mild condition imposed on the second derivative of the Laplace exponent, or equivalently, on…

The main purpose of this chapter is to present some theoretical aspects of parametric estimation of L\'evy processes based on high-frequency sampling, with a focus on infinite activity pure-jump models. Asymptotics for several classes of…

The {\alpha}-stable L\'evy process, commonly used to describe L\'evy flight, is characterized by discontinuous jumps and is widely used to model anomalous transport phenomena. In this study, we investigate the associated exit problem and…

Rayleigh-L\'evy flights are simplified cosmological tools which capture certain essential statistical properties of the cosmic density field, including hierarchical structures in higher-order correlations, making them a valuable reference…