Related papers: L\'evy processes conditioned on having a large hei…

In this paper we study the spectral heat content for various L\'evy processes. We establish the asymptotic behavior of the spectral heat content for L\'{e}vy processes of bounded variation in $\mathbb{R}^{d}$, $d\geq 1$. We also study the…

We find necessary and sufficient conditions for almost sure finiteness of integral functionals of spectrally positive L\'evy processes. Via Lamperti type transforms, these results can be applied to obtain new integral tests on extinction…

We have shown recently that a Markov process conditioned on rare events involving time-integrated random variables can be described in the long-time limit by an effective Markov process, called the driven process, which is given…

We consider a random walk on a Galton-Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive…

We develop a scale-invariant truncated L\'evy (STL) process to describe physical systems characterized by correlated stochastic variables. The STL process exhibits L\'evy stability for the probability density, and hence shows scaling…

A spectrally positive additive L\'evy field is a multidimensional field obtained as the sum $\mathbf{X}_{\rm t}={\rm X}^{(1)}_{t_1}+{\rm X}^{(2)}_{t_2}+\dots+{\rm X}^{(d)}_{t_d}$, ${\rm t}=(t_1,\dots,t_d)\in\mathbb{R}_+^d$, where ${\rm…

In this paper, we present new results on Hunt's hypothesis (H) for L\'{e}vy processes. We start with a comparison result on L\'{e}vy processes which implies that big jumps have no effect on the validity of (H). Based on this result and the…

We show that the law of the overall supremum $\bar{X}_t=\sup_{s\le t}X_s$ of a L\'evy process $X$ before the deterministic time $t$ is equivalent to the average occupation measure $\mu_t(dx)=\int_0^t\p(X_s\in dx)\,ds$, whenever 0 is regular…

We study the maximal degree of (sub)critical L{\'e}vy trees which arise as the scaling limits of Bienaym{\'e}-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree…

For near-critical, transient Markov chains on the non-negative integers in the Lamperti regime, where the mean drift at $x$ decays as $1/x$ as $x \to \infty$, we quantify degree of transience via existence of moments for conditional return…

We use the martingale convergence method to get the weak convergence theorem on general functionals of partial sums of independent heavy-tailed random variables. The limiting process is the stochastic integral driven by $\alpha-$stable…

We establish a dichotomy in the small-time asymptotic behavior of the spectral heat content (SHC) for symmetric, but not necessarily isotropic, L\'evy processes whose L\'evy density satisfies a weak lower scaling condition near zero. This…

This paper constructs a class of martingale transforms based on L\'evy processes on Lie groups. From these, a natural class of bounded linear operators on the $L^p$-spaces of the group (with respect to Haar measure) for $1<p<\infty$, are…

For a continuous-time Bienaym\'e-Galton-Watson process, $X$, with immigration and culling, $0$ as an absorbing state, call $X^q$ the process that results from killing $X$ at rate $q\in (0,\infty)$, followed by stopping it on extinction or…

For a spectrally negative positive self-similar Markov process with an a.s. finite overall supremum we provide, in tractable detail, a kind of conditional Wiener-Hopf factorization at the maximum of the absorption time at zero, the…

This article deals with IDT processes, i.e. processes which are infinitely divisible with respect to time. Given an IDT process $(X_{t},\,t\geq0)$, there exists a unique (in law) L\'evy process $(L_{t}; t\geq0)$ which has the same…

We study a real-valued L\'evy-type process $X$, which is locally $\alpha$-stable in the sense that its jump kernel is a combination of a `principal' (state dependent) $\alpha$-stable part with a `residual' lower order part. We show that…

Given a spectrally negative L\'evy process, we predict, in a $L_1$ sense, the last passage time of the process below zero before an independent exponential time. This optimal prediction problem generalises Baurdoux and Pedraza (2020) where…

The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $\Delta f = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a…

In this paper we give the decomposition of a martingale under the sublinear expectation associated with a $G$-L'evy process X with finite activity and without drift. We prove that such a martingale consists of an Ito integral w.r.t.…