Related papers: Explicit identities for L\'evy processes associate…
Conditioning stable L\'evy processes on zero probability events recently became a tractable subject since several explicit formulas emerged from a deep analysis using the Lamperti transformations for self-similar Markov processes. In this…
Let $X=\{X(t),t\in R_+\}$ be a real-valued symmetric L\'{e}vy process with continuous local times $\{L^x_t,(t,x)\in R_+\times R\}$ and characteristic function $Ee^{i\lambda X(t)}=e^{-t\psi(\lambda)}$. Let…
We study sums of independent and identically distributed random velocities in special relativity. We show that the resulting one-dimensional velocity distributions are not only stable under relativistic velocity addition but define a…
By killing a stable L\'{e}vy process when it leaves the positive half-line, or by conditioning it to stay positive, or by conditioning it to hit 0 continuously, we obtain three different positive self-similar Markov processes which…
We classify L\'evy processes according to the solution spaces of the associated parabolic PIDEs. This classification reveals structural characteristics of the processes and is relevant for applications such as for solving PIDEs numerically…
We analyze confining mechanisms for L\'{e}vy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump - type processes are…
Multistable L\'evy motions are extensions of L\'evy motions where the stability index is allowed to vary in time. Several constructions of these processes have been introduced recently, based on Poisson and Ferguson-Klass-LePage series…
We consider a class of L\'evy-type processes derived via a Doob-transform from L\'evy processes conditioned by a control function called potential. These processes have position-dependent and generally unbounded components, with stationary…
Taking account of recent developments in the representation of $d$-dimensional isotropic stable L\'evy processes as self-similar Markov processes, we consider a number of new ways to condition its path. Suppose that $\Omega$ is a region of…
We obtain a representation of an inhomogeneous Levy process in a Lie group or a homogeneous space in terms of a drift, a matrix function and a measure function. Because the stochastic continuity is not assumed, our result generalizes the…
We introduce a notion of geometric tempering using exponentially-dampened Mittag-Leffler tempering functions and closely investigate the univariate case. Characteristic exponents and cumulants are calculated, as well as spectral densities.…
We characterise, in terms of their transition laws, the class of one-dimensional L\'evy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is exhibited by a broad class of infinite…
We compare two definitions of multistable L\'evy motions. Such processes are extensions of classical L\'evy motion where the stability index is allowed to vary in time. We show that the two multistable L\'evy motions have distinct…
We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump L\'evy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1<p<\infty$, of the corresponding…
In this paper, we consider a long-time behavior of stable-like processes. A stable-like process is a Feller process given by the symbol $p(x,\xi)=-i\beta(x)\xi+\gamma(x)|\xi|^{\alpha(x)},$ where $\alpha(x)\in(0,2)$, $\beta(x)\in\R$ and…
We consider a class of L\'evy-type processes on which spectral analysis technics can be made to produce optimal results, in particular for the decay rate of their survival probability and for the spectral gap of their ground state…
The periodic homogenization problem of integro-differential equations of the alpha stable L{\'e}vy operators is studied in this paper. Thanking to the symmetry of the L{\'e}vy density, we can use the method of the formal asymptotic…
We analyze a specific class of random systems that are driven by a symmetric L\'{e}vy stable noise. In view of the L\'{e}vy noise sensitivity to the confining "potential landscape" where jumps take place (in other words, to environmental…
In this paper we consider Harnack inequalities with respect to a symmetric $\alpha$-stable L\'evy process $X$ in $\mathbb{R}^d$, $\alpha \in (0,2)$, $d\geq 2$. We study the example from the article \cite{bg-sz-1}. There, the authors have…
We show on- and off-diagonal upper estimates for the transition densities of symmetric Levy and Levy-type processes. To get the an-diagonal estimates we prove a Nash type inequality for the related Dirichlet form. For the off-diagonal…