Related papers: A Donsker Theorem for L\'evy Measures
In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an…
This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known L\'evy process: The compound Poisson process. The semi-Markov extension of…
This paper investigates the problem to determine whether a given stochastic process generates a sampled Brownian filtration. A fairly general sufficient condition is obtained by applying the Frank H. Clarke contraction criteria to a…
Consider a system of $n$ weakly interacting particles driven by independent Brownian motions. In many instances, it is well known that the empirical measure converges to the solution of a partial differential equation, usually called…
We develop a stochastic integration theory for predictable integrands with respect to a L\'evy basis. Our approach is based on decoupling inequalities for tangent sequences and reduces the construction of the stochastic integral essentially…
We consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group $\Gamma$ and a random walk $(S_n)_ {n\ge1}$ driven by a certain type of…
We first introduce and derive some basic properties of a two-parameters family of one-sided Levy processes. Their Laplace exponents are given in terms of the Pochhammer symbol. This family includes, in a limit case, the family of Brownian…
Consider a path of the reflected Brownian motion in the half-plane $\{y \ge 0\}$, and erase its part contained in the interior $\{y > 0\}$. What is left is, in an appropriate sense, a path of a jump-type stochastic process on the line $\{y…
In this article a fractional cross-diffusion system is derived as the rigorous many-particle limit of a multi-species system of moderately interacting particles that is driven by L\'{e}vy noise. The form of the mutual interaction is…
The classical notion of L\'evy process is generalized to one that takes as its values probabilities on a first order model equipped with a commutative semigroup. This is achieved by applying a convolution product on definable probabilities…
The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner's generalized Dirichlet distribution (R.H. Lochner, A Generalized…
The index Whittaker convolution operator, recently introduced by the authors, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. In this paper, we introduce…
We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit…
This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein's method with the Malliavin calculus of variations…
Barrier crossing is a widespread phenomenon across natural and engineering systems. While an abundant cross-disciplinary literature on the topic has emerged over the years, the stochastic underpinnings of the process are yet to be linked…
In this paper, we study the compressibility of random processes and fields, called generalized L\'evy processes, that are solutions of stochastic differential equations driven by $d$-dimensional periodic L\'evy white noises. Our results are…
Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the so-called contraction method to…
We study Fourier multipliers which result from modulating jumps of L\'evy processes. Using the theory of martingale transforms we prove that these operators are bounded in $L^p(\Rd)$ for $1<p<\infty$ and we obtain the same explicit bound…
Comb geometry, constituted of a backbone and fingers, is one of the most simple paradigm of a two dimensional structure, where anomalous diffusion can be realized in the framework of Markov processes. However, the intrinsic properties of…
An application of Levy's continuity theorem and Hankel transform allow us to establish a law limit theorem for the sequence $V_n=f(U)\sin(n U)$, where $U$ is uniformly distributed in $(0,1)$ and $f$ a given function. Further, we investigate…