Related papers: L\'evy Processes in a Step 3 Nilpotent Lie Group
Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential…
Let $\mathfrak{g}$ be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows to define a generalized multiplication $f \# g = (f^{\vee} * g^{\vee})^{\wedge}$ of two…
The probabilistic symbol is the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginals of a stochastic process. This object, as long as the derivative exists, provides crucial information…
In this paper we study the Cauchy problem for diffusion equations associated to a class of strongly hypoelliptic pseudo-differential operators on graded Lie groups. To do so, we develop a global complex functional calculus on graded Lie…
In this paper we study pseudo-processes related to odd-order heat-type equations composed with L\'evy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be…
We study the regularity of solutions to the integro-differential equation $Af-\lambda f=g$ associated with the infinitesimal generator $A$ of a L\'evy process. We show that gradient estimates for the transition density can be used to derive…
We suppose that a L\'evy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the L\'evy-Khinchine characteristics as the number of observations…
In this article, we develop a calculus of Shubin type pseudodifferential operators on certain non-compact spaces, using a groupoid approach similar to the one of van Erp and Yuncken. More concretely, we consider actions of graded Lie groups…
In the present paper we show that the Ito representation of the infinitesimal generator $L$ for Levy processes can be written in a convolution type form. Using the obtained convolution form and the theory of integral equations with…
For a L\'evy process $\xi=(\xi_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows \[{\rm{I}}_{\xi}=\int_0^{\infty}e^{\xi_t} dt.\] Under mild conditions on $\xi$, we show that the following…
We present the theory of the Dirichlet problem for nonlocal operators which are the generators of general pure-jump symmetric L\'evy processes whose L\'evy measures need not be absolutely continuous. We establish basic facts about the…
In this paper, we consider the exponential functional \(A_{\infty}=\int_0^\infty e^{-\xi_s}ds\) of a L{\'e}vy process \(\xi_s\) and aim to estimate the characteristics of \(\xi_{s}\) from the distribution of \(A_{\infty}\). We present a new…
When is it possible to interpret a given Markov process as a L\'evy-like process? Since the class of L\'evy processes can be defined by the relation between transition probabilities and convolutions, the answer to this question lies in the…
This note is devoted to representation of some evolution semigroups. The semigroups are generated by pseudo-differential operators, which are obtained by different (parametrized by a number $\tau$) procedures of quantization from a certain…
We define a class of not necessarily linear $C_0$-semigroups $(P_t)_{t\geq0}$ on $C_b(E)$ (more generally, on $C_\kappa(E):=\frac1\kappa C_b(E)$, for some bounded function $\kappa$, which is the pointwise limit of a decreasing sequence of…
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 describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $\mathfrak{g}$. The Lie algebra generators are represented…
We adopt the $p$-group generation algorithm to classify small-dimensional nilpotent Lie algebras over small fields. Using an implementation of this algorithm, we list the nilpotent Lie algebras of dimension at most~9 over $\F_2$ and those…
Consider a third order linear differential equation $L(f)=0$, where $L\in\mathbb{Q}(z)[\partial_z]$. We design an algorithm computing the Liouvillian solutions of $L(f)=0$. The reducible cases devolve to the classical case of second order…
A particularly easy, even if for long overlooked way is presented for defining globally arbitrary Lie group actions on smooth functions on Euclidean domains. This way is based on the appropriate use of the usual parametric representation of…