A Decomposition of Multiple Wiener Integrals by the Lévy Process and Lévy Laplacian

Logarithmic Sobolev inequalities: regularizing effect of Lévy operators and asymptotic convergence in the Lévy-Fokker-Planck equation

In this paper we study some applications of the Lévy logarithmic Sobolev inequality to the study of the regularity of the solution of the fractal heat equation, i. e. the heat equation where the Laplacian is replaced with the fractional Laplacian. It is also used to the study of the asymptotic behaviour of the Lévy-Ornstein-Uhlenbeck process.

Universal Malliavin Calculus in Fock and Lévy-Itô Spaces

We review and extend Lindsay’s work on abstract gradient and divergence operators in Fock space over a general complex Hilbert space. Precise expressions for the domains are given, the L2-equivalence of norms is proved and an abstract version of the Itô-Skorohod isometry is established. We then outline a new proof of Itô’s chaos expansion of complex Lévy-Itô space in terms of multiple Wiener-Lé...

Quantum Stochastic Process Associated with Quantum Lévy Laplacian

A noncommutative extension of the Lévy Laplacian, called the quantum Lévy Laplacian, is introduced and its relevant properties are studied, in particular, a relation between the classical and quantum Lévy Laplacians is studied. We construct a semigroup and a quantum stochastic process generated by the quantum Lévy Laplacian.

A Functional Limit Theorem for Stochastic Integrals Driven by a Time-changed Symmetric Α-stable Lévy Process

Under proper scaling and distributional assumptions, we prove the convergence in the Skorokhod space endowed with the M1-topology of a sequence of stochastic integrals of a deterministic function driven by a time-changed symmetric α-stable Lévy process. The time change is given by the inverse β-stable subordinator.

Lévy processes, the Wiener-Hopf factorisation and applications: part II