arXiv subjects

David Applebaum

Publications and source records attributed to David Applebaum.

22 records · Page 2Linked to original sources

arXiv2011

Aspects of Recurrence and Transience for Levy Processes in Transformation Groups and Non-Compact Riemannian Symmetric Pairs

We study recurrence and transience for Lévy processes induced by topological transformation groups. In particular the transience-recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of non-compact type are transient. We show that we also have "harmonic transience", i.e. local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli's Lévy-Khinchine formula.

math.PR↗

arXiv2011

Pseudo Differential Operators and Markov Semigroups on Compact Lie Groups

We extend the Ruzhansky-Turunen theory of pseudo differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N.Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by Lévy processes. We study a family of Lévy-type linear operators on general Lie groups that are pseudo differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms.

math.PR↗

arXiv2009

Cylindrical Levy processes in Banach spaces

Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito decompositions and an associated Levy-Khintchine formula. If the process is weakly square integrable, its covariance operator can be used to construct a reproducing kernel Hilbert space in which the process has a decomposition as an infinite series built from a sequence of uncorrelated bona fide one-dimensional Levy processes. This series is used to define cylindrical stochastic integrals from which cylindrical Ornstein-Uhlenbeck processes may be constructed as unique solutions of the associated Cauchy problem. We demonstrate that such processes are cylindrical Markov processes and study their (cylindrical) invariant measures.

math.PR↗

arXiv2008

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 $L^2$-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-Lévy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform now identifies Lévy-Itô space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general Lévy process. In particular we establish maximal domains of these operators and obtain the Itô-Skorohod isometry on its maximal domain.

math.PR↗