Cylindrical Lévy processes in Banach spaces

Abstract

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 Lévy processes. These have (weak) Lévy-Itô decompositions and an associated Lévy-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 Lévy 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.

Infinitely divisible cylindrical measures on Banach spaces

“…The notion cylindrical Lévy process appears the first time in Peszat and Zabczyk [16] and it is followed by the works Brzeźniak, Goldys et al [3], Brzeźniak and Zabzcyk [4] and Priola and Zabczyk [17]. The first systematic introduction of cylindrical Lévy processes appears in our work Applebaum and Riedle [1].…”

“…This approach in [1] is inspired by the analogue definition for cylindrical Wiener processes, see Kallianpur and Xiong [10], Metivier and Pellaumail [14] or Riedle [18]. In the same way as cylindrical Wiener processes are related to the class of Gaussian cylindrical measures, the introduction of cylindrical Lévy processes in [1] leads to the new class of infinitely divisible cylindrical measures which have not been considered so far. Since the article [1] is focused on cylindrical Lévy processes and their stochastic integral, no further properties of infinitely divisible cylindrical measures are derived.…”

“…The introduction of cylindrical Lévy processes in [1] are based on the theory of cylindrical or generalised processes and cylindrical measures, see for example Schwartz [20] or Vakhaniya et al [21]. This approach in [1] is inspired by the analogue definition for cylindrical Wiener processes, see Kallianpur and Xiong [10], Metivier and Pellaumail [14] or Riedle [18].…”

Cylindrical fractional Brownian motion in Banach spaces

“…The implication (a) ⇒ (b) can be proved as Theorem 4.8 in [2]. For establishing the implication (b) ⇒ (a), it is immediate that the right hand side of (4.1) converges.…”

“…In the same way, one can introduce cylindrical Wiener processes, see for instance [16,18,24], and recently, this approach has been accomplished in [2] to give the first systematic treatment of cylindrical Lévy processes. Definition 4.1.…”