Cylindrical Wiener Processes

Abstract

In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach space under consideration. Finally, we relate this integral to well-known stochastic integrals in literature.

Cylindrical Lévy processes in Banach spaces

“…The purpose of this paper is to begin a systematic study of cylindrical Lévy processes in Banach spaces with particular emphasis on stochastic integration and applications to SPDEs. It can be seen as a successor to an earlier paper by the second-named author (see [17]) in which some aspects of this programme were carried out for cylindrical Wiener processes. The organisation of the paper is as follows.…”

“…Obviously, a weakly cylindrical Wiener process is an example of a weakly cylindrical Lévy process. The characteristic function of W is given by [17] for more details on cylindrical Wiener processes.…”

“…may exhibit similar behaviour to Q 2 in that it might be discontinuous; see [17] for an example. Consequently, we call L a strongly cylindrical Lévy process of weak order 2 if both Q 1 and Q 2 map to U .…”

Cylindrical fractional Brownian motion in Banach spaces

“…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.…”

Infinitely divisible cylindrical measures on Banach spaces

“…Moreover, according to Remark 4.5 the quadratic form q is of the form q(a) = a, Qa for all a ∈ U * and for a symmetric, positive operator Q : U * → U * * . If Q(U * ) ⊆ U then W is a cylindrical Wiener process in a strong sense as it is usually considered in the literature, see Riedle [18].…”