Modelling Lévy space‐time white noises

Abstract

In this work, we compare Lévy space-time white noises and cylindrical Lévy processes. Lévy space-time white noises are defined as infinitely divisible independently scattered random measures and cylindrical Lévy processes are defined by means of the theory of cylindrical processes. It is shown that Lévy space-time white noises correspond to an entire sub-class of cylindrical Lévy processes, which is completely characterised by the characteristic functions of its members. We embed the Lévy space-time white noise, or the corresponding cylindrical Lévy process, in the space of general and tempered distributions. For the latter case, we show that this embedding is possible if and only if a certain integrability condition is satisfied. We establish that both embedded cylindrical processes are induced by genuine Lévy processes in the space of general or tempered distributions. We complete the picture by establishing Lévy space-time white noise as the weak derivative of Lévy additive sheets.

Stochastic integration in Hilbert spaces with respect to cylindrical martingale-valued measures

“…This is precisely the case of Example 3.4 below where a real-valued martingale-valued measure is constructed from an independently scattered cylindrical Lévy process. In particular, as pointed out in Remark 2.4 in Griffiths and Riedle (2021), the classical definition of Gaussian space-time white noise fits naturally in the setting of the Lévy-valued random martingale measures, and hence also satisfies all the conditions in Definition 3.1.…”

“…As shown in Griffiths and Riedle (2021), the above concept can be related with that of cylindrical Lévy processes. Indeed, for any given locally finite Borel measure…”

“…, L(t)1 An are independent for each t ≥ 0. In such case, Theorem 4.5 in Griffiths and Riedle (2021) shows that the prescription…”

“…where N is a Poisson random measure on R + × O × R with intensity leb ⊗ ζ ⊗ µ for a Lévy measure µ on B(R) satisfying R β 2 µ(dβ) < ∞. This cylindrical process is independently scattered as shown in Example 4.8 in Griffiths and Riedle (2021), has mean-zero and second moments given by…”

Stochastic wave equation with Lévy white noise

“…We refer the reader to the excellent monograph Peszat and Zabczyk (2007) and the references therein. The recent papers Jakubowski and Riedle (2017); Kosmala and Riedle (2021) contain significant advances on stochastic integration with respect to cylindrical Lévy noise, while Griffiths and Riedle (2021) gives a comparison between the cylindrical approach and the random field approach, which complements the similar comparison that was done by Dalang and Quer-Sardanyons (2011) in the Gaussian case.…”