Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

Abstract

Let Φ be a nuclear space and let Φ ′ β denote its strong dual. In this work we establish the oneto-one correspondence between infinitely divisible measures on Φ ′ β and Lévy processes taking values in Φ ′ β . Moreover, we prove the Lévy-Itô decomposition, the Lévy-Khintchine formula and the existence of càdlàg versions for Φ ′ β -valued Lévy processes. A characterization for Lévy measures on Φ ′ β is also established. Finally, we prove the Lévy-Khintchine formula for infinitely divisible measures on Φ ′ β .

Semimartingales on duals of nuclear spaces

“…Properties of cylindrical and Φ ′ -valued Lévy processes were studied by the author in [13]. The reader is referred also to [27,40] for other studies on Lévy and additive processes in the dual of some particular classes of nuclear spaces.…”

“…Example 3.11. If X = (X t : t ≥ 0) is a cylindrical Lévy process in Φ ′ satisfying any of the conditions in Remark 3.9, its regularized Φ ′ -valued semimartingale version Y = (Y t : t ≥ 0) is indeed a Φ ′ -valued càdlàg Lévy process (see Theorem 3.8 in [13]). Moreover, for a Φ ′ -valued Lévy process the conditions in Remark 3.9 are always satisfied if Φ is also barrelled (see Corollary 3.11 in [13]).…”

“…In this section we study in detail the canonical decomposition and characteristics of a Φ ′ -valued Lévy process and how they relate with their Lévy-Itô decomposition studied in [13]. Let L = (L t : t ≥ 0) be a Φ ′ -valued càdlàg Lévy process.…”

“…Suppose that for every T > 0, the family (L t : t ∈ [0, T ]) of linear maps from Φ into L 0 (Ω, F , P) is equicontinuous (at the origin) (see Example 3.11). Under the above conditions it is proved in Section 4 in [13] that the random measure µ of the jumps of L is a Poisson Random measure, that we denote by N , and that its (predictable) compensator measure is of the form ν(ω; dt; df ) = dtν(df ), where ν is a Lévy measure on Φ ′ in the following sense (see [13], Theorem 4.12):…”

“…Finally, in Sect. 5 we examine in detail the canonical representation of a Φ ′ -valued Lévy process and its relation with their Lévy-Itô decomposition studied by the author in [13]. It is shown that the predictable characteristics of a Lévy process coincide with those of its Lévy-Khintchine formula and that the particular form of these character-istics distinguish the Lévy process among the Φ ′ -valued semimartingales.…”