The paper under review is to present a characterization of Lévy measures on strong duals of a nuclear space and provide conditions for the existence of regular versions to cylindrical Lévy processes in the dual space. There are few studies of general Lévy processes in the dual of a nuclear space, and this paper is the first to address the correspondence between Lévy processes and infinitely divisible measures on the dual of general nuclear spaces.
Section 2 starts with the introduction of nuclear spaces and their strong duals. The strong topology on the strong dual space is generated by the family of semi-norms with bounded sup-normal, and a locally convex space is nuclear if its topology is generated by a family of Hilbertian semi-norms through Hilbert-Schmidt canonical inclusions (examples are given in Section 6). A subset of a strong dual space \(\Phi'\) with respect to \(M\subset \Phi'\) \[ Z(\phi_1, \dots, \phi_n; A; M) = \{f\in \Phi': (f[\phi_i])_{1\le i \le n} \in A, \phi_i\in M\} = \pi^{-1}_{\phi_1,\dots,\phi_n} (A), \] is a cylindrical set, where a linear map \(\pi_{\phi_1,\dots,\phi_n}: \Phi' \to \mathbb{R}^n\) is given by \(\pi_{\phi_1,\dots,\phi_n}(f) = (f[\phi_i])_{1\le i \le n}\). The author in previous work [J. Theor. Probab. 31, No. 2, 867–894 (2018; Zbl 1405.60005)] characterized regular \(\Phi'\)-valued random variables and cylindrical processes in \(\Phi'\).
Section 3 devotes to the infinitely divisible measures and convolution semi-groups in the strong dual, it shows that for any \(\mu\) infinitely divisible measure in a strong quasi-complete dual space, there exists a unique continuous convolution semi-group \(\{\mu_t\}_{t\ge 0}\) with \(\mu_1 = \mu\) in Theorem 3.2 based on results of E. Siebert [Z. Wahrscheinlichkeitstheor. Verw. Geb. 28, 227–247 (1974; Zbl 0264.60004); Ann. Probab. 4, 433–443 (1976; Zbl 0338.60012)]. Conversely, for any continuous convolution semi-group \(\{\mu_t\}_{t\ge 0}\) in a strong quasi-complete dual space, then \(\{\mu_t\}_{t\in [0, T]}\) is uniformly tight for any \(T>0\). Every \(\Phi'\)-valued Lévy process determines a cylindrical Lévy process in the strong dual space in Lemma 3.7, where \(L=\{L_t\}_{t\ge 0}\) is a cylindrical Lévy process if for any \(n\), \(\{(L_t(\phi_1), \dots, L_t(\phi_n))\}_{t\ge 0}\) is a \(\mathbb{R}^n\)-valued Lévy process. Theorem 3.8 provides sufficient conditions for a cylindrical Lévy process to be one in the strong dual. Its proof relies on the author’ thesis and a characterization of regular random variables, the Prokhorov theorem, as well as the weak topology. For a cylindrical Lévy process in a strong dual space, Theorem 3.9 shows that there is a \(\Phi'\)-valued, regular, cádlág version of the cylindrical Lévy process, unique up to indistinguishable versions. Sufficient conditions for the existence of a cádlág version of the Lévy process with a finite \(n\)-th moment in some Hilbert spaces are given in Theorem 3.12, that will play an important role in the later Lévy-Itô decomposition. One of the main results, Theorem 3.14, describes the existence of a \(\Phi'\)-valued, regular, cádlág cyclindrical Lévy process for infinitely divisible measures on strong dual of a barrelled nuclear space.
Section 4 starts with Poisson random measures and Poisson processes on the strong dual space summarized in Proposition 4.2, where the distribution, characteristic function, mean and variation are given. The Lévy measure is defined in Definition 4.7 to resemble the property that characterizes Lévy measures on Hilbert spaces in [K. R. Parthasarathy, Probability measures on metric spaces. Probability and Mathematical Statistics. A Series of Monographs and Textbooks. New York-London: Academic Press (1967; Zbl 0153.19101)]. The main result Theorem 4.11 on Lévy measures of a Lévy process relies on the Minlo’s lemma due to X. Fernique [Invent. Math. 3, 282–292 (1967; Zbl 0173.16104)] for the inclusion norm on semi-norms. The main result Theorem 4.17 on the Lévy-Itô decomposition occupies subsection 4.5, first to give characteristic functions for a \(\Phi'\)-valued zero mean, regular and square-integrable, cádlág Lévy process in Theorem 4.13, and then to characterize a \(\Phi'\)-valued zero-mean and covariance functional indistinguishable version of Wiener process of the functional version, and the proof of Theorem 4.17 combines previous theorems and cylindrical Lévy process from the definition in a strong dual space. Theorem 4.18 gives Lévy-Khintchine theorem for a \(\Phi'\)-valued, regular, cádlág cyclindrical Lévy process with the characteristic functional on the strong dual. The final results on the Lévy-Khintchine theorem for infinitely divisible measures are given in Theorem 5.1, Section 5.
Examples, remarks and comparisons are explored in Section 6, literature reviews on Lévy process, Lévy-Itô decomposition, Lévy-Khintchine theorem on a strong dual of some particular nuclear spaces are listed, and various related results are mentioned. Other than first considering regularization of cyclindrical Lévy process in the paper, it is still interesting to see if this setup on the strong dual can help us understand Lévy process deeper in the nuclear space.