Stochastic Evolution Equations with Lévy Noise in the Dual of a Nuclear Space

Abstract

In this article we give sufficient and necessary conditions for the existence of a weak and mild solution to stochastic evolution equations with (general) Lévy noise taking values in the dual of a nuclear space. As part of our approach we develop a theory of stochastic integration with respect to a Lévy process taking values in the dual of a nuclear space. We also derive further properties of the solution such as the existence of a solution with square moments, the Markov property and path regularity of the solution. In the final part of the paper we give sufficient conditions for the weak convergence of the solutions to a sequence of stochastic evolution equations with Lévy noises.

Almost Sure Uniform Convergence of Stochastic Processes in the Dual of a Nuclear Space

“…. , let (A n (t) : t ≥ 0) ⊆ L(Φ, Φ) such that for each t ≥ 0, A n (t) is the infinitesimal generator of a (C 0 , 1)semigroup (S n t (s) : s ≥ 0) on Φ (see Section 4.1 in [10] and references therein). Assume that there exists an increasing sequence (q m : m ≥ 0) of norms generating the topology on Φ such that the following two conditions hold: (1) For every k ≥ 0, there exists m ≥ k such that, for each t ≥ 0 and n ≥ 0, A n (t) has a continuous linear extension form Φ qm into Φ q k (also denoted by A n (t)) and the mapping t → A n (t) is L(Φ qm , Φ q k )-continuous.…”

“…Finally, in Section 5 we apply the tools developed in Section 4 to provide sufficient conditions for the almost surely convergence to the sequence of (generalized) Langevin equations dY n t = A n (t) ′ Y n dt + dL n t , where for each n ∈ N, (A n (t) ′ : t ≥ 0) is the family of dual operators to a family (A n (t) : t ≥ 0) of continuous linear operators which generates a backward evolution system (U (s, t) : 0 ≤ s ≤ t < ∞) of continuous linear operators on Φ and (L n t : t ≥ 0) is a Φ ′ -valued Lévy process. Existence, uniqueness and weak convergence of solutions in the Skorokhod space of such (generalized) Langevin equations were studied by the author in [10].…”

“…In [10], the author introduced a theory of existence and uniqueness, path properties, and weak converge of linear stochastic evolution equations with Lévy noise. In the following paragraphs we summarize the main results we will need for our result in the next section.…”

“…A weak solution to (5.3) is a Φ ′ -valued regular adapted process Y = (Y t : t ≥ 0) satisfying that for any given t ≥ 0, for each φ ∈ Φ we have Suppose that the mapping t → A(t)ψ is continuous from [0, ∞) into Φ for every φ ∈ Φ. By Theorem 5.7 in [10], the generalized Langevin equation ( 5.3) has a unique (up to indistinguishable versions) weak solution (X t : t ≥ 0) which is regular and has càdlàg paths, this solution satisfies P-a.e. X t , φ = η , U (0, t)φ + t 0 L s , A(s)U (s, t)φ ds + L t , φ , ∀ t ≥ 0, φ ∈ Φ.…”

“…Moreover if (L t : t ≥ 0) has continuous paths, then (X t : t ≥ 0) has continuous paths too. For further details and examples the reader is referred to [10].…”