Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications

Abstract

Let Φ ′ β denotes the strong dual of a nuclear space Φ and let D T (Φ ′ β ) be the Skorokhod space of right-continuous with left limits (càdlàg) functions from [0, T ] into Φ ′ β . In this article we introduce the concepts of cylindrical random variables and cylindrical measures on D T (Φ ′ β ), and prove analogues of the regularization theorem and Minlos theorem for extensions of these objects to bona fide random variables and probability measures on D T (Φ ′ β ) respectively. Later, we establish analogues of Lévy's continuity theorem to provide necessary and sufficient conditions for uniform tightness of families of probability measures on D T (Φ ′ β ) and sufficient conditions for weak convergence of a sequence of probability measures on D T (Φ ′ β ). Extensions of the above results to the space D ∞ (Φ ′ β ) of càdlàg functions from [0, ∞) into Φ ′ β are also given. Afterwards, we apply our results to study weak convergence of Φ ′ β -valued càdlàg processes and in particular to Lévy processes. We finalize with an application of our theory to the study of tightness and weak convergence of probability measures on the Skorokhod space D ∞ (H) where H is a Hilbert space.

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

“…, from an application of Fatou's Lemma we can check that r is lower-semicontinuous. Then Proposition 5.7 in [17] shows that r is continuous on Ψ. As Ψ is a nuclear space, there exists a continuous Hilbertian seminorm q on Ψ such that r ≤ q and i r,q is Hilbert-Schmidt.…”

“…where for each n ∈ N, d n γ is the pseudometric defined in (2.1) for T = n. The Skorokhod topology in D ∞ (Φ ′ ) is a completely regular topology generated by the family of pseudometrics (d ∞ γ : γ ∈ Γ). For further details see [17,21].…”

“…Since Y is càdlàg there exists Ω 0 ⊆ Ω such that P(Ω 0 ) = 1 and for each ω ∈ Ω we have that s → X s (ω) ∈ D ∞ (Ψ ′ ). As Ψ is barrelled, given any t > 0 and ω ∈ Ω 0 there exists a continuous seminorm p on Ψ such that s → Y s (ω) ∈ D t (Ψ ′ p ) (Remark 3.6 in [17]). Hence, we have…”

“…The process (Y t : t ≥ 0) is hence adapted. Now, given ω ∈ Ω we have s → X s (ω) ∈ D t (Ψ ′ ) and because Ψ is barrelled, there exists a continuous seminorm p = p(t, ω) on Ψ such that s → X s (ω) ∈ D t (Ψ ′ p (see Remark 3.6 in [17]). Since [0, t] ∋ r → G(r, t)ψ is continuous, the set {G(r, t)ψ : 0 ≤ r ≤ t} is bounded in Ψ, hence under the seminorm p. Then, for each ψ ∈ Ψ, it follows that…”

Stochastic integration with respect to cylindrical semimartingales

“…In that case, for P-a.e. ω ∈ Ω, by Remark 3.6 in [15] for each T > 0 there exists a continuous Hilbertian seminorm p on Φ such that t → Y t (ω) is càdlàg from [0, T ] into Φ p . In particular we have…”