科研成果详情

We fix a rich probability space (Ω, ℱ P). Let (ℍ, ∥·∥) be a separable Hilbert space and let μ be the canonical cylindrical Gaussian measure μ on ℍ. Given any abstract Wiener space (ℍ, double-sturck B sing, μ) over ℍ, and for every Hilbert-Schmidt operator T: ℍ ⊂ double-sturck B sing, → ℍ which is (|·|, ∥·∥)-continuous, where |·| stands for the (Gross-measurable) norm on double-sturck B sing, we construct an OrnsteinUhlenbeck process ξ: (Ω ℱ P) × [0, 1] → (double-sturck B sing, |·|) as a pathwise solution of the following infinite-dimensional Langevin equation dξt=db + T(ξ)dt with the initial data ξ = 0, where b is a double-sturck B sing-valued Brownian motion based on the abstract Wiener space (ℍ, double-sturck B sing, μ). The richness of the probability space (Ω, ℱ, P) then implies the following consequences: the probability space Ω is independent of the abstract Wiener space (ℍ, double-sturck B sing, μ) (in the sense that (Ω, ℱ, P) does not depend on the choice of the Gross-measurable norm |·|) and the space C consisting of all continuous double-sturck B sing-valued functions on [0,1] is identical with the set of all paths of ξ. Finally, we present a way to obtain pathwise continuous solutions ξ: dξ = √|α| β db + α · ξ dt with initial data ξ = 0. where α, β ε R, α ≠ 0 and 0 < β.