A Laplace transform approach to C-semigroups on a T_{ε, λ}-complete random normed module

2025/03/05 by Xia Zhang, Zhang, Xia, Leilei Wei +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #45R05 #46H25 #FOS: Mathematics #Functional Analysis (math.FA) #Fuzzy Systems and Optimization #Probability and Risk Models #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2503.03188

openalex publication_date 2025/03/05 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/29

Abstract

In this paper, we first introduce the notion of the Laplace transform for an abstract-valued function from [0, ∞) to a T_{ε, λ}-complete random normed module S. Then, combining respective advantages of the (ε, λ)-topology and the locally L⁰-convex topology on S, we prove the differentiability, Post-Widder inversion formula and uniqueness of such a Laplace transform. Second, based on the above work, we establish the Hille-Yosida theorem for an exponentially bounded C-semigroup on S, considering both the dense and nondense cases of the range of C, respectively, which extends and improves several important results. Finally, we also apply such a Laplace transform to abstract Cauchy problems in the random setting.

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