White noise differential equations for vector-valued white noise functionals
- PDF / 2,322,177 Bytes
- 27 Pages / 439.37 x 666.142 pts Page_size
- 73 Downloads / 226 Views
Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00088-5 ORIGINAL PAPER
White noise differential equations for vector‑valued white noise functionals Un Cig Ji1 · Peng Cheng Ma2 Received: 30 March 2020 / Accepted: 18 August 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, a framework of vector-valued white noise functionals has been constructed as a Gel’fand triple W𝛼 ⊗ E ⊂ 𝛤 (H) ⊗ K ⊂ (W𝛼 ⊗ E)∗ . Base on the Gel’fand triple, a new notion of Wick product of vector-valued white noise functionals induced by a bilinear mapping 𝔅 ∶ E∗ × E∗ → E is introduced and called a 𝔅-Wick product. We establish the unique existence of solution of an abstract functional differential equation base on the Gel’fand triple. For our purpose, we improve slightly the well-known analytic characterization theorem for S-transform, and the convergence theorem in terms of S-transform for a sequence of vector-valued generalized white noise functionals. As an application, we study the Wick type differential equations for vector-valued white noise functionals, and as examples we discuss the Wick type differential equations for certain operator algebra valued white noise functionals which naturally includes random matrices of white noise functionals. Keywords White noise theory · Gaussian white noise functional · S-transform · White noise integral equation · Wick product · Wick type differential equation. Mathematics Subject Classification 60H40 · 60H20 · 46F25
Communicated by Jan van Neerven. * Un Cig Ji [email protected] Peng Cheng Ma [email protected] 1
Department of Mathematics, Institute for Industrial and Applied Mathematics, Chungbuk National University, Cheongju 28644, Korea
2
Department of Mathematics, Chungbuk National University, Cheongju 28644, Korea
Vol.:(0123456789)
U. C. Ji, P. C. Ma
1 Introduction Since the white noise theory [10, 11, 26, 27] initiated by Hida [8, 9] to give precise mathematical meaning of (Gaussian) white noise as the time derivative of Brownian motion, the white noise calculus has been developed extensively and successfully with wide applications to stochastic calculus [11, 26], mathematical finance, quantum stochastic calculus [12, 13, 17, 19, 21], quantum filed theory [20, 22] and so on. Recently, the infinite dimensional stochastic, especially stochastic partial, differential equations have been refocused and becomes more important with extensive applications [3, 5, 32, 33]. On the other hand, Obata [28, 29] studied the operator calculus on the space of vector-valued Gaussian white noise functionals, and then little attention has been given to the calculus on the vector-valued Gaussian white noise functionals. Main purpose of this paper is to study an abstract white noise differential equation based on the framework of the vector-valued white noise functionals. Then as an application, we study the Wick type differential equations. Over the past decades a considerable number of studies have been made on Wick calculus [7, 11, 14, 15, 23, 26, 31]
Data Loading...
