The Generalized Analogue of Wiener Measure Space and Its Properties

Abstract

Abstract. In this note, we introduce the definition of the generalized analogue of Wiener measure on the space C [a, b] of all realvalued continuous functions on the closed interval [a, b], give several examples of it and investigate some important properties of it -the Fernique theorem and the existence theorem of scale-invariant measurable subsets on C [a, b]. PreliminariesIn 1923, Wiener proved the existence theorem of the meaningly measure on the space C 0 [a, b], the space of all real-valued continuous functions on a closed bounded interval [a, b] which vanish at a, the so-called Wiener space in [9]. This is based on the properties of Brownian motion of a single small particle. In 2002, the author and Dr. Im presented the definition and properties of analogue of Wiener measure on the space C [a, b], the space of all real-valued continuous functions on [a, b] in [3]. This is the theory of many particles, moving along the law of Brownian motion.In this note, we introduce the definition of the generalized analogue of Wiener measure space, which is more generalized concept of Wiener measure space and we give several examples. Furthermore, we investigate important theorems -the Fernique theorem on C [a, b] and the existence theorem of scale-invariant measurable subsets on C [a, b].In this section, we give some notations, definitions and facts which are needed to understand the subsequent sections.

Measurable functions similar to the itô integral and the Paley-Wiener-Zygmund integral over continuous paths

“…Remark 2.8. Some results of Corollaries 2.4, 2.7 and Theorems 2.5, 2.6 were proved by Ryu using Theorem 2.1 [11,12].…”

“…The Borel σ-algebra B(C[0, T]) of C[0, T] with the supremum norm, coincides with the smallest σ-algebra generated by I and there exists a unique positive finite measure w α,β;ϕ on B(C[0, T]) with w α,β;ϕ (I) = m α,β;ϕ (I) for all I ∈ I. This measure w α,β;ϕ is called an analogue of a generalized Wiener measure on (C[0, T], B(C[0, T])) according to ϕ [11,12].…”

“…Let C[0, T] denote an analogue of a generalized Wiener space [4,11,12], the space of continuous realvalued functions on the interval [0, T]. On the space C[0, T], we introduce a finite measure w α,β;ϕ and investigate its properties, where α, β : [0, T] → R are appropriate functions such that β is strictly increasing, and ϕ is an arbitrary finite measure on the Borel class B(R) of R. Using this finite measure w α,β;ϕ , we also introduce two measurable functions on C[0, T]; one of them is similar to the Itô type integral I α,β ( ) for ∈ L 2 α,β [0, T], where L 2 α,β [0, T] is the L 2 -space with respect to the Lebesgue-Stieltjes measure induced by α and β, and the other is similar to the PWZ integral.…”

A Banach Algebra Similar to Cameron-Storvick’s One with Its Equivalent Spaces

“…The Borel -algebra B( [0, ]) of [0, ] with the supremum norm coincides with the smallest -algebra generated by C and there exists a unique positive finite measure , ; on B( [0, ]) with , ; ( ) = , ; ( ) for all ∈ C. This measure , ; is called an analogue of a generalized Wiener measure on ( [0, ], B( [0, ])) according to [8,9].…”

The Translation Theorem on the Generalized Analogue of Wiener Space and Its Applications

“…The theories of the generalized Wiener measure space (C 0 [7]. This is the generalization of the concrete Wiener measure space, including both concepts of the generalized of the concrete Wiener measure space and the analogue of Wiener measure space.…”