Integration With Respect to Analogue of Wiener Measure Over Paths in Wiener Space and Its Applications

Abstract

Abstract. In 1992, the author introduced the definition and the properties of Wiener measure over paths in abstract Wiener space and this measure was investigated extensively by some mathematicians. In 2002, the author and Dr. Im presented an article for analogue of Wiener measure and its applications which is the generalized theory of Wiener measure theory. In this note, we will derive the analogue of Wiener measure over paths in abstract Wiener space and establish two integration formulae, one is similar to the Wiener integration formula and another is similar to simple formula for conditional Wiener integral. Furthermore, we will give some examples for our formulae.

Applications of an analogue of conditional Wiener integrals

“…In [13], the authors presented a simple formula for expressing conditional Wiener integrals in terms of ordinary Wiener integrals. Also, in [3,16], the authors introduced a simple formula for the analogue of conditional Wiener integral which is essentially similar to Park and Skoug's simple formula on Wiener space; in other words,…”

“…Furthermore, in [5,6,17,20], the authors studied various topics pertaining to this analogue of the Wiener measure, together with related topics concerning an analogue of Wiener space. In particular, the authors derived a simple formula for the conditional w ϕ -integral and then used this formula to evaluate the conditional w ϕ -integral of various functionals [3,16].…”

Wiener measure for Heisenberg group

“…Over the recent years, there are still much work on Wiener measure. See [5], [6], [7]. In this paper, we study the Wiener space on Heisenberg group.…”

The Analogue of Wiener Space With Values in Orlicz Space

“…In 1992, the authour showed the existence of m B , by the different way from Kuelbs and LePage's method, and found the Wiener integration formuala for it [9]. In 2002, the author and Dr.Im defined the analogue of Wiener measure on C[0, T ], associated with the Borel measure φ on R [11] and the author proved the existence theorem of analogue of Wiener measure space over paths in abstract Wiener space B, associated with the Borel measure on B [10].…”