| Summary: | 碩士 === 國立成功大學 === 數學系 === 89 === In this paper, the Brownian Bridge is represented as a Brownian Functional on the classical Wiener space C=C[0,1] in such a way that
Br(x,t)=(x,beta(t))=(x,alpha(t))-t(x,alpha(1))
for all x belongs to C where beta(t)(s)=s^t-st and alpha(t)(s)=s^t. Being motivated from the above representation, we study the abstract Wiener space structure for the path space C_br={x belongs to C : x(0)=x(1)=0} of Brownian Bridge. Let C'_br={x belongs to C':x(0)=x(1)=0} and define
T(xi)(t) = xi(t)-[xi(1)] for all xi belongs to C,
we show that (i,C'_br,C_br) forms an abstract Wiener space and the transformation T carries the abstract Wiener pair (C',C) onto the abstract Wiener pair (C'_br, C_br). Generalizations of the above results are obtained.
1 Introduction 2
2 Brownian Bridge on Wiener space 3
2.1 Definition of Brownian Motion and Brownian Bridge......3
2.2 Wiener measure associated with Brownian Motion.........3
2.3 Wiener measure associated with Brownian Bridge.........4
2.4 Preliminaries in Abstract Wiener Space.................4
2.5 The Classical Wiener Space.............................5
2.6 Fuctional Representation of Brownian Motion and
Brownian Bridge on Classical Wiener Space..............5
3 The Abstract Wienr Space Structure of Brownian Bridge 7
3.1 The path space C_br....................................7
3.2 Representation of C*_br...............................11
4 Transforms that preserve AWS structure 13
4.1 The transform that carries Brownian Motion to
Brownian Bridge.......................................13
4.2 Homeomorphisms between AWS............................16
References 20
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