分數布朗運動的表現是以分數微積分作基礎。在本篇文章裡我們 會先討論到分數微積分的定義以及性質，包括存在性、函數的微分與 積分的不同表示法、互為反算子、以及半群性質。再來，利用維納積 分定義分數布朗運動，並且利用分數微積分展示出分數布朗運動的核 的不同的表示法。文章的末了我們也會討論一些有關分數布朗運動的 一些性質。 In this thesis, we discuss the representations of fractional Brownian motion (fBm) and properties of fBm. We name it “fractional” because the kernels can be represented by fractional calculus. My thesis has three contributions. First, we show some properties about fractional calculus, including existence, various expressions of derivative and integral of function, inverse operators, and semigroup property. Second, we propose to use fractional calculus to indicate fBm and different representations of kernels. Finally, we show some properties of fBm in the end of this thesis.
