A Short Note About Markov Process, Transition Semi-group etc.

Markov process 的機率特性 (意指 pdf, fdd 等) 由 transition probability 完全決定。所以先介紹 transition probability 的前身 -- measure kernal ：

Def. Measure Kernal
Let (X,A) and (Y,B) be measurable spaces. A measure kernal k is a function form X,B to R

that is a measure on B given a point in X, and an A-measurable given a event in B.
When the measure is a probability, k is called a probability kernal.

當 (Y,B) = (X,A) 的時候， probability kenals 可以`相乘’：
Def. Product of Probability Kernals
\mu and \nu are probability kernals, their product is defined as

where x is a point in X and B is a event in \mathcal{B}, sorry for the bad notations.
Remark. 我們可以把 k(x,B) 想像成一個 partical 從 x 出發，最後落到 B  的機率。上面的乘法 (\mu \nu)(x,B) 可以解釋成一個 partical 從 x 出發，先經過 y ，再落到 B 上的機率。
Remark2. 注意一下乘法的順序， \mu\nu 就好像是 \mu 作用在 \nu 上面。的確，按照定義 \mu 是當作 measure。

Example. When \mu is the delta distribution concentrated on x

then it is an identity

當然， Probability kernals 還可以跟 X 上的 measure 相乘：

可以解釋為 \pi 作用在 \mu 上面 (因為 \pi 作為 measure)。另外一個對偶的解釋是 \mu 作用在 \pi 上變成 X 上一個新的 measure 。

Def. Transition Semi-group
A transition semi-group is a collection of probability kernals
indexed by two time variable s and t (s<=t) such that
1) for all s equals to t

2) for all s<=t<=u
 
A transition semi-group is called time-homogeneous if

for all s and t.
Remark. Semi-group 就是有乘法、有結合律的集合。
Remark2. 實際上這是一個 Monoid ，因為 delta 扮演 identity 的角色。

我們知道給定 Markov process 有對應的 transition semi-group ，那給定 transition semi-group 是否有對應的 Markov process 呢？答案是肯定的。以下定理如是說：

Thm. Existence of Markov Process Give a Transition Semi-group
Given a transition semi-group and a collection of distribution \pi_t on (X,A) such that

for all s and t, then there exists a Markov process X(t) having the same probability kernals and distributions. The converse is also true that a Markov process defines a transition semi-group via

Remark. 證明存在就用 Kolmogorov ExtensionTheorem 。因為給定 transition semi-group 和一個初始 distribution 就足以決定所有的 fdd (finite dimensional distribution) ，且由於 transition semi-group 的定義，這些 fdd 一定 consistent 。

Thm. Markov property is preserved under coraser filter
If a Markov process is adapted to {g_t}, then it is still Markovian for all coraser filters.