# 查普曼-科尔莫戈罗夫等式

${\displaystyle p_{i_{1},\ldots ,i_{n}}(f_{1},\ldots ,f_{n})}$

${\displaystyle p_{i_{1},\ldots ,i_{n-1}}(f_{1},\ldots ,f_{n-1})=\int _{-\infty }^{\infty }p_{i_{1},\ldots ,i_{n}}(f_{1},\ldots ,f_{n})\,df_{n}}$

（注意这里各随机变量的顺序不重要）.

## 特化为马尔可夫链

${\displaystyle p_{i_{1},\ldots ,i_{n}}(f_{1},\ldots ,f_{n})=p_{i_{1}}(f_{1})p_{i_{2};i_{1}}(f_{2}\mid f_{1})\cdots p_{i_{n};i_{n-1}}(f_{n}{\mid }f_{n-1}),}$

（其中条件概率${\displaystyle p_{i;j}(f_{i}{\mid }f_{j})}$${\displaystyle i>j}$时间的转移概率。Chapman-Kolmogorov等式简化为：

${\displaystyle p_{i_{3};i_{1}}(f_{3}{\mid }f_{1})=\int _{-\infty }^{\infty }p_{i_{3};i_{2}}(f_{3}\mid f_{2})p_{i_{2};i_{1}}(f_{2}\mid f_{1})df_{2}.}$

${\displaystyle P(t+s)=P(t)P(s)\,}$

（其中${\displaystyle P(t)}$是转移矩阵，${\displaystyle X_{t}}$t时间的系统状态），则对于系统状态空间中的任意两个点ij

${\displaystyle P_{ij}(t)=P(X_{t}=j\mid X_{0}=i).}$

## 参考文献

• The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.