馬爾可夫算子從哪裡來?


5

我在Chaos, Fractals, and Noise by Lasota and Mackey這本書中找到了"馬爾可夫運算符"的定義。

根據度量$ \ mu $,用$ L ^ 1(\ mu)$表示Lebesgue可積函數的空間。然後,將運算符$$ P:L ^ 1(\ mu)\ to L ^ 1(\ mu)$$稱為馬爾可夫運算符$$ f \ geq 0 \ Rightarrow Pf \ geq 0 $$和$$ \ int(Pf)(x)\ mu(dx)= \ int f(x)\ mu(dx)$$對於L ^ 1(\ mu)$中的任何$ f都成立。

但這是從哪裡來的?誰發明的?

5

This is a continuous analog of (transposed) stochastic matrix, the transition matrix in a Markov chain with discrete set of outcomes. These were introduced in 1906 by Markov apparently to disprove Nekrasov's claim that central limit theorems only applied to independent events, but later found many practical applications. Entries of a stochastic matrix are probabilities covering a complete set of outcomes in each row. So they have to be positive and add up to $1$. A transposed matrix will have positive entries and column sums $1$, which means that it transforms positive vectors into positive vectors, and preserves their entry sums.

The continuous version of Markov chains replaces vectors with functions, matrices with operators, and sums with integrals, the stochasticity conditions are transformed accordingly. These were first introduced by Kolmogorov in 1931 based on the analysis of Markov's discrete case.