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I found this definition of "Markov operator" in the book Chaos, Fractals, and Noise by Lasota and Mackey.

Denote by $L^1(\mu)$ the space of Lebesgue integrable functions according to the measure $\mu$. Then, an operator $$ P: L^1(\mu) \to L^1(\mu)$$ is called Markov Operator when $$ f\geq 0 \Rightarrow Pf \geq 0$$ and $$ \int (Pf)(x) \mu(dx) = \int f(x) \mu(dx) $$ holds for any $f \in L^1(\mu)$.

But where does this originate from? Who invented it?

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  • $\begingroup$ One might think Markov, but no. Kolmogorov. $\endgroup$ – Conifold Apr 30 '15 at 21:58
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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.

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  • $\begingroup$ I see that Kolmogorov extended Markov chains for continuous time in gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002278235 (1936), but he was still working on a finite set of outcomes. Thus, matrices became time-dependet but not operators and sums stayed as sums, but time-depending. Who generalized the concept to continuous outcomes and actually invented the Markov operator? $\endgroup$ – Adam May 5 '15 at 12:24
  • $\begingroup$ @Adam Good catch. The answer is still Kolmogorov, and even earlier than I thought. en.wikipedia.org/wiki/Kolmogorov_equations#History $\endgroup$ – Conifold May 6 '15 at 20:58

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