Here the author uses something called stochastic mapping. My guess would be that a stochastic mapping from $A$ to $B$ is a function from $A$ to probability spaces with $B$ as its set of elementary events. A friend of mine suggested that it can also mean a probability space with functions from $A$ to $B$ as its elementary events. I am not sure whether these two definitions are interchangeable.

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    $\begingroup$ I would rather take $f:A\leadsto B$ as something like a probability function $P_f:A\times B\to [0,1]$ such that $\sum_bP_f(a,b)=1$ for all $a$. Either, it is possible that a Markov decision process was understood there right away. $\endgroup$
    – Berci
    Oct 8, 2017 at 21:14

1 Answer 1


In a finite dimensional space a stochastic map is a matrix $S$ that satisfies two properties

$$\forall p_{ij}\in S \text{ one has } p_{ij} \geq 0$$ $$\sum_i p_{ij} = 1 \text{ for every column in } S$$

As Berci in his comment say, the notation $f:A\leadsto B$ can be understood as the probability map $P_f$ with domain in $A \times B$ since the elements in the matrix $S$ are considered the probabilities (of transitions) in the theory. Which explains why the $\sum_b P_f(a,b) = 1$ assumption.


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