# What is stochastic mapping?

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.

• 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. Oct 8, 2017 at 21:14

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.