nLab stochastic map

Idea

In probability theory, a stochastic map between measurable spaces is a (linear) map of their measures which preserves the probability measures among them.

For finite spaces — where a measure is just a tuple (vector) of non-negative real numbers and a probability measure is such a tuple whose sum is 1 – a stochastic map is given by a matrix all whose columns sum to $1$, called a stochastic matrix.

If, on compact measureable spaces, a stochastic map also preserves the uniform distribution, then it is called a doubly stochastic map.

For the case over finite spaces this means that the stochastic matrix has not just columns but also rows which sum to unity, then called a doubly stochastic matrix.

Relate entries

• quantum channel

References

See also:

• Wikipedia, Stochastic matrix

• James Fullwood, Arthur Parzygnat, Section 4 of: The information loss of a stochastic map, Entropy 2021 23(8) 1021 [arXiv:2107.01975, doi:10.3390/e23081021]