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.
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]
Created on September 17, 2023 at 14:27:54. See the history of this page for a list of all contributions to it.