Markov chains and processes with a prescribed invariant measure

Let (E, ) be a measurable space and let [eta] be a probability measure on . Denote by I([eta]) the set of Markov kernels P over (E, ) for which [eta] is an invariant measure: [eta] = [eta]P. We characterize the extreme points of I([eta]) in this paper. When E is a finite set, I([eta]) is a compact, convex set of Markov matrices over E and our characterization generalizes the Birkhoff-von Neumann theorem, which asserts that if [eta] is the uniform distribution on E the extreme points of I([eta]) are the (# E)! permutation matrices. The number of extreme points of I([eta]) depends in a complicated manner on the entries of [eta]; the case # F = 3 is enumerated explicitly and general results are given on the maximum and minimum numbers of extreme points. For finite E a similar treatment is given of the convex cone I*([eta]) of all generator matrices of Markov processes for which [eta] is invariant: its extremal rays are identified.