  For different reversible Markov kernels on finite state spaces, we look for
families of probability measures for which the time evolution almost remains in
their convex hull. Motivated by signal processing problems and metastability
studies we are interested in the case when the size of such families is smaller
than the size of the state space, and we want such distributions to be with
small overlap among them. To this aim we introduce a squeezing function to
measure the common overlap of such families, and we use random forests to build
random approximate solutions of the associated intertwining equations for which
we can bound from above the expected values of both squeezing and total
variation errors. We also explain how to modify some of these approximate
solutions into exact solutions by using those eigenvalues of the associated
Laplacian with the largest absolute values.
