I am wondering if, instead of running MCMC and hoping that it has "mixed" ("achieved stationarity"), there are approaches based on computing (or approximating) the principal left eigenvector of the transition matrix.

Of course, in continuous spaces, this "matrix" has as many entries as S^2, where S is the space our parameters live in... so our "principal eigenvector" becomes the "principal eigenfunction". Functional analysts, how do you compute this?

If it helps, we might want to choose a sparse proposal (such as the one corresponding to Gibbs sampling, in which all transitions changing more than one parameter have probability density zero)