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Given a function $[0,1]\to[0,1]\times[0,1]$ on the reals, such that the function is "stochastic" (probably an abuse of vocabulary: defined such that integrating along any vertical line gives $1$), approximate this function with a finite but very large ($10^{6+}$ entries) stochastic matrix $M$, and then find the eigenvectors of $M$ (specifically that eigenvector with eigenvalue $1$).

What field of mathematics should I be looking in to learn how to solve this and similar problems? (High dimensionality Markov chains come to mind - is this intuition accurate?) What can be said about the eigenvectors of differently sized approximations to the same function? Is this a case where the solution can be taken to the limiting infinite matrix, or will something break on me?

... and, would this question be better asked as an algorithm question on the computer science SE?

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