“Controlled” Versions of the Collatz–Wielandt and Donsker–Varadhan Formulae

A variational characterization of the optimal exit rate for controlled diffusions

“…A discrete counterpart for Markov chains appears in [19]. Subsequently, the present authors and their collaborators extended it to controlled Markov chains [1] and diffusions [3,4,10]; see also [5] for an overview. These articles also point out that these results are abstract counterparts of the Collatz-Wielandt formula for the Perron-Frobenius eigenvalue of irreducible non-negative matrices [17,34].…”