Markov properties for systems described by PDEs and first-order representations

The relation between Markovianity and representability by means of first-order PDEs is investigated. We consider two versions of the Markovian property, weak and strong-Markovianity. The weak version has been introduced in [J.C. Willems, State and first-order representations, in: V.D. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems & Control Theory, Princeton University Press, Princeton, NJ, 2004, pp. 54–57] and conjectured to correspond to first-order representations. We provide a counterexample to this conjecture. For finite-dimensional behaviors, strong-Markovianity is proven to be indeed equivalent to the representability by means of first-order PDEs with a special structure.