Existence proof for orthogonal dynamics and the Mori-zwanzig formalism

We study the existence of solutions to the orthogonal dynamics equation, which arises in the Mori-Zwanzig formalism in irreversible statistical mechanics. This equation generates the random noise associated with a reduction in the number of variables. If L is the Liouvillian, or Lie derivative associated with a Hamiltonian system, and P an orthogonal projection onto a closed subspace of L², then the orthogonal dynamics is generated by the operator (I - P)L. We prove the existence of classical solutions for the case where P has finite-dimensional range. In the general case, we prove the existence of weak solutions.