What is a stochastic Hamiltonian process on finite graph? An optimal transport answer

Both the paper and the candidate solution choose a constant co-state ψ so that u*(0)=0 and ∂xu*(0)=1 reduce the Euler–Lagrange system to the reference master equation; selecting boundary data from the periodic reference trajectory yields a Hamiltonian process with the same T-periodic marginal. The paper’s proof of Theorem 4.1 states the correct critical-point system and the same ψ-constant reduction, though it contains a minor slip in writing the Hamiltonian as ∑ij mij(t)ρi; the correct Hamiltonian for the general SBP is H(ρ,ψ,t)=∑i∑j u*(ψj−ψi)mij(t)ρi, already given earlier in the paper. The model’s solution adheres to this correct Hamiltonian and explicitly shows the generator fits Definition 3.1.