Diffusion Processes: entropy, Gibbs states and the continuous time Ruelle operator

The paper defines admissible tilts via g and the associated generator L̃ = L + Γ(g,·), fixes the stationary initial law µ̃ with density e^{2g}/γ̃, and proves the variational principle P(V) = λ_V (Proposition 2.4) by computing H(P̃_{µ̃}|P_{µ̃}) and completing the square, with equality iff g = log F + const; hence the equilibrium state is precisely the Gibbs state built from the principal eigenpair (F, λ_V) (Definition 2.1 and Proposition 2.4). These definitions, statements, and the proof steps appear explicitly in the PDF, including the Girsanov/Γ calculus set-up for L̃ (and its invariant density), the entropy-rate formula, and the square-completion argument leading to the bound and its equality case . The candidate solution rederives the same result using the Doob h-transform L_V = L + Γ(u,·) with u = log F (equivalent to the paper’s 1/F(L+V)(F·) − λ_V form) and the identity V + L u + (1/2)Γ(u,u) = λ_V, then writes g = u + h to obtain the same inequality H + ∫V ≤ λ_V and the same equality condition h ≡ const. This is the same core proof in Γ-calculus notation, with minor strengthening via symmetry/Dirichlet-form remarks about uniqueness that are consistent with the paper’s invariant density construction for L_V and its identification of the maximizing pair as the Gibbs state .