Stationary measures for stochastic differential equations with degenerate damping

The paper proves existence of a stationary measure (with all polynomial moments) for dX_t = B(X_t,X_t) dt − A X_t dt + σ dW_t under the finite-jet instability condition on the conservative flow, using a time-averaged coercivity framework and short-time control near ker A; see Theorem 1.1 and condition (1.8), together with Lemma 2.1–2.2 and Proposition 3.1 (r = 1) in 2206.02240.pdf . The candidate model’s solution reaches the same conclusion but via a different, hypocoercive Lyapunov construction built from finitely many jets of the conservative flow and cross terms designed to make L_B dissipative on higher jets and L_{−A} coercive on the base mode. This jet–Lyapunov scheme is compatible with the paper’s algebraic condition and energy identity (Appendix A) , but it is a distinct proof strategy. The paper’s argument is complete and carefully justified; the model’s proof is a plausible outline that would require filling in technical estimates (e.g., bounding L_{−A}G_k, diffusion terms, and coefficient choices), but does not contradict the paper.