Invariant measures for stochastic Burgers equation on unbounded domains

Under (H1), the growth bound |σ(u)| ≤ l|u| holds for all u, so in particular σ(0) = 0. With the mild formulation of (2.1) in the paper, starting from u0 = 0 gives u(t) ≡ 0 for all t, hence the Dirac measure δ0 is invariant for the Markov semigroup for any k (no smallness condition needed). This is immediate from the paper’s equation and assumptions, see (2.1) and (H1)–(H2) in the setup and definitions of the semigroup/invariance . The main theorem (Theorem 2.6) asserts existence of an invariant measure only under k > 0 and al2 < 3/7 k, obtained via Feller + tightness and Krylov–Bogolioubov (Sections 4.1–4.3), which is correct but unnecessary for mere existence since δ0 is already invariant; see the statement of Theorem 2.6 and the subsequent Krylov–Bogolioubov passage to an invariant measure . Therefore, the model’s solution is correct and strictly stronger for the existence claim, while the paper’s main result is incomplete in formulation (it does not exclude the trivial invariant measure nor state nontriviality).