On the pitchfork bifurcation for the Chafee-Infante equation with additive noise

The paper proves finite-time Lyapunov and k-volume bifurcation statements for the SPDE du = (Δu + αu − u^3)dt + dW via an energy estimate for the upper bounds and an invariant-cone argument (extended to wedge spaces) for the lower bounds on positive-probability events; see Theorem A and Theorem B and their proofs, including Proposition 2.6 and Lemma 3.2/3.5 for the cone method and the construction of the smallness event in H1 (hence L∞ in 1D) . The candidate solution obtains the same results: the same energy estimate for upper bounds and, for lower bounds, a variation-of-constants plus a reverse Grönwall estimate once the random equilibrium is kept uniformly small on [0,T]; this is ensured by the same type of positive-probability event (small H1/L∞ norm) constructed in the paper’s Appendix using parabolic regularization and a singular Grönwall lemma . Minor notational care is needed in the wedge-lift (distinguishing Λk of the operator vs. the induced generator), but the approach is sound and matches the paper’s quantitative bounds.