Uncertainty quantification analysis of bifurcations of the Allen–Cahn equation with random coefficients

The paper proves that bifurcations from the trivial branch of the stationary Allen–Cahn equation with random coefficient occur at p*_i(y) = −λ_i^g(y) (Proposition 3.2), and in the spatially homogeneous case g(x,y)=g(y) one has p*_i(y)=−λ_i−g(y) with density ρ_{p*_i}(z)=ρ_g(−λ_i−z) (Proposition 4.1); it also shows that nontrivial branches are horizontal shifts of a deterministic reference branch and that the mean branch equals the reference when E[g]=0 (Proposition 4.2 and Definition 3.1). The candidate solution re-derives these statements via the same linearization and Crandall–Rabinowitz framework, adding standard functional-analytic details (smoothness of the Nemytskii map, selfadjointness with compact resolvent, and transversality). The proofs align in structure and conclusions; the model’s argument slightly expands on regularity/measurability assumptions but remains consistent with the paper’s setup. See Proposition 3.2 and (3.6)-(3.7) , Proposition 4.1 and 4.2 , and Definition 3.1 on mean bifurcation curves .