Strong averaging principle for nonautonomous multi-scale SPDEs with fully local monotone and almost periodic coefficients

The paper proves two strong averaging results for the nonautonomous slow–fast SPDE system (1.3): convergence to the first averaged equation (1.6) under A1–A5, B1–B4 (Theorem 2.4, eq. (2.6)), and, under additional almost periodic assumptions C1–C3, convergence to a time-averaged, autonomous limit (1.8) (Theorem 2.10) . The proof strategy matches the candidate’s: Khasminskii time-discretization with a frozen fast process and an evolution system of measures, a priori bounds in the variational setting, and a second (time) averaging step when coefficients are uniformly almost periodic. The paper constructs the evolution system of measures for the frozen equation and defines F̄(t,x)=∫F(t,x,y)μ^x_t(dy) (eq. (1.5)) to obtain (1.6) ; it develops blockwise auxiliary processes Ŷ^ε with estimates (3.3)–(3.6) and uses stopping times to control nonlinearities . A key correlation estimate yields a block error of order (ε/δ)^{1/2}+δ^{1/2} (see (4.13)), leading to a local rate ε^{1/6} after choosing δ=ε^{2/3} (Proposition 4.2) . For the second averaging, the paper proves uniform almost periodicity of the evolution measures μ^x_t (Lemma 4.3) and of F̄(t,·) (Lemma 4.5), enabling Bohr means and the autonomous limit (1.8) (eq. (1.7) discussion) . The candidate’s solution outlines the same two-phase approach and obtains the same convergence conclusions. Minor differences are technical: the candidate sketches an “O(ε) per block” bound (leading to an indicative O(δ^{1/2}+ε/δ) rate), whereas the paper’s rigorous estimate gives a root-type term (ε/δ)^{1/2}. Both yield convergence; the paper’s use of stopping times and correlation/BDG is a careful, variationally robust version of the candidate’s sketch.