Operator dependent para-controlled calculus, periodic homogenisation and singular PDEs

TL;DR

This paper introduces an operator-dependent para-controlled calculus framework based on spectral decompositions of elliptic operators, enabling analysis of homogenisation and singular PDEs with spectral localization challenges.

Contribution

It develops a novel para-controlled calculus using generalised Besov spaces from spectral decompositions, and applies it to periodic homogenisation and singular PDEs like $ ext{Φ}^4_3$ and KPZ.

Findings

Established decay estimates for high-frequency components of spectrally-localised products.

Proved uniform bounds for para-products and commutators in operator-dependent Besov spaces.

Demonstrated convergence of solutions and fluxes in periodic homogenisation for complex stochastic PDEs.

Abstract

We develop a variant of the para-controlled distributions framework based on operator dependent (generalised) Besov spaces. These spaces were introduced by Kerkyacharian-Petrushev ([KP15]). In contrast to the Fourier decomposition in the classical setting, they are built from spectral decomposition of general self-adjoint elliptic operators. A major difference is that products of two spectrally localised functions in general have ``frequencies" spread over the whole spectrum. We obtain a decay estimate on ``high frequency" component of the product of two spectrally-localised functions. This enables us to obtain uniform bounds for the naturally associated para-product and commutator operations over a class of operators. This in particular includes the family of periodic homogenisation operators uniform in the oscillation parameter. Next, we show convergence properties of these operations in the periodic homogenisation setting. A key ingredient is the convergence of the generalised Littlewood-Paley block operators with quantitative dependence on the spectrum level. The proof combines a novel clustering argument to group nearby eigenvalues, Weyl's asymptotic formula that implies quantitative bounds on the sizes and locations of the eigenvalue clusters, together with a result by Kenig-Lin-Shen ([KLS13]) on convergence of individual eigenvalues. Finally, we apply this framework to periodic homogenisation problems for dynamical $\Phi^4_3$ and KPZ equations. Assuming the convergence of stochastic objects to their homogenised limit, we show that the solutions and fluxes also converge to the corresponding limits. The convergence of the explicit stochastic terms with suitable renormalisations will be treated in a separate note.