High Order Paracontrolled Calculus

Abstract

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm some singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators. We illustrate the efficiency of this elementary approach on the example of the 3-dimensional generalised parabolic Anderson model equation pBt`Lqu " f puqζ, and the generalised KPZ equation pBt`Lqu " f puqζ`gpuqpBuq 2 , driven by a p1`1q-dimensional space/time white noise.

Quasilinear generalized parabolic Anderson model equation

“…Tremendous progress has been made recently in the application of rough path ideas to the construction of solutions to singular stochastic partial differential equations (PDEs) driven by time/space rough perturbations, in particular, using Hairer's theory of regularity structures [12] and the tools of paracontrolled calculus introduced by Gubinelli, Imkeller and Perkowski in [11]. We refer the reader to the works [2,6,8,14,15,16] for a tiny sample of the exponentially growing literature on the subject. The (generalised) parabolic Anderson model equation itself was studied from both points of view in different settings in [5,6,7,11,12,13].…”

Paracontrolled quasilinear SPDEs

“…Even if, in practice, this is not a big issue, and the calculus is still able to deal with a large class of problems, it makes the paracontrolled approach less appealing for a general theory of singular SPDEs. Let us remark that recently Bailleul and Bernicot [5] developed an higher order version of the paracontrolled calculus. However, apart from these recent development, whose impact is still to be assessed, the most general theory for singular SPDEs has been developed by Hairer [19,20,13] under the name of regularity structures theory.…”

An Introduction to Singular SPDEs

“…For that we need to understand the relation between the two paraproducts * ≺ * and * * and also some commutator estimates involving * * . All this is worked out in [BB16b], where also the nonlinear case of gPAM with ξ of regularity ξ ∈ C α−2 with α < 2/3 is treated. In that case we also need a higher order version of the paralinearisation result (6), but this is relatively easy to derive.…”

“…Lemma 5.2 ([BB16b], formula(3.8)) Let α, β, γ, δ ∈ R be exponents such that α + β + γ + δ > 0, then there exists a four-linear map C (2) : C α × C β × C γ × C δ → C α+β+γ+δ such that if f, g, h, ζ are smooth functions we have C (2) (f, g, h, ζ) = C(f ≺g, h, ζ) − f C(g, h, ζ).…”