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 examples of the 3-dimensional generalised parabolic Anderson model equation and the generalised KPZ equation driven by a (1+1)-dimensional space/time white noise.