High Order Paracontrolled Calculus

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of 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, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations E and F . We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation eqnarray([STIX]x2202ₜ+L)u=f(u)[STIX]x1D701,eqnarray on a 3-dimensional closed manifold, and the generalized KPZ equation eqnarray([STIX]x2202ₜ+L)u=f(u)[STIX]x1D701+g(u)([STIX]x2202u)²,eqnarray driven by a $(1+1)$ -dimensional space/time white noise.