Higher order paracontrolled calculus
Résumé
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, with the benefit that there is no need to introduce the algebraic apparatus inherent to the latter theory. 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 generalised parabolic Anderson model equation $$ (\partial_t + L) u = f(u)\zeta $$ for a spacial 'noise' $\zeta$ of Hölder regularity $\alpha-2$, with $\frac{2}{5}< \alpha \leq \frac{2}{3}$, and the generalized KPZ equation $$ (\partial_t + L) u = f(u)\zeta + g(u)(\partial u)^2, $$ in the relatively mild case where $\frac{1}{2}<\alpha\leq \frac{2}{3}$.
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