Geodesic
HIGH ORDER PARACONTROLLED CALCULUS
Forum of Mathematics, Sigma, Tome 7 (2019)

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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 $\mathsf{E}$ and $\mathsf{F}$. We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation

$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701},\end{eqnarray}$

on a 3-dimensional closed manifold, and the generalized KPZ equation

$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701}+g(u)(\unicode[STIX]{x2202}u)^{2},\end{eqnarray}$

driven by a $(1+1)$-dimensional space/time white noise. 
@article{10_1017_fms_2019_44,
     author = {ISMA\"EL BAILLEUL and FR\'ED\'ERIC BERNICOT},
     title = {HIGH {ORDER} {PARACONTROLLED} {CALCULUS}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {7},
     year = {2019},
     doi = {10.1017/fms.2019.44},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.44/}
}
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ISMAËL BAILLEUL; FRÉDÉRIC BERNICOT. HIGH ORDER PARACONTROLLED CALCULUS. Forum of Mathematics, Sigma, Tome 7 (2019). doi: 10.1017/fms.2019.44

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