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HIGH ORDER PARACONTROLLED CALCULUS

Part of: Stochastic analysis Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  02 December 2019

ISMAËL BAILLEUL  and
FRÉDÉRIC BERNICOT
ISMAËL BAILLEUL
Affiliation:
Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000Rennes, France; ismael.bailleul@univ-rennes1.fr
FRÉDÉRIC BERNICOT
Affiliation:
Laboratoire de Mathématiques Jean Leray, CNRS – Université de Nantes, 2 Rue de la Houssinière 44322 Nantes Cedex 03, France; frederic.bernicot@univ-nantes.fr

Abstract

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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.

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations 35R01: Partial differential equations on manifolds
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e44
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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