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HIGH ORDER PARACONTROLLED CALCULUS
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- Journal: Forum of Mathematics, Sigma / Volume 7 / 2019
- Published online by Cambridge University Press: 02 December 2019, e44
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View abstract
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 on a 3-dimensional closed manifold, and the generalized KPZ equation
$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701},\end{eqnarray}$$driven by a
$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701}+g(u)(\unicode[STIX]{x2202}u)^{2},\end{eqnarray}$$
$(1+1)$-dimensional space/time white noise.
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Where does randomness lead in spacetime?
- Journal: ESAIM: Probability and Statistics / Volume 14 / 2010
- Published online by Cambridge University Press: 22 May 2008, pp. 16-52
- Print publication: 2010
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