Introduction to regularity structures

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.

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Hopf-algebraic Deformations of Products and Wick Polynomials

International Mathematics Research Notices ◽

10.1093/imrn/rny269 ◽

2018 ◽

Vol 2020 (24) ◽

pp. 10064-10099 ◽

Cited By ~ 1

Author(s):

Kurusch Ebrahimi-Fard ◽

Frédéric Patras ◽

Nikolas Tapia ◽

Lorenzo Zambotti

Keyword(s):

Hopf Algebra ◽

Random Variables ◽

Recent Theory ◽

Linear Functionals ◽

Regularity Structures ◽

Convolution Products ◽

Algebra Deformation

Abstract We present an approach to cumulant–moment relations and Wick polynomials based on extensive use of convolution products of linear functionals on a coalgebra. This allows, in particular, to understand the construction of Wick polynomials as the result of a Hopf algebra deformation under the action of linear automorphisms induced by multivariate moments associated to an arbitrary family of random variables with moments of all orders. We also generalize the notion of deformed product in order to discuss how these ideas appear in the recent theory of regularity structures.

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