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Journées équations aux dérivées partielles

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Ismaël Bailleul
Paracontrolled calculus
Journées équations aux dérivées partielles (2016), Exp. No. 1, 11 p., doi: 10.5802/jedp.642
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Class. Math.: 60H15, 35R60, 35R01

Résumé - Abstract

At the same time that Hairer introduced his theory of regularity structures, Gubinelli, Imkeller and Perkowski developed paracontrolled calculus as an alternative playground where to study a number of singular, classically ill-posed, stochastic partial differential equations, such as the $2$ or $3$-dimensional parabolic Anderson model equation (PAM)

$$ \partial _t u = \Delta u + u\zeta , $$

the $\Phi ^4_3$ equation of stochastic quantization

$$ \partial _t u = \Delta u - u^3 + \zeta , $$

or the one dimensional KPZ equation

$$ \partial _t u = \Delta u + (\partial _x u)^2 + \zeta , $$

to name but a few examples. In each of these equations, the letter $\zeta $ stands for a space or time/space white noise who is so irregular that we do not expect any solution $u$ of the equation to be regular enough for the nonlinear terms, or the product $u\zeta $, in the equations to make sense on the sole basis of the regularizing properties of the heat semigroup. Like Hairer’s theory of regularity structures, paracontrolled calculus provides a setting where one can make sense of such a priori ill-defined products, and finally give some meaning and solve some singular partial differential equations. We present here an overview of paracontrolled calculus, from its initial form to its recent extensions.


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