Paracontrolled calculus

Bailleul, Ismaël

Journées équations aux dérivées partielles (2016), Exposé no. 1, 11 p.

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 ${\Phi }_3^4$ equation of stochastic quantization

$${\partial }_{t}u=\Delta u-u^3+\zeta ,$$

or the one dimensional KPZ equation

$${\partial }_{t}u=\Delta u+{\left({\partial }_{x}u\right)}^2+\zeta ,$$

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.

Classification : 60H15, 35R60, 35R01

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