Paracontrolled calculus was developed by Gubinelli, Imkeller and Perkowski as a way of treating singular stochastic PDEs such as KPZ, $\Phi_3^4$ or PAM, around the same time regularity structures were introduced by Hairer.

Regularity structures are now the standard treatment of singular stochastic PDEs although I have been reading that there are perhaps potentional benefits to paracontrolled calculus. This paper for example says:

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm some singular partial differential equations with the same efficiency as regularity structures, with the benefit that there is no need to introduce the algebraic apparatus inherent to the latter theory.

I'm not sure that this is such a large benefit. Is this the only reason to work with paracontrolled calculus, or are there other benefits? Is there anything you can do in paracontrolled calculus that you can't in regularity structures?


1 Answer 1


I don't think that the reason given in the paper by Bailleul and Bernicot is a good one. Basically, they treat an example which is simple enough so that it is still manageable to describe the various bits and pieces needed to control their solutions "by hand" instead of combining them into a single object in a more coherent way.

This being said, there are certainly situations in which paracontrolled calculus is more efficient than regularity structures thanks to the fact that it provides a more "global" decomposition of the solution. (Think of Fourier decomposition vs Taylor expansion.) For example, it is not clear how one would implement the strategy of proof for the result of this article by Weber and Mourrat in the framework of regularity structures. It also has the advantage of relying solely on concepts that are quite well-known and have been used in other context for a long time, which makes it possible to reuse more existing results. (For example it makes use of classical function spaces, so that embedding theorems and functional inequalities can just be taken "off the shelf". In the theory of regularity structures there are analogues of most function spaces that share many properties with their classical counterparts, but this always has to be proven first...)


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