We develop in this note the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant differential operators. We describe in particular the renormalised equation for a very large class of spacetime dependent renormalization schemes.
Let $mathscr{T}$ be the regularity structure associated with a given system of singular stochastic PDEs. The paracontrolled representation of the $sf Pi$ map provides a linear parametrization of the nonlinear space of admissible models $sf M=(g,Pi)$
on $mathscr{T}$, in terms of the family of para-remainders used in the representation. We give an explicit description of the action of the most general class of renormalization schemes presently available on the parametrization space of the space of admissible models. The action is particularly simple for renormalization schemes associated with degree preserving preparation maps; the BHZ renormalization scheme has that property.
We give a short essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and its applications to the study of singular stochastic PDEs.
We develop further in this work the high order paracontrolled calculus setting to deal with the analytic part of the study of quasilinear singular PDEs. A number of continuity results for some operators are proved for that purpose. Unlike the regular
ity structures approach of the subject by Gerencser and Hairer, and Otto, Sauer, Smith and Weber, or Furlan and Gubinelli study of the two dimensional quasilinear parabolic Anderson model equation, we do not use parametrised families of models or paraproducts to set the scene. We use instead infinite dimensional paracontrolled structures that we introduce here.
We prove a general equivalence statement between the notions of models and modelled distributions over a regularity structure, and paracontrolled systems indexed by the regularity structure. This takes in particular the form of a parametrisation of t
he set of models over a regularity structure by the set of reference functions used in the paracontrolled representation of these objects. A number of consequences are emphasized. The construction of a modelled distribution from a paracontrolled system is explicit, and takes a particularly simple form in the case of the regularity structures introduced by Bruned, Hairer and Zambotti for the study of singular stochastic partial differential equations.
We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type $$ dX_t = V(X_t,mathcal{L}(X_t))dt + F(X_t,mathcal{L}(X_t))dW_t $$ where $W$ is a random rough pa
th and $mathcal{L}(X_t)$ is the law of $X_t$. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper [1] for solving mean field rough differential equations and in particular upon a corresponding version of the It^o-Lyons continuity theorem. The rate of convergence is obtained by a coupling argument developed first by Sznitman for particle systems with Brownian inputs.
We define kinetic Brownian motion on the diffeomorphism group of a closed Riemannian manifold, and prove that it provides an interpolation between the hydrodynamic flow of a fluid and a Brownian-like flow.
We start in this work the study of the relation between the theory of regularity structures and paracontrolled calculus. We give a paracontrolled representation of the reconstruction operator and provide a natural parametrization of the space of admissible models.
We define in this work a notion of Young differential inclusion $$ dz_t in F(z_t)dx_t, $$ for an $alpha$-Holder control $x$, with $alpha>1/2$, and give an existence result for such a differential system. As a by-product of our proof, we show that a b
ounded, compact-valued, $gamma$-Holder continuous set-valued map on the interval $[0,1]$ has a selection with finite $p$-variation, for $p>1/gamma$. We also give a notion of solution to the rough differential inclusion $$ dz_t in F(z_t)dt + G(z_t)d{bf X}_t, $$ for an $alpha$-Holder rough path $bf X$ with $alphain left(frac{1}{3},frac{1}{2}right]$, a set-valued map $F$ and a single-valued one form $G$. Then, we prove the existence of a solution to the inclusion when $F$ is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.
We give an elementary proof that Davies definition of a solution to a rough differential equation and the notion of solution given by Bailleul in (Flows driven by rough paths) coincide. This provides an alternative point on view on the deep algebraic
insights of Cass and Weidner in their work (Tree algebras over topological vector spaces in rough path theory).
