Renormalising SPDEs in regularity structures

Abstract

The formalism recently introduced in [BHZ ] allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was then shown in [CH ] that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT.The present work completes this programme by constructing an action of the renormalisation group onto a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular nonlinear SPDEs.

Comparing the stochastic nonlinear wave and heat equations: a case study

“…At the second equality, we performed the Wick renormalization: 2 . It is easy to see that and have regularities 8 −α− and −2α−, respectively (see Lemma 3.1 below). Then, thanks to one degree of smoothing from the wave Duhamel integral operator, we expect that v has regularity 1 − 2α−.…”

“…Over the last decade, we have seen a tremendous development in the study of singular stochastic PDEs, in particular in the parabolic setting [32,33,28,10,36,39,12,11,8,9]. Over the last few years, we have also witnessed a rapid progress in the theoretical understanding of nonlinear wave equations with singular stochastic forcing and/or rough random initial data [51,29,30,31,44,48,41,43,46,49,47,42,7].…”

“…We also point out that the paper [22] handles noises with rougher temporal regularity than the space-time white noise and Theorem 1.3 is a subcase of the main result in [22]. 8 In the following, we restrict our attention to spatial regularities. Moreover, we use a− (and a+) to denote a − ε (and a + ε, respectively) for arbitrarily small ε > 0.…”

Paracontrolled calculus and regularity structures I

“…Starting with his groundbreaking work [12], M. Hairer has developed with his co-authors [7,8,6] a theory of subcritical singular stochastic partial differential equations (PDEs) that provides now an automated blackbox for the basic understanding of a whole class of stochastic PDEs. Equations of this class all share the common feature of involving ill-defined products of distributions with functions or distributions.…”

“…It happens then that one can reformulate the formal ill-posed equation into the space of jets as a well-posed, model-dependent, fixed point equation in a well-chosen space of jets. For the random model built from a renormalisation procedure in [8], the space-time function/distribution associated with the solution of the fixed point equation on the jet space can be shown to be the limit in probability of solutions of a family of well-posed space-time stochastic PDEs driven by regularized noises, as the regularization parameter tends to 0 -this is the content of [6]. The fact that some of the terms in these modified and regularized stochastic PDEs blow up as the regularization parameter goes to 0 is a feature of the singular nature of the initial equation.…”