An analytic BPHZ theorem for regularity structures

By A. Chandra, Martin Hairer. 2016

184 citations recorded by the source corpus.

We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated framework for studying these problems but previous works had to expend significant effort to obtain these stochastic estimates in an ad-hoc manner. In contrast, the main result of this article operates as a black box which automatically produces these estimates for nearly all of the equations that fit within the scope of the theory of regularity structures. Our approach leverages multi-scale analysis strongly reminiscent to that used in constructive field theory, but with several significant twists. These come in particular from the presence of "positive renormalizations" caused by the recentering procedure proper to the theory of regularity structure, from the difference in the action of the group of possible renormalization operations, as well as from the fact that we allow for non-Gaussian driving fields. One rather surprising fact is that although the "canonical lift" is of course typically not continuous on any Holder-type space containing the noise (which is why renormalization is required in the first place), we show that the "BPHZ lift" where the renormalization constants are computed using the formula given in arXiv:1610.08468, is continuous in law when restricted to a class of stationary random fields with sufficiently many moments.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/12a5e91656331930374dee1f23cb5d4c49859d1f

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Citations in the corpus

The infinitesimal generator of the stochastic Burgers equation (2018, background): “Although by now there are very sophisticated and powerful black box type tools available in regularity structures that should handle the complicated expansion automatically [2,4,7].”
A diagram-free approach to the stochastic estimates in regularity structures (2021, background): “They provide a framework for local well-posedness for a large class of semilinear SPDEs, as worked out in [20, 5, 8, 4].”
Algebraic renormalisation of regularity structures (2016, result): “m Πˆ(ε deﬁned as in (6.22) should converge to a limiting model, prov ided that the ξ(ε) t converge in a suitable sense as ε →0. This is indeed the case, as shown in the companion “analytical” article [CH16]. It is also possible to veri fy that the renormalisation procedures that were essentially “gu”
A regularity structure for rough volatility (2017, methodology): “Example 3.6), then the renormalization 𝒞𝜀 = 𝒦𝜀(⋅, ⋅) that was applied in Theorem 3.14 and thus in Definition 3.24 is a constant, which is the familiar concept one encounters in the study of singular SPDE (Chandra & Hairer, 2016; Hairer, 2013, 2014).”
HIGH ORDER PARACONTROLLED CALCULUS (2016, methodology): “oherent enhancements is the task of renormalisation of stochastic singular partial diﬀerential equations. This has been implemented in the setting of regularity structures in the groundbreaking works [7, 10] of Hairer and co-authors. This work has not been done yet in the paracontrolled setting, but”