An analytic BPHZ theorem for regularity structures (1612.08138v5)

Abstract: 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 H\"older-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.

An Analytic BPHZ Theorem for Regularity Structures

The paper by Chandra and Hairer presents a comprehensive exploration into the analytic BPHZ theorem within the context of regularity structures, aimed at providing a streamlined framework for tackling stochastic convergence in models derived from nonlinear stochastic PDEs. Unlike previous methodologies that required intricate ad-hoc efforts, this paper introduces a generalized approach that efficiently automates the generation of stochastic estimates across various equations fitting within the theory of regularity structures.

Key Contributions

1. Automated Stochastic Estimates: Central to the paper is the introduction of a "black box" framework that automates the computation of stochastic estimates for models formulated within the scope of regularity structures, effectively reducing the laborious manual efforts previously required.
2. Renormalization Techniques: A robust multi-scale analysis akin to those employed in constructive field theory is leveraged. However, the intricacies due to positive renormalizations from recentering procedures and the operations of non-Gaussian driving fields introduce significant modifications.
3. BPHZ Lift's Continuity: A pivotal result asserts the continuity of the BPHZ lift in law when constrained to a certain class of stationary random fields with ample moments. This underscores a surprising trait wherein, despite the canonical lift not exhibiting continuity within a H\"older-type space encapsulating the noise, the BPHZ lift achieves continuity.

Analytical Framework and Methodology

The authors extend the regularity structures’ theory to accommodate automatic existence results for a broad spectrum of parabolic SPDEs. The use of scale decomposition to navigate the issue of overlapping divergences exemplifies a particularly elegant solution. This methodology ensures that overlapping divergent structures are treated distinctly across different scale assignments. The essence of the approach lies in asserting that these divergences do not overlap in phase-space, a concept borrowed from field theory but adapted to accommodate the singularities inherent in stochastic PDEs.

Implications and Future Directions

The findings presented hold substantial theoretical implications for the development of stochastic PDE solutions, potentially simplifying the derivation of these solutions in practical scenarios. Particularly, the seamless translation of renormalization procedures from quantum field theory to stochastic PDE analysis paves the way for further exploration into non-Gaussian fields and models beyond Gaussian benchmarks.

The authors hint at extending their framework to accommodate even more singular driving noises, governed by power-counting constraints, implying that future research might breach the current limitations, achieving deeper insights into subcritical and possibly critical regimes.

Conclusion

Chandra and Hairer’s paper significantly advances the understanding of stochastic PDEs through the lens of regularity structures. By systematically integrating BPHZ renormalization, they set a new benchmark in automated existence results for stochastic PDEs, promising more fluid integration of stochastic estimates into complex model analyses. The innovative approach and significant theoretical assertions offered in this research will serve as fundamental stepping stones for subsequent inquiries into non-linear SPDE landscapes and their solutions.