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PreprintCC BY 4.0arXiv

A general paracontrolled ansatz for singular SPDEs

  • Yvain Bruned
  • Nicolas Moench
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Abstract

We provide a general ansatz for the paracontrolled approach introduced by Gubinelli, Imkeller, Perkowski for treating singular SPDEs. The ansatz proposed covers a large class of equations. It is described via decorated trees that encode its coefficients and its stochastic data. The main novelty is the recursive definition of the paracontrolled stochastic iterated integrals that involve a well-chosen combinatorial set of words. The paralinearisation is performed via a lift of iterated paraproducts to modelled distributions. The main idea is to integrate localised iterated paraproducts in order to get a global object with a good regularity. We also provide a fixed point argument and the renormalised equation with this ansatz when one uses the preparation map formalism that encompasses the BPHZ renormalisation. The last main result is the convergence of the renormalised stochastic data via the BPHZ renormalisation using an adaptation of the spectral gap approach. One obtains in the end a general solution theory for parabolic singular SPDEs via the paracontrolled approach.

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Cite

BibTeXRIS
BibTeX
@misc{bruned2026general,
  title = {{A general paracontrolled ansatz for singular SPDEs}},
  author = {Yvain Bruned and Nicolas Moench},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.09700},
  eprint = {2608.09700},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.09700},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A general paracontrolled ansatz for singular SPDEs
AU  - Yvain Bruned
AU  - Nicolas Moench
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2608.09700
UR  - https://arxiv.org/abs/2608.09700
AB  - We provide a general ansatz for the paracontrolled approach introduced by Gubinelli, Imkeller, Perkowski for treating singular SPDEs. The ansatz proposed covers a large class of equations. It is described via decorated trees that encode its coefficients and its stochastic data. The main novelty is the recursive definition of the paracontrolled stochastic iterated integrals that involve a well-chosen combinatorial set of words. The paralinearisation is performed via a lift of iterated paraproducts to modelled distributions. The main idea is to integrate localised iterated paraproducts in order to get a global object with a good regularity. We also provide a fixed point argument and the renormalised equation with this ansatz when one uses the preparation map formalism that encompasses the BPHZ renormalisation. The last main result is the convergence of the renormalised stochastic data via the BPHZ renormalisation using an adaptation of the spectral gap approach. One obtains in the end a general solution theory for parabolic singular SPDEs via the paracontrolled approach.
ER  -