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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Geometric stochastic heat equations
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by Y. Bruned, F. Gabriel, M. Hairer and L. Zambotti PDF
J. Amer. Math. Soc. 35 (2022), 1-80

Abstract:

We consider a natural class of ${\mathbf {R}}^d$-valued one-dimensional stochastic partial differential equations (PDEs) driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on ${\mathbf {R}}^d$. This class contains in particular the Kardar-Parisi-Zhang (KPZ) equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. We exhibit a one-parameter family of solution theories with the following properties:

  1. For all stochastic partial differential equations (SPDEs) in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using Itô calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgers-type equations), or the Hopf-Cole transform (KPZ equation).

  2. Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid.

  3. Every solution theory satisfies an analogue of Itô’s isometry.

  4. The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation.

In particular, points (2) and (3) show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and Itô interpretations of stochastic differential equations (SDEs) simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the $L^2$-gradient flow for the Brownian loop measure.

References
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Additional Information
  • Y. Bruned
  • Affiliation: University of Edinburgh, Edinburgh, United Kingdom
  • MR Author ID: 952078
  • Email: Yvain.Bruned@ed.ac.uk
  • F. Gabriel
  • Affiliation: École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
  • MR Author ID: 1205076
  • ORCID: 0000-0001-9510-1135
  • Email: franckr.gabriel@gmail.com
  • M. Hairer
  • Affiliation: Imperial College London, London, United Kingdom
  • MR Author ID: 664196
  • ORCID: 0000-0002-2141-6561
  • Email: m.hairer@imperial.ac.uk
  • L. Zambotti
  • Affiliation: LPSM, Sorbonne Université, Université de Paris, CNRS, Paris
  • MR Author ID: 647350
  • Email: lorenzo.zambotti@upmc.fr
  • Received by editor(s): February 7, 2019
  • Received by editor(s) in revised form: September 28, 2020, and January 18, 2021
  • Published electronically: April 30, 2021
  • Additional Notes: The third author gratefully acknowledges financial support from the Leverhulme trust via a Leadership Award, the ERC via the consolidator grant 615897:CRITICAL, and the Royal Society via a research professorship. The first and third authors are grateful to the Newton Institute for financial support and for the fruitful atmosphere fostered during the programme “Scaling limits, rough paths, quantum field theory”.
  • © Copyright 2021 by the authors
  • Journal: J. Amer. Math. Soc. 35 (2022), 1-80
  • MSC (2020): Primary 60H15
  • DOI: https://doi.org/10.1090/jams/977
  • MathSciNet review: 4322389