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Journal of the American Mathematical Society

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



Geometric stochastic heat equations

Authors: Y. Bruned, F. Gabriel, M. Hairer and L. Zambotti
Journal: J. Amer. Math. Soc. 35 (2022), 1-80
MSC (2020): Primary 60H15
Published electronically: April 30, 2021
MathSciNet review: 4322389
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Abstract | References | Similar Articles | Additional Information


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.

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Additional Information

Y. Bruned
Affiliation: University of Edinburgh, Edinburgh, United Kingdom
MR Author ID: 952078

F. Gabriel
Affiliation: École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
MR Author ID: 1205076
ORCID: 0000-0001-9510-1135

M. Hairer
Affiliation: Imperial College London, London, United Kingdom
MR Author ID: 664196
ORCID: 0000-0002-2141-6561

L. Zambotti
Affiliation: LPSM, Sorbonne Université, Université de Paris, CNRS, Paris
MR Author ID: 647350

Keywords: Brownian loops, renormalisation, stochastic PDE
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”.
Article copyright: © Copyright 2021 by the authors