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Non-linear rough heat equations
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  • Published: 03 February 2011

Non-linear rough heat equations

  • A. Deya1,
  • M. Gubinelli2 &
  • S. Tindel1 

Probability Theory and Related Fields volume 153, pages 97–147 (2012)Cite this article

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Abstract

This article is devoted to define and solve an evolution equation of the form dy t  = Δy t dt + dX t (y t ), where Δ stands for the Laplace operator on a space of the form \({L^p(\mathbb R^n)}\), and X is a finite dimensional noisy nonlinearity whose typical form is given by \({X_t(\varphi)=\sum_{i=1}^N \, x^{i}_t f_i(\varphi)}\), where each x = (x (1), … , x (N)) is a γ-Hölder function generating a rough path and each f i is a smooth enough function defined on \({L^p(\mathbb R^n)}\). The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.

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Author information

Authors and Affiliations

  1. Institut Élie Cartan Nancy, Université de Nancy, B.P. 239, 54506, Vanœduvre-lès-Nancy Cedex, France

    A. Deya & S. Tindel

  2. CEREMADE, Université de Paris-Dauphine, 75116, Paris, France

    M. Gubinelli

Authors
  1. A. Deya
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  2. M. Gubinelli
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  3. S. Tindel
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Corresponding author

Correspondence to M. Gubinelli.

Additional information

This research is supported by the ANR Project ECRU (ANR-09-BLAN-0114-01/2).

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Deya, A., Gubinelli, M. & Tindel, S. Non-linear rough heat equations. Probab. Theory Relat. Fields 153, 97–147 (2012). https://doi.org/10.1007/s00440-011-0341-z

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  • Received: 18 December 2009

  • Revised: 15 December 2010

  • Published: 03 February 2011

  • Issue Date: June 2012

  • DOI: https://doi.org/10.1007/s00440-011-0341-z

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Keywords

  • Rough paths theory
  • Stochastic PDEs
  • Fractional Brownian motion

Mathematics Subject Classification (2000)

  • 60H05
  • 60H07
  • 60G15
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