Infinitesimal Invariance for the Coupled KPZ Equations

Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)


This paper studies the infinitesimal invariance for \(\mathbb{R}^{d}\)-valued extension of the Kardar-Parisi-Zhang (KPZ) equation at approximating level.


Invariant Measure Functional Derivative Martingale Problem Euclidean Quantum Wiener Chaos 
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The author thanks Herbert Spohn for suggesting the problem discussed in this paper. He also thanks Jeremy Quastel for helpful discussions and Michael Röckner for pointing out the last comment mentioned in Remark 3.1-(2).


  1. 1.
    A.G. Bhatt, R.L. Karandikar, Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21, 2246–2268 (1993)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    P. Echeverria, A criterion for invariant measures of Markov processes. Z. Wahrsch. Verw. Gebiete 61, 1–16 (1982)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    P.L. Ferrari, T. Sasamoto, H. Spohn, Coupled Kardar-Parisi-Zhang equations in one dimension. J. Stat. Phys. 153, 377–399 (2013)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    T. Funaki, The reversible measures of multi-dimensional Ginzburg-Landau type continuum model. Osaka J. Math. 28, 463–494 (1991)MATHMathSciNetGoogle Scholar
  5. 5.
    T. Funaki, A stochastic partial differential equation with values in a manifold. J. Funct. Anal. 109, 257–288 (1992)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    T., Funaki, J. Quastel, KPZ equation, its renormalization and invariant measures. Stochastic Partial Differential Equations: Analysis and Computations, 3(2), 159–220 (2014)Google Scholar
  7. 7.
    M. Gubinelli, P. Imkeller, N. Perkowski, Paracontrolled distributions and singular PDEs. Forum Math. Pi. Preprint (2014, to appear) [arXiv:1210.2684v3]Google Scholar
  8. 8.
    M. Hairer, A theory of regularity structures. Invent. Math. 198, 269–504 (2014)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Yor, Existence et unicité de diffusions \(\grave{\text{a}}\) valeurs dans un espace de Hilbert. Ann. Inst. Henri. Poincaré Sect. B 10, 55–88 (1974)MATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoKomaba, TokyoJapan

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