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I am a mathematician interested in stochastic PDEs. Recently, Martin Hairer introduced his theory of regularity structures to solve singular stochastic PDEs such as KPZ: $\partial_t u=\partial_{xx} u+(\partial_x u)^2-\infty+\xi$ and dynamical $\Phi_3^4$ model: $\partial_t \Phi=\Delta \Phi+\infty \cdot \Phi-\Phi^3+\xi$ where $\xi$ is white noise.

He has an introduction to the latter in his papers, but it is usually quite brief. Are there any references on the equation that offer the physicist's perspective? I would like to read how the equation is derived, its significance, and how physicists thought (think?) about the solutions before regularity structures. Here is a link to his paper: https://arxiv.org/abs/1508.05261

Is there any reference?

  • What is "$\infty\cdot \phi$"? Is that supposed to indicate that the parameters need to be regularized, and that this coefficient will diverge as $\epsilon\to 0$? – Thomas Apr 03 '18 at 19:18
  • @thomas answering that question was worth a Field's medal :) but essentially yes. –  Apr 03 '18 at 19:26
  • In general, this equation goes by many names, time-dependent Landau-Ginzburg (TDLG), kinetic Ising, model A, etc. In statistical physics it describes critical behavior near a phase transition with a non-conserved order parameter (and no other conserved quantities). – Thomas Apr 03 '18 at 19:26
  • The standard reference for model A (etc.) is Hohenberg and Halperin, Reviews of Modern Physics 49,3 (1977) 435. Model A is defined in equ. (4.1) – Thomas Apr 03 '18 at 19:29
  • I should have stressed that this is dynamical critical behavior, so the theory determines critical slowing down in order parameter relaxation near a phase transition. – Thomas Apr 03 '18 at 19:34
  • Thomas already gave the relevant references, which I've also summarized in an answer to a different, related, question: https://physics.stackexchange.com/questions/139618/describe-ising-model-dynamics-in-stochastic-differential-equation-or-stochastic/154076, which you may or may not find useful. But to answer the question at hand, I guess the situation is very similar as that with Navier-Stokes: Physicists typically don't particularly care if the equations have solutions or are well defined in a mathematical sense if they give useful answers (numerically, and in some limit cases analytically). – alarge Apr 05 '18 at 07:28
  • Maybe you could also take a brief look at the book by Glimm and Jaffe (Quantum Physics: a functional integral point of view). This book is basically the bible of constructive QFT, and the relevance of that equation may become clear, since the authors were the first ones to construct it's invariant measure (one of the major breakthroughs of the 70s in math physics). – shalop Apr 07 '18 at 08:39

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