7
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: not found
      • Article: not found

      Renormalization Group and Stochastic PDEs

      Annales Henri Poincaré
      Springer Nature

      Read this article at

      ScienceOpenPublisher
      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          Related collections

          Most cited references9

          • Record: found
          • Abstract: not found
          • Conference Proceedings: not found

          The Ocean Surface Topography Constellation: The Next 15 Years in Satellite Altimetry

            Bookmark
            • Record: found
            • Abstract: found
            • Article: found
            Is Open Access

            A theory of regularity structures

            We introduce a new notion of "regularity structure" that provides an algebraic framework allowing to describe functions and / or distributions via a kind of "jet" or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a "renormalisation group" which is defined canonically in terms of the regularity structure associated to the given class of PDEs. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the \(\Phi^4_3\) Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of 3-dimensional ferromagnets near their critical temperature.
              Bookmark
              • Record: found
              • Abstract: not found
              • Book: not found

              Gaussian Measures

                Bookmark

                Author and article information

                Journal
                Annales Henri Poincaré
                Ann. Henri Poincaré
                Springer Nature
                1424-0637
                1424-0661
                March 2016
                April 28 2015
                March 2016
                : 17
                : 3
                : 497-535
                Article
                10.1007/s00023-015-0408-y
                1f85cf92-55d4-4d57-9e2c-fc6362ee3096
                © 2016

                http://www.springer.com/tdm

                History

                Comments

                Comment on this article