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Communications in Mathematical Physics

, Volume 360, Issue 1, pp 307–346 | Cite as

A Macroscopic Multifractal Analysis of Parabolic Stochastic PDEs

  • Davar Khoshnevisan
  • Kunwoo Kim
  • Yimin Xiao
Article

Abstract

It is generally argued that the solution to a stochastic PDE with multiplicative noise—such as \({\dot{u}=\frac12 u''+u\xi}\) , where \({\xi}\) denotes space-time white noise—routinely produces exceptionally-large peaks that are “macroscopically multifractal.” See, for example, Gibbon and Doering (Arch Ration Mech Anal 177:115–150, 2005), Gibbon and Titi (Proc R Soc A 461:3089–3097, 2005), and Zimmermann et al. (Phys Rev Lett 85(17):3612–3615, 2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (J Phys A 22(13):2621–2626, 1989; Proc Lond Math Soc (3) 64:125–152, 1992). The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as “stretch factors.” A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Department of MathematicsPohang University of Science and Technology (POSTECH)PohangKorea
  3. 3.Department Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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