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A Macroscopic Multifractal Analysis of Parabolic Stochastic PDEs

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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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Correspondence to Kunwoo Kim.

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Communicated by M. Hairer

Research supported in part by the NSF Grants DMS-1307470, DMS-1608575 and DMS-1607089 [D.K. & Y.X.] and 0932078000 [K.K. through The Mathematical Sciences Research Institute at UC Berkeley], and the National Research Foundation of Korea (NRF-2017R1C1B1005436) and the TJ Park Science Fellowship of POSCO TJ Park Foundation [K.K]. A portion of this material is based upon work supported also by the NSF Grant DMS-1440140 while D.K. was in residence at the Mathematical Sciences Research Institute in Berkeley, CA.

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Khoshnevisan, D., Kim, K. & Xiao, Y. A Macroscopic Multifractal Analysis of Parabolic Stochastic PDEs. Commun. Math. Phys. 360, 307–346 (2018).

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