A Macroscopic Multifractal Analysis of Parabolic Stochastic PDEs

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.