It is generally argued that the solution to a stochastic PDE with multiplicative noise---such as $dot{u}=frac12 u+uxi$, where $xi$ denotes space-time white noise---routinely produces exceptionally-large peaks that are macroscopically multifractal. See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (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 (1989; 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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