  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 (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.
