  Motivated by the modeling of three-dimensional fluid turbulence, we define
and study a class of stochastic partial differential equations (SPDEs) that are
randomly stirred by a spatially smooth and uncorrelated in time forcing term.
To reproduce the fractional, and more specifically multifractal, regularity
nature of fully developed turbulence, these dynamical evolutions incorporate an
homogenous pseudo-differential linear operator of degree 0 that takes care of
transferring energy that is injected at large scales in the system, towards
smaller scales according to a cascading mechanism. In the simplest situation
which concerns the development of fractional regularity in a linear and
Gaussian framework, we derive explicit predictions for the statistical
behaviors of the solution at finite and infinite time. Doing so, we realize a
cascading transfer of energy using linear, although non local, interactions.
These evolutions can be seen as a stochastic version of recently proposed
systems of forced waves intended to model the regime of weak wave turbulence in
stratified and rotational flows. To include multifractal, i.e. intermittent,
corrections to this picture, we get some inspiration from the Gaussian
multiplicative chaos, which is known to be multifractal, to motivate the
introduction of an additional quadratic interaction in these dynamical
evolutions. Because the theoretical analysis of the obtained class of nonlinear
SPDEs is much more demanding, we perform numerical simulations and observe the
non-Gaussian and in particular skewed nature of their solution.
