Dynamical Fractional and Multifractal Fields

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