Dynamical Fractional and Multifractal Fields

Motivated by the modeling of three-dimensional fluid turbulence, we study a stochastic partial differential equation (SPDE) that is randomly stirred by a spatially smooth and uncorrelated in time forcing term. This dynamical evolution includes a linear but nonlocal interaction which is responsible for a cascading transfer of energy towards smaller scales. In the linear and Gaussian framework, the solution develops fractional regularity, for which we derive explicit predictions for the statistical behavior. Intermittency corrections are included drawing inspiration from a known probabilistic model, the Gaussian multiplicative chaos, which motivates the introduction of a quadratic interaction in this model. Through numerical simulations, we observe the non-Gaussian and in particular skewed nature of these solutions, an important feature in the modeling of turbulent velocity fields.