Mixed Third-Order Flux Laws for Dual Cascade in the Stochastic SQG Equation

Abstract: We study dual-cascade flux laws for the stochastic forced--dissipative surface quasi-geostrophic (SQG) equation on a large periodic box. For statistically stationary solutions, under a weak anomalous dissipation assumption, we derive rigorous mixed third-order structure-function laws for the dual cascade: a Yaglom-type law for the direct cascade of surface potential energy (SPE) and an antisymmetrized mixed flux law for the inverse cascade of the Hamiltonian. In particular, the inverse Hamiltonian law appears to be new even as an explicit third-order structure-function relation. We also prove Onsager-type obstruction results showing that sufficiently regular stationary families cannot sustain the corresponding non-zero fluxes: $B^s_{3,\infty}$-regularity above the Onsager threshold $1/3$ rules out the direct SPE flux, while sufficient low-frequency Besov regularity rules out the inverse Hamiltonian flux. These results provide a rigorous formulation of the SQG dual-cascade phenomenology in a stochastic stationary setting.