This paper presents a mathematical analysis of a stochastic generalized Constantin-Lax-Majda-DeGregorio (gCLMG) equation. We focus on the specific case where the nonlinearity in the equation allows an inviscid conserved quantity, i.e., the squared norm of model vorticity. The existence of an invariant measure is proved via the classical Krylov-Bogoliubov argument. The uniqueness of the measure and exponential mixing are proved under a sufficiently large viscosity condition, in which the nonlocal structure of the nonlinear term becomes manifest.
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Fujita et al. (2026) studied this question.
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