Stochastic Euler Equations with Pseudo-differential Noise: Continuous and Discontinuous Perturbations in Compressible and Incompressible Flows

We study stochastic Euler equations in compressible and incompressible regimes on ${\mathbb{R}}^d$ and ${\mathbb{T}}^d$ ($d\ge 2$), driven by genuinely mixed multiplicative noise comprising continuous Stratonovich/Itô and discontinuous Marcus components. The noise amplitudes are modeled by nonlocal pseudo-differential operators. We develop a unified local-in-time theory of classical solutions for both regimes, introducing novel analytical tools to control the delicate interplay between jump discontinuities and nonlocal operators. For the compressible barotropic case, we establish a transformation principle that generalizes the classical Makino transform beyond the standard polytropic $\gamma$-law. This extension accommodates a broad class of physically relevant equations of state, including piecewise-defined $\gamma$-laws, (piecewise-defined) Chaplygin-type laws, and the pressure law for white dwarf stars, many of which have remained unexplored in the stochastic compressible setting even under purely Itô-type forcing. For the incompressible damped case, we identify a hierarchy of damping–noise interactions successively guaranteeing global existence, uniform-in-time bounds, and exponential decay. To capture the long-time statistical behavior without viscous smoothing, we establish a novel ergodic criterion for Markov semigroups satisfying a \emph{restricted Feller property under mismatched metrics}. This robustly circumvents the strict single-topology Feller constraints of the classical Krylov–Bogoliubov theory. As a principal application, we provide the first positive resolution to Shirikyan’s open problem regarding the ergodicity of the 2D damped Euler equations, substantially generalizing the result to all dimensions $d\ge 2$ on both ${\mathbb{T}}^d$ and ${\mathbb{R}}^d$ under fully mixed multiplicative noise.