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

Abstract

We study stochastic Euler equations in compressible and incompressible regimes, on the whole space and the torus, driven by mixed multiplicative noise: continuous Stratonovich/It\^o components and a discontinuous Marcus component. The noise amplitudes are pseudo-differential operators including the transport operator. We develop a local-in-time theory of classical solutions, establishing existence, uniqueness, and a blow-up criterion. Discontinuous Marcus noise requires new analytical tools to control interactions between jump discontinuities and nonlocal operators. For compressible equations, we formulate a transformation-and-extension principle recovering the Makino variable and its symmetric quasilinear formulation. Using transformed variables, compatible nonlinear Sobolev estimates are developed to close high-order stochastic energy bounds. This accommodates broad physically relevant state equations, including piecewise $\gamma$-laws, Chaplygin laws, and the white dwarf pressure law. Many equations remain unexplored in multidimensional stochastic compressible settings, even under pure It\^o forcing. For the incompressible damped case, we identify damping--noise regimes guaranteeing global existence, uniform bounds, and decay. To study statistical behavior, we establish a novel existence criterion for invariant probability measures tailored to Markov semigroups satisfying a \emph{restricted Feller property under mismatched metrics}. Bypassing single-topology Feller continuity robustly extends the Krylov--Bogoliubov theory. We use this to construct invariant measures for singular stochastic evolution systems in Hilbert spaces. For mixed multiplicative noise and $d\ge 2$, we prove existence of invariant measures for stochastic damped Euler equations on $\mathbb{T}^d$ under moderate damping--noise, and uniqueness under strong damping--noise on $\mathbb{T}^d$ and $\mathbb{R}^d$.