On local well-posedness of the stochastic incompressible density-dependent Euler equations

In this paper we study the stochastic inhomogeneous incompressible Euler equations in the whole space ${\text{RR}}^3$. We prove the existence and pathwise uniqueness of local solutions with both additive and multiplicative stochastic noise. Our approach is based on reducing our problem to a random problem and some estimations for type transport equations.