Measure valued solutions to the stochastic Euler equations in $$\mathbb {R}^d$$ R d

Abstract

We establish the existence of measure-valued solutions of the stochastic Euler equations in \(\mathbb {R}^d, \, d \ge 2.\) Our definition of a measure-valued solution is based on that for the deterministic Euler equations. Our first result is the existence over a given probability space in the spirit of Bensoussan and Temam (J Funct Anal 13:195–222, 1973). The main tool is measurable selection established in Stroock and Varadhan (Multidimensional diffusion processes, 1997). As a necessary step for this result, we also present a new existence result on the stochastic Navier–Stokes equations. Our second result is the existence of a martingale measure-valued solution. The main tool is the generalized Skorokhod representation theorem due to Jakubowski (Theory Probab Appl, 42:167–174, 1998).