In this paper, we are concerned with the three-dimensional Euler equations driven by an additive stochastic forcing. First, we construct global Hölder continuous (stationary) solutions in \(C(\mathbb {R};C^{\vartheta })\) space for some \(\vartheta >0\) via a different method from Lü and Zhu (Stoch Process Appl 177, 2024). Our approach is based on applying stochastic convex integration to the construction of Euler flows in De Lellis and Székelyhidi (Invent Math 193:377–407, 2013) to derive uniform moment estimates independent of time. Second, for any divergence-free Hölder continuous initial condition, we show the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in \(L^p_\mathrm{{loc}}([0,\infty );C^{\vartheta '}) \cap C_\mathrm{{loc}}([0,\infty );H_{\sigma }^{-1})\) for all \(p\in [1,\infty )\) and some \(\vartheta '>0\) .
