Non-uniqueness of H\"older continuous solutions for stochastic Euler and Hypodissipative Navier-Stokes equations

We construct infinitely many H\"older continuous, global-in-time, and stationary solutions to the stochastic Euler equations and the hypodissipative Navier-Stokes equations, taking values in the space $C(\mathbb{R};C^{\vartheta})$. For the Euler case, the H\"older exponent $\vartheta$ satisfies $0<\vartheta<\frac{5}{7}\beta$ with $0<\beta< \frac{1}{200}$, while for the hypodissipative Navier-Stokes equations, $\beta$ must additionally satisfy $0<\beta< \min\left\{ \frac{2(1-2\alpha)}{21}, \frac{1}{200}\right\}$. The construction relies on a modified stochastic convex integration scheme, which is central to the analysis. This scheme incorporates Beltrami flows as building blocks and carefully tracks inductive estimates, both pathwise and in expectation. These refinements allow us to achieve improved H\"older regularity for solutions to the underlying stochastic equations, advancing the scope of convex integration techniques in the stochastic setting.