Asymptotic behavior at infinity of solutions of Monge-Ampère equations in half spaces

We prove that any convex viscosity solution of \(\det D^2u=1 \) outside a bounded domain of \(\mathbb{R}^n_+\) tends to a quadratic polynomial at infinity with rate at least \(\frac{x_n}{|x|^{n}}\) if \(u\) is a quadratic polynomial on \(\{x_n=0\}\) and satisfies \( \mu|x|^2\leq u\leq \mu^{-1}|x|^2\) as \(|x|\rightarrow \infty\) for some \(0