Extremes of homogeneous Gaussian random fields

Let $\{X(s,t):s,t\geqslant 0\}$ be a centered homogeneous Gaussian field with a.s. continuous sample paths and correlation function $r(s,t)=Cov(X(s,t),X(0,0))$ such that $$r(s,t)=1-|s|^{\alpha_1}-|t|^{\alpha_2}+o(|s|^{\alpha_1}+|t|^{\alpha_2}), \quad s,t \to 0,$$ with $\alpha_1,\alpha_2\in(0,2],$ and $r(s,t)<1$ for $(s,t)\neq(0,0)$. In this contribution we derive an exact asymptotic expansion (as $u\to \infty$) of $$\mathbb{P}\left(\sup_{(s n_1(u),t n_2(u))\in\left[0,x\right]\times\left[0,y\right]}X(s,t)\leqslant u\right),$$ where $n_1(u)n_2(u)=u^{2/\alpha_1+2/\alpha_2}\Psi(u)$, which holds uniformly for $(x,y) \in [ A , B ]^2$ with $A , B$ two positive constants and $\Psi$ the survival function of an $N(0,1)$ random variable. We apply our findings to the analysis of asymptotics of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally we determine the extremal index of the discretised random field determined by $X(s,t)$.