Asymptotic expansions for a class of singular integrals emerging in nonlinear wave systems

We find asymptotic expansions as $\nu\to 0$ for integrals of the form $\int_{\mathbb{R}^d}F(x)/(\omega^2(x)+\nu^2)\,dx$, where sufficiently smooth functions $F$ and $\omega$ satisfy natural assumptions on their behavior at infinity and all critical points of $\omega$ in the set $\{\omega(x)=0\}$ are nondegenerate. These asymptotic expansions play a crucial role in analyzing stochastic models for nonlinear waves systems. We generalize a result of Kuksin that a similar asymptotic expansion occurs in a particular case where $\omega$ is a nondegenerate quadratic form of signature $(d/2,d/2)$ with even $d$.