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

We find asymptotic expansions as $ν\to 0$ for integrals of the form $\int_{\mathbb{R}^d}F(x)/(ω^2(x)+ν^2) dx$, where sufficiently smooth functions $F$ and $ω$ satisfy natural assumptions on their behavior at infinity and all critical points of $ω$ in the set $\{ω(x)=0\}$ 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 $ω$ is a nondegenerate quadratic form of signature $(d/2,d/2)$ with even $d$.