Laplace Asymptotics near Stratified Minimum Sets

Abstract

We develop a real Laplace method for integrals $$I(z)={\int }_{N}a(x)e^{-zf(x)}{\mathrm{dvol}}_g(x),z\to \infty ,$$ near a compact, possibly stratified, minimum set of the phase. The basic input is an adapted normal-cell disintegration, defined up to null sets, together with piecewise anisotropic dilations and homogeneous limiting models. The framework simultaneously permits finite $C^2$ stratifications; Borel base refinements and $C^1$ adapted charts; measurable base-dependent cells, relative domains, and incident-stratum assignments; different fiber dimensions and weights on different pieces; a continuous phase and a merely measurable amplitude; and limiting models controlled either by ambient coercivity or directly by cellwise exponential tightness. The cells may converge only in weighted measure, or pointwise under an integrable base majorant. To the best of our knowledge, no existing theorem for real Laplace integrals accommodates this full combination of geometric and analytic assumptions. The tradeoff is that the adapted disintegration and limiting models are supplied hypotheses to be verified in each application.