Laplace Asymptotics near Stratified Minimum Sets

Technical Summary: Laplace Asymptotics Near Stratified Minimum Sets

Problem Statement
The paper addresses the asymptotic behavior of real Laplace integrals of the form
$$I(z) = \int_N a(x) e^{-z f(x)} \, d\text{vol}_g(x), \quad z \to \infty,$$
where $(N, g)$ is a Riemannian manifold, $f$ is a continuous phase function bounded below, and $a$ is a measurable amplitude. The focus is on the case where the global minimum set $M = \{x : f(x) = f_0\}$ is compact but possibly stratified (composed of smooth manifolds of varying dimensions meeting at singularities) and the phase $f$ may be non-quadratic or non-smooth near $M$.

Classical Laplace methods rely on a universal local model (typically a non-degenerate quadratic form) to determine the decay rate and leading coefficient. However, outside the smooth, non-degenerate regime, no such universal normal form exists. Leading decay orders can vary (e.g., polynomial, stretched-exponential), and different directions may possess different weights. Existing theorems often fail to accommodate the full combination of geometric complexities (stratification, boundaries, relative domains) and analytic generality (measurable amplitudes, non-coercive models) simultaneously.

Methodology: The Normal-Cell Calculus
The author develops a constructive calculus based on an adapted normal-cell disintegration. Rather than assuming a canonical coordinate system, the framework requires the user to supply a specific geometric decomposition adapted to the problem.

1. Stratified Disintegration: The minimum set $M$ is decomposed into finitely many Borel base pieces $P_\alpha$. For each piece, a $C^1$ coordinate map $F_\alpha(p, s)$ is defined, where $p \in P_\alpha$ is the base variable and $s$ is a fiber (normal) variable.
2. Admissible Cells: The fiber variables are restricted to measurable "admissible cells" $A_\alpha(p) \subset \mathbb{R}^{d_\alpha}$. These cells can be full normal spaces, relative domains (e.g., half-spaces), polar cones, or sets dependent on the base point $p$. Crucially, these cells are chosen to cover a neighborhood of $M$ up to null sets without double counting.
3. Anisotropic Rescaling: The method employs anisotropic dilations $D^\beta_u t = (u^{\beta_1}t_1, \dots, u^{\beta_d}t_d)$ with weights $\beta$. The phase is rescaled to converge to a homogeneous limiting model $G_\alpha(p, t)$, and the amplitude converges to a limit $b_0(p)$.
4. Convergence Regimes: The paper establishes three distinct modes of convergence for the rescaled integrals:
   • Uniform Profile: Requires uniform convergence of cells and an exponential tightness condition (often via ambient coercivity).
   • Pointwise Cell: Allows the rescaled cells to converge only pointwise in the base, provided the rescaled fiber integrals are dominated by an integrable base majorant.
   • Joint Frontier Rescaling: Handles cases where a "collar" variable (tangent to a higher stratum) must be rescaled jointly with normal variables because the fiberwise coefficient becomes non-integrable near a frontier.

Key Contributions and Results

• Generalized Leading Order Theorems: The paper proves leading-order asymptotics where the coefficient is computed by integrating the limiting profile and amplitude over the limiting cell and then over the base piece. This framework recovers the Morse–Bott formula, conic constraints, and Euclidean/radial specializations as special cases.
• Handling Degeneracy and Cancellation:
  • The theory allows for lower-dimensional strata to dominate higher-dimensional ones if the former have sufficiently flat (high-order) phase behavior, resulting in a smaller decay exponent.
  • It provides an additive global comparison theorem that remains valid even when leading coefficients vanish or cancel (e.g., due to signed or complex amplitudes), necessitating higher-order expansions.
• Finite Expansions: The paper derives finite asymptotic expansions by representing scaled cells on a fixed reference cell and expanding the transported density. This includes corrections for moving cell boundaries via "straightening" maps and weighted jet criteria.
• Specific Applications:
  • Rank-Deficient Positive-Semidefinite Cones: In Bayesian estimation of covariance matrices, the method yields a Gaussian solid-angle factor that standard unconstrained formulas miss.
  • Inverse Kinematics: For a planar three-link robot arm with joint limits, the analysis shows that an isolated joint-limit fold (a boundary singularity) contributes to the posterior at the same leading order as a smooth self-motion curve. The limiting posterior is a mixture of a continuous component and an atom.
  • Weibullian Chaos: The framework analyzes tails of homogeneous functionals of random vectors with direction-dependent radial decay. It identifies regimes where the tail is polynomial (due to interior zeros of the cost) versus exponentially small, and derives "fractional" and "integer" frontier laws when the cost vanishes at the boundary of the feasible set.

Significance and Claims
The paper claims that no existing theorem for real Laplace integrals accommodates the full combination of geometric and analytic assumptions presented here. Specifically, it simultaneously permits:

• Compact minimum sets with finite $C^2$ stratifications.
• Borel base refinements, $C^1$ adapted charts, and measurable base-dependent cells.
• Different fiber dimensions and anisotropic weights on different pieces.
• A continuous phase and a merely measurable amplitude.
• Homogeneous limiting models that need not be smooth or ambiently coercive (controlled instead by cellwise exponential tightness).
• Joint rescaling of frontier and normal variables.

Trade-off: The generality comes at the cost of requiring the adapted normal-cell disintegration and the weighted limiting model to be supplied hypotheses to be verified in each application, rather than being consequences of a general stratification theorem. The paper does not assert that such a disintegration exists for arbitrary singular sets but provides the analytical machinery to compute asymptotics once such a structure is supplied.

The work positions itself as complementary to resolution-of-singularities methods (used in singular learning theory); while resolution produces powers and logarithmic multiplicities via log canonical thresholds, this paper's approach assumes a weighted principal model is supplied directly and controls its tails via coercivity or tightness on limiting cells.