Projective Systems of Heat Kernel Measures

• Projective systems of heat kernel measures are organized families of probability measures that ensure compatibility across refined or projected approximations, capturing continuum limits.
• They are applied in diverse contexts—from Riemann surfaces and fractal spaces to infinite-dimensional Lie groups and gauge theories—to model diffusion and quantum processes.
• Their construction employs rigorous methods such as group summation, finite-dimensional projections, and resistance forms to manage analytic challenges in irregular and infinite systems.

A projective system of heat kernel measures is an organized family of probability measures derived from heat kernels across a directed system of spaces—often covering a continuum limit, a refinement process, or an approximation of infinite structures. Heat kernel measures encode diffusion or quantum mechanical processes through fundamental solutions to the heat equation, and projective systems ensure their compatibility under projections, refinements, or group actions. This concept is central in geometric analysis, probability on manifolds, statistical mechanics, and quantum field theory, especially in constructions where the infinite-volume or continuum limit is achieved via refined or exhaustive sequences of finite models.

1. Model Examples: Riemann Surfaces and Universal Coverings

On compact, non-simply-connected Riemann surfaces, projective systems of heat kernel measures arise from lifting local heat kernel constructions to the universal cover and then “tiling” by the action of the covering group. For instance, the explicit heat kernels for the Euclidean plane $\mathbb{R}^2$, hyperbolic plane $\mathbb{H}^2$, and sphere $S^2$ serve as universal models. The heat kernel $K_M(x,y,t)$ on the surface $M$ is obtained by summing over the covering group $G$:

$$K_M(x, y, t) = \sum_{g \in G} K_U(\tilde{x}, g \cdot \tilde{y}, t)$$

where $U$ is the universal cover and $\tilde{x}, \tilde{y}$ are liftings of $x,y$. The compatibility implied by the group action produces a projective system over all finite covers, relevant for path integral measures, large-scale limits, and spectral theory (Jones et al., 2010).

2. Approximations by Finite-Dimensional Projections and Infinite-Dimensional Limits

In the analysis of infinite-dimensional Lie groups (e.g., Heisenberg-like groups), projective systems are constructed by considering finite-dimensional “projection groups” $G_P$ and associated heat kernel measures $v_t^P$, then passing to the limit. The projection/limit procedure is carefully controlled via uniform estimates: generalized curvature–dimension inequalities, Wang-type integrated Harnack inequalities, and explicit quasi-invariance bounds (e.g., Cameron-Martin type results):

$$\left( \frac{d(v_t \circ R_y^{-1})}{dv_t} \right)(x) \leq \exp\left[(1 + 2\|w\|^{1/2})\frac{q d(e, y)^2}{4T}\right]$$

Here, $d(e, y)$ is a sub-Riemannian distance, emphasizing the role of horizontal structures in the non-elliptic setting. Such projective systems are fundamental for the paper of infinite-dimensional diffusion measures, strong Feller and smoothing properties, and the geometry of path spaces (Baudoin et al., 2011).

3. Heat Kernel Measures in Abelian Gauge Theory via Cochain Complexes

For Abelian polyhedral gauge theories, projective systems arise from sequences of discrete complexes (lattices or polyhedral meshes) with refinement maps. Each finite complex $K_i$ is associated to a measure on 1-cochains $C^1(K_i; G)$ (gauge fields) via push-forward of the heat kernel on 2-cochains through the coboundary operator $d_1^G$:

$$\mu_\beta^{K_i} = z_\beta \left(H_\beta^{K_i} \circ d_1^G \right) \cdot \nu_1^{K_i}$$

Projection maps between cochains at different levels are required to be cochain maps (commute with coboundaries), and the measures on orbit spaces $C^1(K_i; G)/C^0(K_i; G)$ are assembled into a consistent projective system. Taking the projective limit yields a continuum or infinite-volume heat kernel measure that can be translation invariant (after suitable Hilbert space constructions). This formalism leads to massless gauge theories with power-law decay of correlation functions, distinct from standard lattice models (Zahariev, 24 Oct 2025).

4. Projective Systems on Fractals, Graphs, and Metric Measure Spaces

In fractal and dendritic spaces equipped with resistance forms, projective systems of heat kernel measures reflect the transition from discrete approximants (finite graphs or trees) to their continuous limits. The resistance metric $R(x, y)$ and local irregularities determine heat kernel asymptotics:

$$p_t(x, x) \approx h^{-1}(t)/t \quad \text{with fluctuations encoded by} \; g(r) = f_i(r)/f_u(r)$$

Corrections arising from non-uniform volume growth or log-logarithmic terms propagate through the projective system and affect short-time behavior. Analogous consistency is achieved in quantum graphs or "manifold-like" spaces, where locality of the heat kernel under compatible Dirichlet forms ensures that projective limits maintain asymptotic expansion and analytic properties (Croydon, 2012, Post et al., 2017).

5. Harmonic Analysis, Quantum Systems, and Representation Theory

In the quantization of symmetric multi-quDit systems, Wigner and Husimi quasi-probability distributions are organized as a projective system of heat kernel smoothed distributions $\mathcal{F}_\rho^{(s)}$ on phase space $\mathbb{C}P^{D-1}$. The generalized heat kernel $\mathcal{K}_{s,s'}$ acts as a smoothing operator between distributions with varying regularity (s-parameter), and in the thermodynamic limit $N \to \infty$ this yields standard Gaussian smoothing:

$$\partial \Delta^{(s)}(z)/\partial s \simeq \frac{1}{2N} \Delta \Delta^{(s)}(z)$$

Young tableaux provide a diagrammatic classification of the projective structure; the consistency of the invariant measure (arising from Haar measure and Laplacian eigenfunctions) is established across projections to subspaces and limit measures (Calixto et al., 20 Jul 2025).

6. Singularities and Compatibility in Projective Limits

Fractal diffusions with full off-diagonal sub-Gaussian heat kernel estimates (walk dimension $\beta>2$) yield energy measures that are singular with respect to the underlying symmetric measure, affecting the construction of projective systems. In contrast, for Gaussian estimates ($\beta=2$), the energy and symmetric measures are absolutely continuous, and the intrinsic metric is non-degenerate. This dichotomy determines the feasibility and structure of projective limits for heat kernel measures and their analytic properties, particularly in fractal and highly irregular spaces (Kajino et al., 2019).

7. Summary Table: Construction Contexts and Key Properties

Context	Limit Object	Projective Consistency Mechanism
Riemann surfaces/universal covers	Heat kernel on base M	Tiling/group summation over covers
Abelian lattice gauge theory	Infinite-volume measure	Renormalized inner products, cochain maps
Infinite-dimensional Lie groups	Limit measure on group	Uniform estimates on projections
Fractal/dendritic spaces	Limit heat kernel	Resistance form, volume corrections
Multi-quDit quantum systems	Thermodynamic smoothing	Haar measure, Laplacian eigenfunction basis
Ultrametric/hierarchical spaces	Limit oscillatory kernel	Ball structure, eigenfunctions

8. Technical Challenges and Limitations

When constructing projective systems of heat kernel measures, key challenges include ensuring compatibility of measures under projection, managing singularities arising in irregular spaces, and selecting appropriate renormalizations (e.g., inner products in gauge theory, or coefficient weights in representation theory). In some settings, measures may fail to be translation invariant or unique unless further structures or Hilbert space limits are imposed. The analytic behavior (e.g., masslessness, oscillatory asymptotics, non-regular decay) is governed by the geometry and scaling properties of the underlying spaces.

9. Significance in Analysis, Physics, and Geometry

Projective systems formalize the passage from finite, often combinatorial, approximations to infinite or continuum models in analysis and mathematical physics—encoding quantum, probabilistic, or spectral information in measures that remain consistent under refinement or projection. They provide the foundation for limit behaviors (scaling, renormalization), explicit computations (e.g., correlation functions in gauge theory), and unifications (e.g., of heat kernel asymptotics across manifolds, fractals, and quantum systems). The existence and explicit construction of such systems enable rigorous paper of large-scale and limiting phenomena not accessible through purely local or finite means.