Elementary Rule 156: Borel Transformations

• Elementary Rule 156 is a framework that defines Borel transformations on random fields to systematically analyze pushforward laws and uniform continuity in distribution.
• It provides precise methods for determining fractal dimensions using metric convergence and local behavior, facilitating rigorous geometric interpretations.
• The approach enables analytic regularization through Borel resummation and constructs non-Gaussian measures, with significant applications in quantum field theory and cosmology.

Borel transformations of random fields constitute a rigorous framework for analyzing how Borel-measurable maps affect the probabilistic structure of random fields defined on complex state spaces. These transformations enable the study of image measures, pushforward laws, fractal properties, and stability in applications ranging from stochastic analysis and probability theory to mathematical physics and cosmology. Key results include precise conditions for uniform continuity in distribution, the fractal dimension theorems for images of random fields, construction of non-Gaussian measures through Borel transformations, and methods such as Borel resummation for regularizing divergent series that naturally arise in physical models.

1. Definitions and Structural Framework

Random fields are formalized as Borel-measurable maps $t: W \to \mathbb{C}^S$ with $W$ a complete separable metric space and $S$ a generic index set. For a Borel probability measure $\eta \in \mathcal{P}(W)$, the induced law is written as $\distr_\eta(t) = \eta \circ t^{-1}$ (Bufetov, 30 Dec 2025). A Borel transformation is a map $g: \Lambda \times V \to W$, where $\Lambda \subset \mathcal{P}(W)$ is compact, $V \subset W$ is Borel with $\eta(V) = 1$ for all $\eta \in \Lambda$, and $g$ is jointly Borel. The pushforward map $g_*: \Lambda \to \mathcal{P}(W)$ is given by $$(g_* \eta)(A) = \int_V \mathbf{1}_A(g(\eta, w)) d\eta(w)$$ for all Borel sets $A \subset W$.

Topological and metric considerations include the Lévy–Prokhorov metric $d_{\mathrm{LP}}$ on $\mathcal{P}(W)$, uniform-in-$\eta$ convergence in probability for Borel maps, and the Tchebycheff–uniform metric on $\Lambda \to \mathcal{P}(W)$ (Bufetov, 30 Dec 2025). The interplay of these metric structures is essential for rigorous analysis of continuity and stability phenomena in random field transformations.

2. Uniform Continuity in Distribution for Borel Transformations

A central theorem (Bufetov) establishes that under simple, abstract conditions—namely, compactness of $\Lambda$, fullness of $V$, and Borel measurability—the mapping $g \mapsto g_*$ is uniformly continuous in distribution. Specifically, convergence of $g_n \to g$ in probability (uniformly over $\Lambda$) implies uniform convergence of the pushforward laws $g_{n*} \eta \to g_* \eta$ in $d_{\mathrm{LP}}$, for all $\eta \in \Lambda$ (Bufetov, 30 Dec 2025).

The proof strategy involves reduction to product spaces ($W = \mathbb{C}^\mathbb{N}$), coordinate-wise convergence, finite-dimensional projection arguments via Prokhorov's theorem, and explicit control of law distances using bounded-Lipschitz metrics. This result serves as a black-box criterion for continuity of empirical functionals in probability and statistics, and underlies stability properties in structural transformations of random fields.

3. Fractal Dimensions of Images Under Borel Transformations

Dimension theorems for Borel transformations of random fields characterize the Hausdorff and packing dimensions of image measures and image sets. For an $(N, d)$ random field $X$, a finite Borel measure $\mu$ on $\mathbb{R}^N$, and an analytic set $E \subset \mathbb{R}^N$, key results include (Shieh et al., 2010): $$\dim_H \mu_X = \min\left\{ d, \frac{1}{H} \dim_H \mu \right\}, \qquad \dim_P \mu_X = \frac{1}{H} \mathrm{Dim}_{Hd} \mu$$ for image measures, and

$$\dim_H X(E) = \min\left\{ d, \frac{1}{H} \dim_H E \right\}, \qquad \dim_P X(E) = \frac{1}{H} \mathrm{Dim}_{Hd} E$$

for image sets, provided appropriate local increment and small-ball conditions (Hölder-continuity and sectorial local nondeterminism) are met.

This "stretching" of Hausdorff and packing dimensions by the field's roughness index $H$ is established for fractional Brownian motion, self-similar stable fields, real harmonizable fractional Lévy fields, and other non-Gaussian processes.

4. Borel Resummation and Physical Applications

Borel transformations provide analytic regularization of divergent series, particularly in quantum field theory and stochastic models. The Borel transform of a formal power series $f(z)=\sum_{k=0}^\infty a_k z^{k+\alpha}$ is

$$\mathcal{B}[f](t) = \sum_{k=0}^\infty \frac{a_k}{\Gamma(k+\alpha)} t^{k+\alpha-1}$$

and the inverse Laplace–Borel integral recovers the analytic continuation (Honda et al., 2023). In stochastic inflation, the secular series for correlation functions exhibit zero convergence radius, but Borel–Padé approximation yields nonperturbative analytic functions for all time, capturing both transient growth and equilibrium saturation.

The Borel–Padé resummation method is robust: given finitely many series coefficients, one forms and analytically continues the truncated Borel transform by Padé approximants, and integrates via the Laplace–Borel formula. Singularities in the Borel plane provide insight into late-time behavior and equilibrium values, with cosmological implications for primordial black hole production and non-Gaussian CMB features.

5. Non-Gaussian Measures via Nonlinear Borel Transformations

Nonlinear Borel transformations enable the construction of genuinely non-Gaussian Borel measures on path spaces. Let $o:\mathbb{R}\to\mathbb{R}$ be smooth and strictly increasing; for a Gaussian field $X_0$ on balls in $\mathbb{R}^d$, the transformed field $X(B) = o(X_0(B))$ induces a pushforward measure $P = T_* P_0$ on the space of continuous, ball-indexed functions (Zahariev, 2021).

Finite-dimensional marginals of $P$ are non-Gaussian, with nonvanishing higher cumulants. The construction ensures probabilistic normalization, positivity, Euclidean covariance, and reflection positivity, enabling the functional realization of operator algebras (Haag–Kastler nets) and physical Hilbert spaces after Osterwalder–Schrader reconstruction. The measure supports rich algebraic structures in Euclidean and Minkowski quantum field theory.

6. Examples, Applications, and Further Directions

Normalized exponential transformations underpin the theory of Gaussian multiplicative chaos (GMC), guaranteeing that small perturbations in the underlying field induce uniformly small changes in the induced law, a property pivotal for universality and stability phenomena (Bufetov, 30 Dec 2025). In statistical inference, the uniform continuity in distribution of empirical functionals is assured under these general conditions.

Borel transformations and resummation techniques extend to higher-order correlators and full probability distribution functions, yielding analytic access to rare fluctuations and sharpened physical predictions in stochastic and cosmological models (Honda et al., 2023). This family of results opens possibilities for studying functional differentiability (Fréchet derivatives) and is adaptable to general Polish spaces and alternative metrics, including Wasserstein and bounded-Lipschitz topologies.

7. Caveats and Critical Considerations

Results are generally valid "almost surely" outside a single null set in the probability space (Shieh et al., 2010). Packing dimension profiles may differ from classical packing dimensions, necessitating careful distinction. Certain analogues and upper bounds require additional conditions on tail exponents and small-ball probabilities. For Lévy processes in high dimensions, some packing dimension formulae break down absent further hypotheses.

In summary, Borel transformations of random fields form a unified, rigorous framework for analyzing both the probabilistic and geometric properties of fields under measurable transformations. Their applications span fractal analysis, stochastic process regularization, quantum field theory construction, and statistical inference, grounded in robust topological and measure-theoretic concepts.