Borel Transformations of Random Fields

• Borel transformations are measurable maps that push forward probability measures, rigorously altering the distribution and properties of random fields.
• They facilitate the construction of non-Gaussian measures and resummation techniques, proving essential in statistical mechanics and quantum field theory.
• The uniform continuity of pushforward maps under compactness conditions guarantees stability in probabilistic analysis and fractal dimension studies.

A Borel transformation of a random field refers to the application of a Borel-measurable map to the field, with the resulting pushforward measure characterizing the transformed probabilistic structure. This notion is fundamental in probability theory, modern statistical mechanics, the study of stochastic PDEs, and quantum field theory, where random fields and their non-linear functionals frequently arise. Borel transformations enable the rigorous analysis of random fields under measurable perturbations, support the construction of non-Gaussian measures, facilitate dimension-theoretic investigations, and underlie advanced resummation techniques for divergent formal expansions in physical models.

1. Formalism of Borel Transformations of Random Fields

Let $(W, d)$ be a complete separable metric space (state space), and denote by $\mathcal{P}(W)$ the set of Borel probability measures on $W$, equipped with the weak topology. A random field indexed by $S$ is a Borel-measurable map $t: W \rightarrow \mathbb{C}^S$, and induces a distribution $\eta \circ t^{-1} \in \mathcal{P}(\mathbb{C}^S)$ for $\eta \in \mathcal{P}(W)$.

A Borel transformation is a map $g: \Lambda \times V \rightarrow W$ (with $\Lambda \subset \mathcal{P}(W)$ compact, $V \subset W$ Borel, $\eta(V)=1$ for all $\eta \in \Lambda$), Borel-measurable in both arguments. The associated pushforward is

$$g_*(\eta) = (g(\eta, \cdot))_*(\eta) \in \mathcal{P}(W),$$

meaning, for all bounded continuous $\varphi$,

$$\int_W \varphi\,d(g_*\eta) = \int_V \varphi(g(\eta,w))\,d\eta(w).$$

The topology on Borel transformations is given via uniform-in-$\eta$ convergence in probability, with neighborhoods defined by

$$U(g_0;\varepsilon, \delta) = \Bigl\{g:\sup_{\eta \in \Lambda} \eta\left\{w: d(g(\eta,w), g_0(\eta,w)) > \varepsilon\right\} < \delta\Bigr\}.$$

The induced metric on $\mathcal{P}(W)$ is the Lévy–Prokhorov distance,

$$d_{\rm LP}(\mu, \nu) \le \epsilon \iff \forall\,\text{closed } K \subset W,\; \mu(K) \le \nu(K_\epsilon) + \epsilon \text{ and } \nu(K) \le \mu(K_\epsilon) + \epsilon,$$

where $K_\epsilon = \{w : d(w, K) \leq \epsilon\}$.

2. Uniform Continuity of Borel Pushforwards

A principal result establishes that the map $g \mapsto g_*$ is uniformly continuous in distribution under compactness and Borel-measurability, with respect to the topologies introduced above. Given any sequence $g_n \to g$ (in uniform-in-$\eta$ convergence in probability), the induced pushforwards $g_{n*} \to g_*$ uniformly over $\eta \in \Lambda$ in the Lévy–Prokhorov metric. The rigorous theorem (Bufetov) (Bufetov, 30 Dec 2025) states: $$g \mapsto g_* : \left(B_\Lambda(\Lambda \times V, W), d_{\rm prob}\right) \to \left(B(\Lambda, \Lambda), d_{\rm unif}\right)$$ is uniformly continuous, requiring only compact parameter space $\Lambda$, “full” support $V$ ($\eta(V)=1$), and mere Borel-measurability.

Key elements of the proof include:

• A reduction to coordinatewise convergence in probability on $W = \mathbb{C}^{\mathbb{N}}$.
• The control of full laws via finite-dimensional marginals, established by Prokhorov compactness arguments.
• Quantitative inequalities relating coordinatewise deviations to the bounded-Lipschitz metric, ultimately bounding the Lévy–Prokhorov distance between the transformed laws.

No moduli of continuity or moment growth conditions beyond compactness and Borel-measurability are needed for this uniformity. This result is crucial for stability analysis and universality in Gaussian multiplicative chaos, as well as for establishing black-box criteria in statistical inference for random fields (Bufetov, 30 Dec 2025).

3. Transformation-Induced Measures and Non-Gaussian Random Fields

Borel transformations allow explicit construction of non-Gaussian measures associated with random fields. Given a canonical Gaussian measure $P_0$ on a function space (typically spaces of continuous functions indexed by balls in $\mathbb{R}^d$), a nonlinear measurable map $T: \Omega \to \Omega$ yields a non-Gaussian Borel measure $P = T_* P_0$. For instance, with transformation $X(B) = o(X_0(B))$ for a smooth strictly increasing function $o$ (e.g., $o(t) = t + \lambda t^3$), the resulting field has nonvanishing higher cumulants and supports the analysis of reflection positivity, covariance, and operator algebra constructions in Euclidean quantum field theory (Zahariev, 2021).

The induced finite-dimensional marginals and cumulants of the transformed field are obtained by pushing forward the Gaussian law under $o$, leading to

$$\kappa_n(B_1, \ldots, B_n) = \mathrm{Cum}_{P}\bigl(X(B_1),\ldots,X(B_n)\bigr),$$

with explicit calculation via Hermite expansions against Gaussian densities.

This rigorous construction provides explicit examples of non-Gaussian fields satisfying the Haag–Kastler axioms through the transformed measure, with Osterwalder–Schrader positivity and reconstruction yielding net structures compatible with quantum field theory (Zahariev, 2021).

4. Dimension Theory for Borel Transformations of Random Fields

For random fields $X = \{X(t) : t \in \mathbb{R}^N\}$, Borel (analytic) transformations $X$ induce image measures and sets with fractal dimension characteristics determined by the regularity of $X$. Under hypotheses on local Hölder behavior (local maximal-increment bounds, small-ball estimates), one obtains explicit almost-sure formulas for the Hausdorff and packing dimensions of transformed measures and sets (Shieh et al., 2010): $$\dim_H \mu_X = \min\left\{ d, \frac{1}{H}\dim_H \mu \right\}, \qquad \dim_P \mu_X = \frac{1}{H} \mathrm{Dim}_{Hd}\, \mu,$$

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

where $H$ is the self-similarity (Hurst) index. These results apply to Gaussian, stable, real-harmonizable fractional Lévy, and Rosenblatt fields, provided suitable local regularity and density conditions are met.

Such "stretching" of fractal dimensions by $1/H$ (up to ambient caps) is fundamental in characterizing the image sets of Borel transforms of random fields, particularly for studying universal scaling, multifractal spectra, and pathwise irregularity.

5. Borel Transformations in Series Resummation and Stochastic Inflation

Borel transformations are central to the resummation of divergent series arising in stochastic field theories. The Borel transform of a formal (possibly divergent) power series $f(z) = \sum_{k=0}^\infty a_k z^{k+\alpha}$ is defined as

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

and inverted via the Laplace–Borel integral

$$S[f](z) = \int_0^\infty dt\,e^{-t/z} \widetilde{\mathcal{B}[f]}(t).$$

In stochastic inflation, perturbative expansions for correlation functions (e.g., $\langle\phi^2\rangle$) often diverge due to secular growth. Borel–Padé methods allow effective resummation by constructing rational approximants to the Borel transform, handling singularities, and delivering analytic continuations capturing both short- and long-time behavior (Honda et al., 2023).

The absence of singularities on the integration contour ensures unambiguous, nonperturbative results for observable correlators, with broad applications to inflationary cosmology, large deviation statistics, and universality in SPDEs.

6. Applications and Further Developments

Borel transformations of random fields have applications in:

• Gaussian multiplicative chaos: Ensuring stability and universality under perturbations, as uniform continuity in law for normalized exponential functionals guarantees small probabilistic deformations propagate continuously to entire law changes (Bufetov, 30 Dec 2025).
• Quantum field theory: Construction of non-Gaussian Euclidean field theories consistent with Osterwalder–Schrader axioms and rigorous operator algebraic structures (Zahariev, 2021).
• Stochastic processes and dimension theory: Determining fractal properties and multifractal spectra of image sets and measures under random field mappings (Shieh et al., 2010).
• Resummation in mathematical physics: Analytical control of secular divergences and nonperturbative physics in self-interacting fields, crucial for cosmology and stochastic analysis (Honda et al., 2023).

Future directions include: extension to Polish target spaces, consideration of alternative metrics (e.g., Wasserstein, bounded-Lipschitz), and investigations of Fréchet differentiability and functional delta methods in statistical inference and functional analysis.

7. Summary Table of Core Results

Paper Title	Principal Concept	arXiv id
Uniform Continuity in Distribution for Borel Transformations of Random Fields	Uniform continuity of Borel pushforwards	(Bufetov, 30 Dec 2025)
Borel resummation of secular divergences in stochastic inflation	Borel–Padé resummation of divergent series	(Honda et al., 2023)
Set-indexed random fields and algebraic Euclidean quantum field theory	Non-Gaussian measure construction, QFT	(Zahariev, 2021)
Hausdorff and packing dimensions of the images of random fields	Dimension theorems for image measures	(Shieh et al., 2010)

These foundational results establish rigorous frameworks and analytic techniques at the interface of probability, measure theory, analysis, and mathematical physics, underpinning the modern theory and applications of Borel transformations of random fields.