Occupation Measures & Potentials

• Potentials of occupation measures are defined as integrals of kernel functions over empirical time measures, capturing the spatial and temporal structure of stochastic processes.
• They provide analytic tools for assessing regularity, convergence, and variational properties in settings like Markov processes, random fields, and mean-field interactions.
• Applications include mean-field models, trajectory optimization, and collision analysis, linking potential theory with probabilistic geometry and scaling limits.

Potentials of occupation measures are central objects in stochastic analysis, probability theory, and the calculus of variations. They arise in multiple contexts including Markov process theory, variational problems involving energies of distributions, interacting particle systems, and the study of regularity and scaling limits for random fields. The concept unifies analysis of functionals such as the Coulomb potential, Riesz and Bessel potentials, and general Markovian potentials associated with additive functionals. Potentials encode both the spatial and temporal structure of the underlying occupation measure, providing crucial quantitative tools for understanding large deviations, mean-field behavior, and pathwise regularity.

1. Formal Definitions and Classes of Potentials

The foundational notion is the occupation measure of a stochastic process, typically defined as the empirical measure of time spent in subsets of the state space. Given a process $X=(X_t)_{t\geq0}$ on a state space $S$, the time-$t$ occupation measure is

$$\mu_t(A) = \frac{1}{t} \int_0^t \mathbf{1}_A(X_s)\ ds,$$

for Borel $A\subset S$ (Koenig et al., 2015). For random fields $X(t)$ indexed by $t$ in Euclidean domains, the occupation measure is

$$\mu_X(B) = \mathcal{L}^k\{t\in[0,1]^k:\ X(t)\in B\}$$

with $\mathcal{L}^k$ the Lebesgue measure (Hinz et al., 16 Dec 2025).

Given a Borel measure $\mu$ on $\mathbb{R}^n$ and a potential kernel $K(x,y)$—for instance, the Newtonian kernel $|x-y|^{-1}$ in $\mathbb{R}^3$, or the Riesz kernel $|x-y|^{\alpha-n}$—the potential is defined by

$U\mu(x) = \int K(x,y)\ \mu(dy).$

Prominent examples include:

• Coulomb potentials: $$A(\mu)(x) = \int_{\mathbb{R}^3} |x-y|^{-1}\ \mu(dy)$$, related to electrostatics and mean-field interactions (Koenig et al., 2015).
• Riesz potentials: $$U^\alpha\mu(x) = \int_{\mathbb{R}^n}|x-y|^{\alpha-n}\ \mu(dy)$$, for $0<\alpha<n$ (Hinz et al., 16 Dec 2025).
• Bessel potentials: Using the Bessel kernel $g_\alpha$ (Fourier dual to $\langle\xi\rangle^{-\alpha}$), $$G_\alpha\mu(x) = \int_{\mathbb{R}^n}g_\alpha(x-y)\,\mu(dy)$$ (Hinz et al., 16 Dec 2025).
• Markov process potentials: Given a heat kernel $p(t,x,y)$, $$U\mu(x) = \int_0^\infty \int_S p(t,x,y)\ \mu(dy)\ dt$$ (Noda, 22 Oct 2025).

These potentials are used both as analytic tools and as components of variational energies, governing concentration properties and regularity.

2. Analytical and Probabilistic Properties

Potentials of occupation measures display intricate regularity and integrability properties, often linked to the underlying geometry and stochasticity. Key results include:

• Continuity and boundedness: For a measure $\mu$ with Fourier transform $\hat\mu$, a sufficient criterion for potential continuity is $$\int_{\mathbb{R}^n} |\hat\mu(\xi)||\xi|^{-\alpha}d\xi<\infty$$ for the Riesz potential $U^\alpha\mu$, ensuring $U^\alpha\mu\in C_0(\mathbb{R}^n)$ (Hinz et al., 16 Dec 2025).
• Hölder continuity: For Gaussian random fields $X$ with local nondeterminism, if certain determinants of increment covariances are integrable in a manner weaker than the classical Pitt condition, the associated Bessel or Riesz potential is almost surely Hölder continuous (Hinz et al., 16 Dec 2025).
• Large deviation tube properties: For occupation measures arising from three-dimensional Brownian paths with mean-field Coulomb self-interaction, the entire profile of $A(\mu_t)$ is shown to concentrate, in uniform norm, near the manifold of translates of the maximizer’s potential with exponentially small deviation probabilities (Koenig et al., 2015).
• Additive functionals and Markovian structure: For Markov processes, the Revuz correspondence provides an isomorphism between positive continuous additive functionals (PCAFs) and their associated “Revuz measure” so that the potential operator $U$ is the $0$-resolvent of the process (Noda, 22 Oct 2025).

The table below summarizes some key regularity criteria for occupation potentials.

Setting	Criterion for $U^\alpha\mu$ Continuity	Reference
General measure	$\int |\hat\mu(\xi)||\xi|^{-\alpha} d\xi < \infty$	(Hinz et al., 16 Dec 2025)
Fractional Brownian field (k,n,H)	$n - k/(2H) < \alpha < n$	(Hinz et al., 16 Dec 2025)
Markov process with heat kernel	Uniform decay of $R_p^\alpha\mu$ as $\alpha\to\infty$	(Noda, 22 Oct 2025)

3. Potentials in Variational Problems and Mean-Field Interactions

Occupation measure potentials serve as variational objects in problems involving self-interaction or external fields. For Brownian motion in $\mathbb{R}^3$ subject to exponential tilt by Coulomb energy, the Donsker–Varadhan variational formula determines the leading asymptotics of the partition function: $$\lim_{t\to\infty} \frac{1}{t}\log Z_t = \sup_{\mu \in M_1(\mathbb{R}^3)} \left\{ H(\mu) - I(\mu) \right\},$$ where $H(\mu)$ is Coulomb energy and $I(\mu)$ is Dirichlet energy for $\mu(dx)=\psi^2(x)dx$ (Koenig et al., 2015). The maximizers (up to translation) are explicitly characterized, and the occupation measure—under the tilted path measure—concentrates around these maximizers. The induced Coulomb potential concentrates in supremum norm, not just weak topology, with pathwise Hölder continuity. This rigorously justifies mean-field pictures in models such as the strong-coupling polaron.

In the context of delay differential equations and ODEs, occupation measures underpin convex relaxations of nonconvex trajectory optimization problems, allowing infinite-dimensional linear programming and semidefinite relaxations with rigorous outer approximations (Miller et al., 2023).

4. Occupation Potentials in Markov Process Theory

For continuous-time Markov processes, the occupation potential encodes the solution to elliptic problems and harmonic analysis on the process’s state space. Let $X$ be a Markov process with heat kernel $p(t,x,y)$, and let $\mu$ be a smooth measure (e.g., via the Revuz correspondence). Then the occupation potential

$$U\mu(x) = \int_0^\infty \int_S p(t,x,y) \mu(dy)\ dt$$

is the $0$-resolvent applied to $\mu$. This framework is extended to space–time occupation measures (STOMs), random measures on $S\times[0,\infty)$ defined by

$\Pi(G) = \int_0^\infty G(X_s,s)\, dA_s,$

where $A$ is a PCAF (Noda, 22 Oct 2025). Potentials govern additive functionals and, via their decay, control tightness and convergence in Gromov–Hausdorff-type scaling limits.

Uniform decay conditions on the potentials, such as

$$\lim_{\alpha\to\infty} \|R_p^\alpha \mu\|_\infty = 0,$$

are pivotal for establishing convergence of both PCAFs and STOMs as the underlying metric-measure structures vary (Noda, 22 Oct 2025).

5. Collision Measures and Scaling Limits

Potentials of occupation measures play a crucial role in the analysis of collision events for independent stochastic processes. For processes $X^1, X^2$ on $S$, the collision measure on $S \times [0, \infty)$ is described as the projection of the STOM of the product process to the diagonal: $$\Pi(dx\,dt) = \sum_{y\in S} 1_{\{X^1_t = X^2_t = y\}}\, \delta_y(dx)\, dt.$$ Under structural convergence of the underlying spaces and measures (e.g., for critical Galton–Watson trees or Erdős–Rényi graphs), collision measures converge in distribution, with uniform decay of associated occupation potentials crucial for the proof (Noda, 22 Oct 2025). This yields concrete scaling limits in random graph models and resistance metric spaces, connecting stochastic process theory, metric geometry, and potential theory.

6. Strengths, Limitations, and Generalizations

Potentials of occupation measures offer several advantages over classical “local time” analysis:

• Existence in higher dimensions: While local times typically exist only under restrictive dimension regimes, potentials may be continuous and bounded in much broader settings, with weaker integrability or moment constraints (Hinz et al., 16 Dec 2025).
• Weaker regularity assumptions: Pitt-type and Fourier analytic criteria for continuity are less restrictive than those for local times, enabling treatment of more singular random fields.
• Variance-minimizing properties: In variational problems, occupation potentials encode self-interaction energies, optimal resource allocation, and energy-minimizing behavior.
• Flexible frameworks: Occupation potentials extend naturally to space–time measures, delay systems, and generalized energy functionals (Noda, 22 Oct 2025, Miller et al., 2023).

Limitations include potential computational complexity in high-dimensional optimization problems and the need for good moment or decay estimates of potentials to ensure convergence in scaling limits or semidefinite programming approximations (Miller et al., 2023).

7. Representative Applications and Broader Impact

Occupation measure potentials are instrumental in:

• Mean-field models: Justifying asymptotics in polaron models and other strong-coupling limits (Koenig et al., 2015).
• Variational calculus and fractal minimizers: Proving existence and regularity of non-trivial minimizers with explicit potential-theoretic tools (Hinz et al., 16 Dec 2025).
• Trajectory optimization in dynamical systems: Enabling convex relaxations and rigorous outer approximations for peak estimates, safety constraints, and delay systems (Miller et al., 2023).
• Scaling limits and collision analysis: Deriving concrete laws for collision times and locations in random walks on critical random graphs and trees, with explicit link to potential decay (Noda, 22 Oct 2025).

These connections highlight occupation potentials as a unifying concept bridging stochastic processes, potential theory, analysis, and probabilistic geometry.