Dual Gaussian-Particle Representation

• Dual Gaussian-particle representation is a framework that constructs Gaussian processes as fluctuation limits of particle systems, linking microscopic dynamics to macroscopic covariance structures.
• It recovers classical and novel processes, including fractional Brownian motion, sub-fractional, and negative sub-fractional Brownian motions through dual pairing with test functions.
• The approach offers a transparent physical interpretation of increment correlations and long-range dependencies, making it valuable for statistical physics and spatial statistics.

The dual Gaussian-particle representation unifies analytic and physical perspectives by constructing Gaussian processes and fields as fluctuation limits of large-scale particle systems, pairing distribution-valued random fields (in the sense of $S'$-valued processes) with test functions to recover well-known and novel Gaussian processes—including fractional Brownian motion (fBm), sub-fractional Brownian motion (sub-fBm), and negative sub-fractional Brownian motion (nsfBm). This approach yields both mathematical generality and a physically transparent interpretation of increment structure, dependence, and covariance, directly linking the microscale dynamics of symmetric $\alpha$-stable Lévy particle systems (and their empirical measures) to the covariance structure of non-Markovian Gaussian processes over the full range of admissible Hurst parameters.

1. Construction of Gaussian Processes from Particle Systems

The foundational mechanism proceeds via a double limiting procedure:

1. Empirical Process Fluctuation Limit: Consider a high-density system of independent particles, possibly with critical binary branching, starting from homogeneous Poisson or quasi-homogeneous initial configurations. Each particle follows an independent symmetric $\alpha$-stable Lévy motion. Define the empirical process $N_T(t)$ as the particle count in a region up to time $t$, and normalize by scaling factor $V_T$:

$X_T(t) = \frac{N_T(t) - \mathbb{E}[N_T(t)]}{V_T}$

As $T \to \infty$, $X_T$ converges (in the sense of finite-dimensional distributions in $S'$) to a centered $S'$-valued Gaussian process $X$ (the "density process" or fluctuation field).

1. Pairing with Test Functions: For a fixed time, the process $X(1)$ (the limit at time $1$) is a $S'$-valued random variable. By pairing this with indicator test functions $u_t$,

$Y(t) = \langle X(1), u_t \rangle,$

and scaling $u_t$ according to a shrinking spatial parameter, one recovers, in the limit, real-valued Gaussian processes. Specific choices of $u_t$ select the target process:

| $u_t(x)$ | Limit Process | Covariance Structure | |----------------------------------------|-------------------|------------------------------------------------| | $1_{[0,t]}(x)$ | fBm | $\sim \tfrac{1}{2}(s^{2H} + t^{2H} - |t-s|^{2H})$ | | $1_{[0,t]}(x) - 1_{[-t,0]}(x)$ | sub-fBm (odd part)| $$\sim (1-H)(s^{2H} + t^{2H}) - (s+t)^{2H} + |t-s|^{2H}$$ | | $1_{[-t,t]}(x)$ | nsfBm | $\sim (s+t)^{2H} - |t-s|^{2H}$ |

This step yields processes possessing prescribed long-range dependence, increment correlation structure (positive or negative), and covariance scaling governed by the Hurst parameter $H$ (with full admissible range, excluding endpoints).

2. Physical Interpretation and the Particle Picture

A notable achievement is the explicit "particle picture" for sub-fBm and nsfBm:

• Representation of Increments: Sub-fBm is associated with negative correlations of increments due to antisymmetric test functions ($u_t$), realized in the spatial antisymmetry of particle occupation. Such interpretation was previously unavailable (particularly for the negative sub-fractional case) except for narrow parameter ranges.
• Robustness Across Initial Configurations: The limit holds for a broad class of initial configurations, including nearly deterministic ones, provided they produce suitable covariance structure (parameter dependence on $\mathrm{Var}(\theta_0) - \mathbb{E}[\theta_0]$). This clarifies physical origins of correlation sign changes in increments.
• Odd Part of fBm: The "odd part" process (per Dzhaparidze and van Zanten) is formally constructed for the first time with a transparent physical interpretation.

The approach reveals that the essence of long-range dependence and increment correlation in these Gaussian processes is embedded in the spatial organization and fluctuation of the underlying particle system.

3. Mathematical Structure: $S'$-Valued Process and Dual Pairing

All target processes are realized as dual pairings between an $S'$-valued Gaussian random variable (arising as the empirical fluctuation process) and elements of the Schwartz class (or indicators thereof):

$Y(t) = \langle X(1), u_t \rangle,$

where

• $X(1)$ is a centered Gaussian process in the $S'$ space,
• $u_t$ is a deterministic test function (indicator-type),
• The covariance $$\mathbb{E}[\langle X, \varphi \rangle \langle X, \psi \rangle]$$ is characterized via integrals involving the semigroup of the underlying stable process and initial configuration parameters (see equation (2.5) in the source).

This dual analytic-probabilistic construction generalizes the white-noise approach (pairing with $1_{[0,t]}$ yields classical Brownian motion) to fractional and sub-fractional classes, and is straightforwardly extended to multidimensional random fields using product test functions.

4. Generalization, Simplification, and Parameter Dependence

The dual representation encompasses and generalizes prior work in the following ways:

• Simplified Limit Arguments: The convergence of functionals $\langle X(1), u_t \rangle$ to target Gaussian processes is established directly at the level of finite-dimensional distributions, due to the universal structure of the density field and explicit scaling relationships.
• Extensive Parameter Ranges: Coverage of Hurst parameter ranges is full (excluding trivial endpoints), overcoming restrictions of earlier particle-based constructions.
• Extension to Multidimensional Fields: Processes of the form $$\langle X(1), \prod_{i=1}^d 1_{[0,t_i]}(x_i) \rangle$$ produce multidimensional random fields with covariance determined by the same density process.
• Explicit Role of Initial Configuration: The sign of the covariance parameter $(\mathrm{Var}(\theta_0) - \mathbb{E}[\theta_0])$ determines the sign of increment correlations, providing a structural explanation for the appearance of different types of Gaussian processes within the same framework.

5. Emergence of New Processes and S′-Valued Density Fields

The method naturally generates new classes of Gaussian processes and random fields:

• Novel Gaussian Processes: By varying $u_t$ or underlying particle dynamics (e.g., by adjusting branching rates or initial measure parameters), a wide range of covariance structures is attainable.
• $S'$-Valued Density Processes: The limiting empirical fields extend the classical density process constructions of Martin-Löf, yielding processes that encompass classical, sub-fractional, and negative sub-fractional Brownian motions as projections.
• Connection to Physical Random Fields: This analytical structure provides a blueprint for constructing and interpreting physically meaningful random fields in statistical mechanics and other domains where long-range correlated Gaussian processes are used for modeling.

6. Summary Table: Key Elements of Dual Gaussian-Particle Representation

Element	Probabilistic/Physical	Analytical/Distributional
Particle system (empirical/occupation field)	Microscopic trajectories	Fluctuation density process $X(1)$ in $S'$
High-density limit, normalization	Scaling control, fluctuation	Centered Gaussian field in $S'$
Pairing with test function $u_t$	Observables, process mapping	Duality: $Y(t) = \langle X(1), u_t \rangle$
Covariance functional	Increment correlations	Explicit formulas via $X(1)$ covariance, $u_t$
Process obtained	fBm, sub-fBm, nsfBm, odd part	Full range of Hurst parameter, negative/positive correlations

7. Implications and Applications

The dual Gaussian-particle framework forms a rigorous bridge between particle systems and Gaussian process theory, enabling:

• Transparent interpretation of increment correlation and long-range dependence in terms of physical randomness and occupation/empirical fluctuation.
• Unified and streamlined derivation of non-Markovian Gaussian process classes via limit theorems for empirical particle fields.
• Generalization to multidimensional settings and construction of new, physically interpretable random fields.
• Insights into the microscopic origin of process covariance features (e.g., negative increments), critical for applications in statistical physics, spatial statistics, and beyond.

The representation is both robust and adaptable, allowing analytic tractability and interpretability across an array of domains requiring long-range correlated, scaling-variant Gaussian processes (Bojdecki et al., 2011).