Koopman & Perron-Frobenius Operators

• Koopman and Perron-Frobenius operators are infinite-dimensional linear operators that linearize nonlinear dynamics to enable rigorous spectral and modal analysis.
• They are approximated numerically using methods like Ulam’s method, EDMD, and tensor decompositions, facilitating reliable prediction and control in complex systems.
• Their applications span optimal control, Bayesian filtering, and reduced-order modeling, proving essential in modern dynamical systems research.

Koopman and Perron-Frobenius operators are two intimately linked, infinite-dimensional linear operators central to the spectral analysis, statistical mechanics, and ergodic properties of dynamical systems. The Perron-Frobenius operator (sometimes called the transfer operator) propagates densities under the dynamics, while the Koopman operator advances observables via composition with the system’s flow or map. These operators linearize the nonlinear flow of deterministic, stochastic, or even parameter-dependent dynamical systems, providing a rigorous framework for modal decomposition, control, filtering, and reduced-order modeling in high- and infinite-dimensional state spaces. Advances in their theoretical and computational treatment have expanded from the classical theory of matrices and functional analysis to modern operator-theoretic dynamics, numerical approximation, and data-driven control.

1. Operator Definitions and Functional Frameworks

Let $(X, \mathcal{B}, \mu)$ be a measurable space, and $\varphi: X \to X$ a measurable transformation (deterministic or stochastic).

• Perron-Frobenius Operator (PF):

Propagates densities (or measures). For $f \in L^1(\mu)$,

$$[\mathcal{P}f](y) = \sum_{x \in \varphi^{-1}(y)} \frac{f(x)}{|\det D\varphi(x)|}$$

in the smooth, invertible case. More generally, for a Markov operator $P$, $[\mathcal{P}f](x) = \int p(x, y) f(y) \, d\mu(y)$ where $p(x, y)$ is a Markov (transition) kernel.

• Koopman Operator (K):

Advances observables by composition. For $g \in L^\infty(\mu)$,

$[\mathcal{U}g](x) = g(\varphi(x))$

The major dichotomy is that $\mathcal{U}$ acts on observables, and $\mathcal{P}$ on densities/measures, linked via duality: for $g \in L^\infty$, $f \in L^1$,

$$\int g \cdot \mathcal{P}f\, d\mu = \int \mathcal{U}g \cdot f\, d\mu$$

In stochastic systems, $\mathcal{U}$ is defined as $$[\mathcal{U}g](x) = \mathbb{E}[g(\varphi(x, \omega))]$$.

These operators may be defined and analyzed on Hilbert spaces (e.g., $L^2$), Banach lattices, or more generally, reproducing kernel Banach spaces (RKBS), each setting imparting specific properties regarding boundedness, spectrum, and adjointness (Ikeda et al., 2022). For composition or transfer operators in symbolic dynamics and smooth ergodic systems, the choice of “test space” is essential, leading to the use of Gelfand triplets for generalized spectra (Chiba et al., 2021).

2. Spectral Properties and Generalized Perron-Frobenius Theory

Classically, the Perron-Frobenius theory concerns positive irreducible matrices: the spectral radius is a simple eigenvalue, the associated eigenvector is strictly positive, and, in the finite-dimensional setting, any other eigenvalue lies strictly inside the spectral circle. Extensions to infinite-dimensional Banach lattices and positive (or eventually positive) operators are significant for Koopman/PF theory:

• Irreducibility and Positivity: For a positive, ideal (or band) irreducible operator $T$ with $r(T)$ a pole of the resolvent, the dominant eigenvalue $r(T)$ is simple and has a strictly positive eigenvector. Analogous uniqueness and simplicity results hold for the adjoint operator. Comparison theorems guarantee that if $0 \le S \le T$ and $r(S) = r(T)$, then $S = T$ under certain irreducibility and compactness or “pole of the resolvent” conditions (Gao, 2012).
• Peripheral Spectrum Structure: The spectrum at modulus $r(T)$ is cyclic under irreducibility. When $T$ or some power is compact, each point of the peripheral spectrum is a simple pole, and the action on the corresponding spectral subspace is permutation-like. The spectrum of the Koopman operator (in deterministic discrete time) possesses a lattice (multiplicative) structure: if $\lambda, \eta$ are in the spectrum, so is $\lambda\eta$ (Bramburger, 31 Jul 2025). This lattice property fails for stochastic systems.
• Extensions to Eventually Positive Operators: If $T$ is not positive but some power $T^n$ becomes positive (“eventual positivity”), one can still ensure $r(T)$ is in the spectrum, and under suitable resolvent pole or compactness, there exists a positive eigenvector and cyclic peripheral spectrum (Glück, 2016).
• Complex and Rotational Positivity: Complexification introduces “rotational strong positivity,” for which analogous principal eigenvalue results hold with respect to complex cones. The semigroup generated by such an operator preserves the cone subject to a phase rotation (1803.02060).
• Generalized Spectrum for Symbolic Dynamics: For shift systems, the PF operator’s standard spectrum may be continuous on $|z|=1$. Using a Gelfand triplet, an analytic continuation of the resolvent identifies generalized (discrete) spectrum with precise asymptotic mixing rates (Chiba et al., 2021).

3. Numerical Approximation: Ulam's Method, EDMD, and Tensor Approaches

Practical use of Koopman and Perron-Frobenius operators for prediction, model reduction, and control hinges on finite-dimensional approximation.

• Ulam’s Method: Partition the state-space into boxes; represent the PF operator as a stochastic matrix via probability of transition between boxes. The leading eigenvector approximates the invariant density. For a partition $\{\mathbb{B}_i\}$:

$$p_{ij} = \frac{\mu(\varphi^{-1}(\mathbb{B}_j) \cap \mathbb{B}_i)}{\mu(\mathbb{B}_i)}$$

Monte Carlo estimation is commonly used (Klus et al., 2015).

• Extended Dynamic Mode Decomposition (EDMD): Approximates the Koopman operator by projecting into a function dictionary (e.g., polynomials, radial basis functions). For data $(x_l, y_l)$ where $y_l = \varphi(x_l)$, the matrix is obtained by:

$$K^\top = \Psi_Y \Psi_X^+,\quad A = \frac{1}{m} \sum_l \Psi(y_l)\Psi(x_l)^\top,\quad G = \frac{1}{m} \sum_l \Psi(x_l)\Psi(x_l)^\top$$

EDMD recovers Ulam’s method if indicator functions are chosen as basis (Klus et al., 2015).

• Tensor-Based Approximations: For high-dimensional systems, Ulam/EDMD matrices become exponentially large. Representing the operator as a low-rank tensor—using, e.g., tensor-train (TT) decomposition—enables scalable computation and storage (Klus et al., 2015). Tensorized eigenproblems:

$\mathcal{P} \mathcal{V} = \lambda\mathcal{V}$

with low-rank tensor $\mathcal{V}$, permit efficient extraction of leading eigenfunctions, especially for weakly coupled or separable dynamics.

4. Applications: Control, Sensitivity, Filtering, and Bayesian Inference

• Optimal Control via Operator Methods: The PF operator “lifts” nonlinear dynamics into a linear operator framework, facilitating the use of convex optimization for controller synthesis. Using Lyapunov measures and duality with the Koopman operator, stabilizing controls are constructed via linear programs in measure space, with positivity and the Markov property guaranteeing physical feasibility (Das et al., 2018, Huang et al., 2020).
• Filtering and Data Assimilation: The PF operator provides an infinite-dimensional linear framework for Bayesian filtering, where state densities are propagated via the PF operator and updated by a likelihood operator. Finite-dimensional approximations using Ulam’s method permit sequential updates, yielding improved convergence over particle filtering, especially in high dimensions and for non-Gaussian measures (Liu et al., 2023).
• Differentiability and Linear Response: For SDEs with parametric drift, the dependence of PF and Koopman operators on the drift is Fréchet differentiable in strong operator topology. Explicit formulae are given via Girsanov’s theorem, enabling sensitivity analysis and optimization with respect to model perturbations (Koltai et al., 2018).
• Spectral Decomposition for Mixing Rates: For expanding interval maps (e.g., beta-maps), non-leading eigenvalues of the PF operator (on BV) are generically present, Hölder continuous but nowhere differentiable in system parameters, and their eigenfunctionals are not representable by measures. Such spectral properties govern mixing rates and decay of correlations (Suzuki, 1 Oct 2024).

5. Extensions: Functional Spaces, Sparsity, and Operator Algebra

• Reproducing Kernel Banach Spaces: Koopman and PF operators generalize naturally to RKBSs, permitting extended symmetry (commutation and intertwining) results, closedness, boundedness, and explicit formulas for generators, both in discrete and continuous time (Ikeda et al., 2022).
• Sparse and Factorizable Dynamics: For sparsely coupled or decomposable systems, the operator, its spectrum, and invariant measures can be built from those of subsystems via intertwining relations. This enables computational reduction in eigenfunction calculation (e.g., via EDMD), as principal subsystem eigenfunctions induce global eigenfunctions on the composite system (Schlosser et al., 2021).
• Koopman Operator Spectrum as a Lattice: The Koopman operator for deterministic discrete-time dynamics over $L^2$ has a spectrum closed under multiplication (lattice property); this enables combination of spectral modes and demonstrates an intrinsic algebraic structure. For stochastic Koopman operators (arising from Markov processes), this property fails—distinguishing deterministic and random dynamics fundamentally (Bramburger, 31 Jul 2025).

6. Neural and Data-Driven Approaches

• Neural Network Methods: Physics-informed neural networks (PINNs) and their variational extensions (RVPINNs) are deployed to solve fixed-point and power series problems associated with non-expansive PF operators under $L^p$ norms. These methods leverage flexibility of neural parameterizations, can resolve singular solutions with higher accuracy than fixed-grid Ulam’s method, and are accompanied by rigorous stability-dependent a priori error estimates (Udomworarat et al., 8 May 2025). For operator equations $u - \alpha P u = f_0$, the solution $u$ is approximated efficiently via minimization of network residuals, and a convergence theory relates network expressivity and optimizer tolerance to error constants dependent on the contraction parameter $\alpha$.

7. Interplay with Conservation Laws and Geometric Structures

• Conservation Laws and Exterior Calculus: The Reynolds transport theorem and Liouville equation, when reinterpreted with probability densities, yield a unifying framework for the evolution of densities and observables—embedding PF and Koopman operators into the structure of geometric and physical conservation. This formalism, extended by exterior calculus, enables the description of statistical and geometric dynamics in time, space, phase space, and parameter space (Niven et al., 2018).
• Modal Decomposition and Reduced Order Modeling: The linearity of both operators allows for decomposition of dynamics into modes, providing a spectral framework for the prediction, control, and data-driven modeling of complex, high-dimensional systems. Modal structures—coherent sets, metastable states, and principal eigenfunctions—define invariant or slowly decaying structures fundamental to low-order descriptions.

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Koopman and Perron-Frobenius operator theory thus provides a unifying, rigorous, and flexible platform for both theoretical analysis and algorithmic implementation across deterministic, stochastic, and data-driven settings. Their spectral properties, finite-dimensional approximations, and functional flexibility underpin much of the modern approach to dynamical systems, control, filtering, and statistical prediction, with new developments in non-Hilbertian geometries, tensor numerics, neural parameterizations, and operator-theoretic learning continuing to enlarge the domain of relevance.