Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation

1. Problem Statement

The paper addresses the challenge of computing Koopman eigenfunctions for Stochastic Differential Equations (SDEs). While Koopman operator theory provides a powerful linear framework for analyzing nonlinear dynamical systems, existing kernel-based methods (specifically those developed for deterministic systems in a companion paper [1]) rely on first-order hyperbolic transport equations and the method of characteristics.

These deterministic methods fail for stochastic systems because:

1. The Koopman generator for SDEs includes a second-order diffusion term, transforming the governing equation from a first-order hyperbolic PDE to a second-order elliptic-parabolic PDE.
2. The method of characteristics is no longer applicable.
3. Standard numerical approaches often suffer from poor conditioning and lack rigorous error bounds separating approximation errors from sampling errors.

The goal is to extend the unified kernel framework (Variational, Green's function, and Resolvent) to stochastic settings, prove their equivalence, and develop robust computational algorithms.

2. Methodology

The authors propose a unified framework based on Reproducing Kernel Hilbert Spaces (RKHS) and the Feynman–Kac formula.

A. Mathematical Formulation

For an SDE $dX_t = G(X_t)dt + \sigma(X_t)dW_t$, the Koopman eigenfunction equation $K\phi = \lambda\phi$ becomes:
$$G(x) \cdot \nabla \phi(x) + \frac{1}{2}\text{Tr}[a(x)\nabla^2 \phi(x)] = \lambda \phi(x)$$
where $a(x) = \sigma(x)\sigma(x)^\top$. The authors decompose the eigenfunction $\phi(x) = w^\top x + h(x)$, where $w$ is a left eigenvector of the linearized drift $A = \nabla G(0)$, and $h(x)$ is a nonlinear correction satisfying a linear second-order PDE with a source term.

B. Three Equivalent Kernel Constructions

The paper extends three deterministic constructions to the stochastic case:

1. Lions-Type Variational Principle: Defines a Hilbert space with a graph norm involving the stochastic operator $L_{stoch} = G \cdot \nabla + \frac{1}{2}\text{Tr}[a\nabla^2] - \lambda$. The kernel is the Riesz representer of point evaluation in this space.
2. Green's Function via Feynman–Kac: Replaces the deterministic Green's function with a probabilistic representation. The kernel is the time-integrated transition density (discounted by $e^{-\lambda t}$), derived directly from the Feynman–Kac formula.
3. Stochastic Resolvent Kernel: Defined as the Laplace transform of the stochastic flow, expressed as an expectation over stochastic trajectories: $k_{res}(x,y) = \mathbb{E}_x[\int_0^\infty e^{-\lambda t} \delta(y - X_t) dt]$.

C. Computational Framework

• Collocation Method: The PDE is solved by approximating $h(x)$ as a linear combination of kernel functions at collocation points. This requires explicit formulas for the Drift Matrix (first derivatives) and the Diffusion Matrix (second derivatives/Hessian trace).
• Monte Carlo Integration: For cases where closed-form derivatives are unavailable, the authors use Monte Carlo sampling of the Feynman–Kac formula to estimate the solution at collocation points, followed by kernel ridge regression.
• Hypoelliptic Extension: The framework is generalized to degenerate diffusions (where the diffusion matrix is singular) satisfying Hörmander's condition (e.g., underdamped Langevin dynamics), utilizing subelliptic regularity theory.

3. Key Contributions

1. Unification of Stochastic Kernels: The authors prove that under uniform ellipticity and mild regularity assumptions, the three stochastic kernel constructions (Variational, Feynman–Kac, and Resolvent) yield identical reproducing kernels. This unifies the probabilistic and variational perspectives.
2. Rigorous Error Analysis: They derive error bounds that explicitly separate the RKHS approximation error (dependent on kernel smoothness and collocation density) from the Monte Carlo sampling error (dependent on the number of paths).
3. Conditioning Analysis: A critical finding is that moderate diffusion acts as an elliptic regularizer. The diffusion term introduces a negative-definite diagonal shift in the collocation matrix, significantly improving the condition number compared to the deterministic (purely advective) case, thereby enhancing numerical stability.
4. Algorithmic Implementation: They provide a complete algorithm for computing stochastic Koopman eigenfunctions, including explicit formulas for Gaussian RBF kernel derivatives and a combined Collocation-Monte Carlo approach.
5. Hypoelliptic Generalization: The theory is extended to systems with degenerate noise (Hörmander's condition), addressing the reduced regularity ($H^\varepsilon$ vs $H^1$) and providing guidelines for mollified point evaluations.

4. Results

Numerical experiments were conducted on:

• Ornstein–Uhlenbeck Process: The method recovered the exact linear eigenfunction with machine precision ($<10^{-14}$).
• Nonlinear SDEs (Quadratic Nonlinearity): The method successfully captured nonlinear corrections. Crucially, increasing the diffusion coefficient ($\sigma$) from 0 to 0.5 reduced the condition number of the system matrix by a factor of ~3.7 (from $3.79 \times 10^6$ to $1.03 \times 10^6$) while maintaining low PDE residuals and semigroup errors (<10%).
• 2D Linear Systems: Achieved machine-precision accuracy.
• Semigroup Verification: In all tests, the computed eigenfunctions satisfied the semigroup property $\mathbb{E}[\phi(X_t)] = e^{\lambda t}\phi(x_0)$ with relative errors below 10%.

5. Significance

This work bridges a critical gap between stochastic analysis and data-driven dynamical systems.

• Theoretical Impact: It establishes that the powerful unification of kernel methods for deterministic transport equations holds for stochastic systems, provided the Feynman–Kac formula replaces the method of characteristics.
• Practical Impact: The discovery that diffusion improves numerical conditioning is counter-intuitive but vital. It suggests that adding controlled noise (or leveraging inherent physical noise) can stabilize the computation of Koopman eigenfunctions, making them more robust for applications in molecular dynamics, climate modeling, and financial mathematics where stochasticity is inherent.
• Broader Applications: The framework connects naturally to Generator EDMD, Diffusion Maps, and Kernel Analog Forecasting, offering a rigorous path to compute eigenfunctions for complex, high-dimensional, and noisy systems.

In summary, the paper provides a mathematically rigorous, computationally efficient, and numerically stable framework for analyzing stochastic dynamical systems via Koopman eigenfunctions, leveraging the regularizing power of diffusion to overcome traditional numerical instabilities.