We study a finite-sample change-detection problem for one-dimensional noisy dynamical systems using partition-based empirical approximations of stationary behaviour. Given observations from an interval-valued process, we partition the state space into finitely many intervals and estimate a transition matrix from observed transitions between partition elements. After a small Doeblin-type regularisation, the resulting matrix has a unique stationary distribution. This stationary distribution is used as a finite-dimensional approximation of the invariant density, or stationary law, of the observed regime. Using an initial reference segment, we compute a baseline empirical stationary distribution bπ0,ρ. For each subsequent sliding window, we compute a window-based empirical stationary distribution bπt,ρ and define the score St = bπt,ρ bπ0,ρ 1. Large values of St indicate that the stationary behaviour of the observed regime has changed relative to the baseline. The statistic is therefore a detector of changes in stationary behaviour. It is not, by itself, a detector of all possible changes in transition dynamics that preserve the invariant density.

We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems.

In this section, we outline the'relaxation' or'decay' of the spectral components of Each trajectory is simulated for 100000 steps. A separate testing set was generated in an identical manner but with a different random seed. The original data was obtained upon request from DE Shaw Research, and details about the simulations are available in the original publication [19]. Figures 8, 9 and 10, show conditional distributions generated by CG-SE3-ITO models and comparisons of MD with ITO simulations on the fast folders Trp-Cage, BBA, and Villin, respectively. Reference value and observables We compute observables using Markov state models.

Molecular dynamics (MD) is a broadly adopted technique to approximate such quantities.

Deep neural networks (DNNs) achieve remarkable performance on a wide range of tasks, yet their mathematical analysis remains fragmented: stability and generalization are typically studied in disparate frameworks and on a case-by-case basis. Architecturally, DNNs rely on the recursive application of parametrized functions, a mechanism that can be unstable and difficult to train, making stability a primary concern. Even when training succeeds, there are few rigorous results on how well such models generalize beyond the observed data, especially in the generative setting. In this work, we leverage the theory of stochastic Iterated Function Systems (IFS) and show that two important deep architectures can be viewed as, or canonically associated with, place-dependent IFS. This connection allows us to import results from random dynamical systems to (i) establish the existence and uniqueness of invariant measures under suitable contractivity assumptions, and (ii) derive a Wasserstein generalization bound for generative modeling. The bound naturally leads to a new training objective that directly controls the collage-type approximation error between the data distribution and its image under the learned transfer operator. We illustrate the theory on a controlled 2D example and empirically evaluate the proposed objective on standard image datasets (MNIST, CelebA, CIFAR-10).

We develop a general theory of semantic dynamics for large language models by formalizing them as Continuous State Machines (CSMs): smooth dynamical systems whose latent manifolds evolve under probabilistic transition operators. The associated transfer operator $P: L^2(M,μ) \to L^2(M,μ)$ encodes the propagation of semantic mass. Under mild regularity assumptions (compactness, ergodicity, bounded Jacobian), $P$ is compact with discrete spectrum. Within this setting, we prove the Semantic Characterization Theorem (SCT): the leading eigenfunctions of $P$ induce finitely many spectral basins of invariant meaning, each definable in an o-minimal structure over $\mathbb{R}$. Thus spectral lumpability and logical tameness coincide. This explains how discrete symbolic semantics can emerge from continuous computation: the continuous activation manifold collapses into a finite, logically interpretable ontology. We further extend the SCT to stochastic and adiabatic (time-inhomogeneous) settings, showing that slowly drifting kernels preserve compactness, spectral coherence, and basin structure.

A general idea to overcome the above problem is to perturb the system dynamics.

In this section, we outline the'relaxation' or'decay' of the spectral components of Each trajectory is simulated for 100000 steps. A separate testing set was generated in an identical manner but with a different random seed. The original data was obtained upon request from DE Shaw Research, and details about the simulations are available in the original publication [19]. Figures 8, 9 and 10, show conditional distributions generated by CG-SE3-ITO models and comparisons of MD with ITO simulations on the fast folders Trp-Cage, BBA, and Villin, respectively. Reference value and observables We compute observables using Markov state models.

Molecular dynamics (MD) is a broadly adopted technique to approximate such quantities.