Generalized Invariance Principles for Stochastic Dynamical Systems and Their Applications
Bibliographic record
Abstract
Investigating long-term behaviors of stochastic dynamical systems often requires to establish criteria that are able to describe delicate dynamics of the considered systems. In this article, we develop generalized invariance principles for continuous-time stochastic dynamical systems. Particularly, in a sense of probability one and by the developed semimartingale convergence theorem, we not only establish a local invariance principle, but also provide a generalized global invariance principle that allows the sign of the diffusion operator to be positive in some bounded region. We further provide an estimation for the time when a trajectory, initiating outside a particular bounded set, eventually enters it. Finally, we use several representative examples, including stochastic oscillating dynamics, to illustrate the practical usefulness of our analytical criteria in deciphering the stabilization or/and the synchronization dynamics of stochastic systems.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".