Small mass limit of expected signature for physical Brownian motion
Bibliographic record
Abstract
Physical Brownian motion describes the dynamics of a Brownian particle experiencing frictional force. It was investigated in the classical work [L. S. Ornstein and G. E. Uhlenbeck, Phys. Rev. 36 (1930)] as a physically meaningful approach to realising the standard ``mathematical'' Brownian motion, via sending the mass $m \to 0^+$ and performing natural scaling. The analysis was extended to a Brownian particle in an external magnetic field in [P. Friz, P. Gassiat, and T. Lyons, Trans. Amer. Math. Soc. 367 (2015)], discovering the new phenomenon that the area process associated to the physical process converges -- but not to Lévy's stochastic area. In this paper, we carry out the singular limit analysis of a generalised stochastic differential equation (SDE) model encompassing the physical Brownian motion as a special case. We show that the expected signature of the solution $\{P_t\}_{t \geq 0}$ for the generalised SDE converges to a nontrivial tensor as $m \to 0^+$, at each degree in the tensor algebra and on each time interval $[0,T]$, through a delicate convergence analysis based on the graded PDE system for the expected signature of Itô diffusion processes. Moreover, explicit solutions exhibiting intriguing combinatorial patterns are obtained when the coefficient matrix $\mathscr{M}$ in our SDE is diagonalisable. In the case of physical Brownian motion, $\{P_t\}_{t \geq 0}$ corresponds to the momentum of the particle (viewed as a rough path), and $\mathscr{M}$ is the stress tensor. Our work appears among the very first endeavours to study the singular limit of expected signature of diffusion processes, especially for nonzero initial datum $p=P_0$.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".