Pith. Sign in

RECORD

Pathwise uniqueness for a degenerate stochastic differential equation

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Richard F. Bass, Krzysztof Burdzy, Zhen-Qing Chen

Probability · Submitted 20 Jan 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv IDmath/0601505 (version 2)
Pith ID7VLK6XGF, Pith's permanent record of this paper and its review
CitationKrzysztof Burdzy, Richard F. Bass, Zhen-Qing Chen. "Pathwise uniqueness for a degenerate stochastic differential equation." arXiv:math/0601505 (2006). https://pith.science/paper/7VLK6XGF
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,596 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.83 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.35 · compares with prior results 0.19 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation \[dX_t=|X_t|^{\alpha} dW_t,\] where $W_t$ is a one-dimensional Brownian motion and $\alpha\in(0,1/2)$. Weak uniqueness does not hold for the solution to this equation. If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists. We also consider a class of stochastic differential equations with reflection.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation \[dX_t=|X_t|^{\alpha} dW_t,\] where $W_t$ is a one-dimensional Brownian motion and $\alpha\in(0,1/2)$. contribution 0.99 0.95 0.29 asserted 1.00
s1 Weak uniqueness does not hold for the solution to this equation. result 0.75 0.57 0.13 asserted 1.00
s2 If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists. result 0.98 0.85 0.49 asserted 1.00
s3 We also consider a class of stochastic differential equations with reflection. contribution 0.99 0.84 0.03 asserted 0.72

What kind of paper the classifier reports this is: method or tool 0.66 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation \[dX_t=|X_t|^{\alpha} dW_t,\] where $W_t$ is a one-dimensional Brownian motion and $\alpha\in(0,1/2)$. Weak uniqueness does not hold for the solution to this equation. If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists. We also consider a class of stochastic differential equations with reflection.

Human review

How Pith checks papers → · Full audit of this paper →

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.