ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A Brownian bridge can cross a barrier missed by its endpoints

Example

Assume AC. Let a scalar Brownian bridge run from $1$ to $1$ over duration $1$ , with barrier $0$ . Both endpoints are above the barrier, but the bridge crosses zero with probability $e^{- 2}$ .

Facts & Assumptions

Given: AC and the scalar Brownian bridge crossing formula for endpoints above a fixed barrier. Scalar Brownian bridge barrier-crossing probability

[A1]

AC is inherited from the bridge conditional-law interface; the endpoint substitution uses no further choice. The Axiom of Choice

Proof

1.1given

Apply the crossing theorem with $x = y = 1$ , $c = 0$ , and $h = 1$ . Its hypotheses $x, y \geq c$ and $h > 0$ hold, and it gives $P {\inf_{0 \leq s \leq 1} B_{s} \leq 0} = \exp (- 2 (1 - 0) (1 - 0) / 1) = e^{- 2}$ .

2.1A1givenstep 1.1algebra∎

Since $e^{- 2} > 0$ , the endpoint-only check “both endpoints exceed zero” does not certify that the path stayed above zero. If either endpoint were exactly zero, the crossing event would instead have probability one by the boundary case of the same theorem.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Proposition 2.7.2 (standard reference, not scraped)