[ChrisRackauckas-Claude]

Summary

The scalar, vector, and inplace OU-Bridge testsets in test/bridge_test.jl compared three independent Monte Carlo estimates to each other at atol = 1e-3 over a 500k-sample loop. The Random.seed!(1234) calls between the samplers were dead code: solve reseeds each NoiseProcess from system entropy (reseed = true is the default), so every realization is an independent draw.

A difference of two 500k-sample estimates carries ~4.3 standard errors of headroom at atol = 1e-3, so the test flaked on a few percent of CI runs on every Julia version. The Inplace OU-Bridge failure on Julia 1.13 CI was this tail event, not a 1.13 regression — the same failure mode has appeared on other versions.

The fix

Each sampler is now compared directly against the exact OU marginal moments — mean μ(1 − e^{−Θt}), variance σ²(1 − e^{−2Θt})/(2Θ) — at the same atol = 1e-3. That gives ~6 standard errors of headroom and is a strictly stronger test: a bias shared by all three samplers can no longer hide in a pairwise comparison.

This also matches the file's existing convention — the Brownian Bridge testset already compares timestep_mean to analytical moments.

The bridge estimator is compared against the unconditional OU moments, which is correct: an OU bridge pinned to an endpoint drawn from the forward process's own distribution reproduces the forward law when marginalizing over the endpoint.

Discriminating-test evidence

This is a flakiness fix, so "fails before / passes after" doesn't apply per-run — instead, evidence that the old formulation was noise-limited and the new one is not:

1. Before (master): the three samplers were compared pairwise; instrumented on Julia 1.13, all three track the analytical moments within ~4×10⁻⁴ (~2.5 SE) — no bias — yet pairwise differences of two estimates reach ~1.5×10⁻³ in tail realizations, exceeding atol = 1e-3. Reproduced failure: Inplace OU-Bridge in https://github.com/SciML/DiffEqNoiseProcess.jl CI (master, Julia 1.13).
2. After: against fixed analytical moments each estimate has ~6 SE of headroom at the same tolerance.

Verification

• test/bridge_test.jl on Julia 1.13.0: all testsets pass — Brownian Bridge 80/80, Scalar/Vector/Inplace OU-Bridge 6/6 each.
• Same file on Julia 1.12.7: all pass.
• GROUP=QA equivalent (test/qa/qa.jl) on 1.13: 21/21 pass.
• Runic format check: clean. typos: clean.

Docs build skipped — test-only change, no docstrings or public API touched.

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