Pathwise uniqueness for stochastic heat equations with coloured noise
Consider the stochastic heat equation
driven by coloured noise, with noise parameter $\alpha$ and coefficient Hölder exponent $\gamma$. **Coloured-noise pathwise uniqueness conjecture.** If\alpha<2(2\gamma-1),
. This strengthens the preceding pathwise-uniqueness result in the white-noise case, where the argument gives the condition ; the conjectured coloured-noise range is not established here.
References
Primary source
Leonid Mytnik and Edwin Perkins, “Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case”, arXiv:0809.0248 (2008).
Progress summary
A 2012 preprint claims to prove the conjectured uniqueness range for coloured-noise heat equations, but the scan found no independent verification.
The conjecture, formulated in 2008, asserts pathwise uniqueness under for the coloured-noise stochastic heat equation. It was presented as unresolved, but a later preprint claims to confirm it.
Known results
- Space-time white noise: pathwise uniqueness for Hölder coefficients with (Mytnik and Perkins, 2008).
- Earlier coloured-noise work established weaker sufficient conditions (Mytnik, Perkins, and Sturm, 2006).
December 18, 2012 claimed proof
Thomas Rippl and Anja Sturm’s preprint, New results on pathwise uniqueness for the heat equation with colored noise, states that its main result confirms the earlier conjecture, covering the claimed coloured-noise range. This is an arXiv claim and remains unverified by the retrieved sources.
Current status (as of September 2026): The white-noise result for is established, while the coloured-noise conjecture has a 2012 claimed proof but no independent verification recorded here.
Sources
- arxiv.org
- dml.mathdoc.fr
- arxiv.org
- semanticscholar.org
- ideas.repec.org
- ui.adsabs.harvard.edu
- hal.science
- digitalscholarship.unlv.edu
- mathoverflow.net
- researchgate.net
- quantamagazine.org
- arxiv.org
- ar5iv.labs.arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- ideas.repec.org
- aimsciences.org
- researchgate.net
- mathoverflow.net
- quantamagazine.org
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