Pathwise uniqueness at the three-quarter Hölder threshold
Let $\sigma : \mathbb{R} \to \mathbb{R}$ satisfy $|\sigma(a) - \sigma(b)| \le C|a - b|^{3/4}$ for all $a, b$, and let $u_0$ be bounded and continuous. Consider the stochastic heat equation $\partial_t u = \frac12 \partial_x^2 u + \sigma(u)\dot W$ on $\mathbb{R}$, with $\dot W$ space–time white noise, in its Itô–Walsh mild form $$u(t, x) = (p_t * u_0)(x) + \int_0^t \int_{\mathbb{R}} p_{t-s}(x - y)\, \sigma(u(s, y))\, W(ds, dy), \qquad p_t(x) = (2\pi t)^{-1/2} e^{-x^2/(2t)}.$$
Mytnik and Perkins proved pathwise uniqueness, among solutions with continuous paths in the continuous functions of subexponential growth, whenever $\sigma$ is Hölder continuous of index $\gamma > 3/4$. Mueller, Mytnik and Perkins proved that for $0 < \gamma < 3/4$ the equation with $\sigma(u) = |u|^\gamma$ and $u_0 = 0$ has a solution that is not identically zero, so that neither pathwise uniqueness nor uniqueness in law holds; by analogy with the Yamada–Watanabe threshold $1/2$ for stochastic differential equations, they guess that uniqueness holds at $\gamma = 3/4$. On the circle and for coefficients bounded away from zero, Han proved uniqueness in law for every Hölder index above $2/3$ (2026, preprint).
Problem. Must any two adapted continuous mild solutions with the same $u_0$ and the same noise be indistinguishable, within the class of solutions satisfying $$\sup_{0 \le t \le T,\ x \in \mathbb{R}} e^{-a|x|} |u(t, x)| < \infty \quad \text{almost surely, for every } a, T > 0?$$ The coefficient may vanish, and solutions may take either sign.
Source: Mueller, Mytnik and Perkins (2014), discussion after Theorem 1.1, on the endpoint γ = 3/4 of Mytnik and Perkins (2011). Stated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, pathwise uniqueness, space-time white noise, stochastic heat equations
References
@article{mueller2014,
author = {Mueller, Carl and Mytnik, Leonid and Perkins, Edwin},
title = {Nonuniqueness for a parabolic {SPDE} with $\frac{3}{4}-\varepsilon$-{H}{\"o}lder diffusion coefficients},
journal = {Ann. Probab.},
volume = {42},
number = {5},
year = {2014},
pages = {2032--2112},
doi = {10.1214/13-AOP870},
eprint = {1201.2767},
archivePrefix = {arXiv},
}
@article{mytnik2011,
author = {Mytnik, Leonid and Perkins, Edwin},
title = {Pathwise uniqueness for stochastic heat equations with {H}{\"o}lder continuous coefficients: the white noise case},
journal = {Probab. Theory Related Fields},
volume = {149},
number = {1-2},
year = {2011},
pages = {1--96},
doi = {10.1007/s00440-009-0241-7},
eprint = {0809.0248},
archivePrefix = {arXiv},
}
@misc{han2026,
author = {Han, Yi},
title = {Stochastic heat equation with nondegenerate {H}{\"o}lder diffusion coefficient: uniqueness below the three-fourth threshold},
year = {2026},
eprint = {2608.01279},
archivePrefix = {arXiv},
}
@misc{aim-stochastic-heat-critical-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Pathwise uniqueness at the three-quarter Hölder threshold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/190-stochastic-heat-critical-uniqueness.md},
note = {Entry 190 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00476, "Pathwise uniqueness at the three-quarter Hölder threshold", clopen.solutions.