Uniqueness for stationary fractional stochastic Burgers below the critical threshold
Let $\mathbb{T} = \mathbb{R}/(2\pi\mathbb{Z})$, let $A = -\partial_x^2$ act on mean-zero functions, and fix $1/2 < \theta < 3/4$. Consider the formal equation $$du_t = -A^\theta u_t\, dt + \partial_x(u_t^2)\, dt + \sqrt{2}\, A^{\theta/2}\, dW_t,$$ where $W$ is a cylindrical Brownian motion on $L^2_0(\mathbb{T})$; its stationary marginal is mean-zero spatial white noise, with covariance the identity on $L^2_0$. Fix $T > 0$ and let $\Pi_N$ keep the Fourier modes $0 < |k| \le N$. A stationary controlled solution is a process $u$ with continuous paths in $\mathcal{S}'(\mathbb{T})$ such that $u_t$ is white noise for every $t \in [0, T]$; for each smooth mean-zero test function $f$ the limit $$B_t(f) = \lim_{N \to \infty} \int_0^t \langle \partial_x((\Pi_N u_s)^2), f \rangle\, ds$$ exists in probability in $C([0,T])$, $B$ has a continuous distribution-valued version with $B_0 = 0$, and each $B(f)$ has zero quadratic variation; $$M_t(f) = u_t(f) - u_0(f) + \int_0^t u_s(A^\theta f)\, ds - B_t(f)$$ is a continuous martingale in the natural filtration of $u$, with quadratic variation $2t \|A^{\theta/2} f\|_2^2$; and the same martingale condition holds for $\hat u_t = u_{T-t}$ and $\hat B_t = -(B_T - B_{T-t})$ in the natural backward filtration. Gubinelli and Jara constructed controlled solutions, in their formulation, for every $\theta > 1/2$ and proved pathwise uniqueness for $\theta > 5/4$; Gubinelli and Perkowski sketched how their method gives uniqueness in law for $\theta > 3/4$.
Problem. For every $T > 0$, do all stationary controlled solutions have the same law on $C([0,T], \mathcal{S}'(\mathbb{T}))$?
Source: Gubinelli and Jara (2013), who introduce controlled solutions, construct them for θ > 1/2 and prove pathwise uniqueness for θ > 5/4; Gubinelli and Perkowski (2020, Section 5.2) sketch uniqueness in law for θ > 3/4. Stated for 1/2 < θ < 3/4 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, energy solutions, singular SPDEs, stochastic Burgers equations
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{gubinelli2013,
author = {Gubinelli, Massimiliano and Jara, Milton},
title = {Regularization by noise and stochastic {B}urgers equations},
journal = {Stoch. Partial Differ. Equ. Anal. Comput.},
volume = {1},
number = {2},
year = {2013},
pages = {325--350},
doi = {10.1007/s40072-013-0011-5},
eprint = {1208.6551},
archivePrefix = {arXiv},
}
@article{gubinelli2020,
author = {Gubinelli, Massimiliano and Perkowski, Nicolas},
title = {The infinitesimal generator of the stochastic {B}urgers equation},
journal = {Probab. Theory Related Fields},
volume = {178},
number = {3-4},
year = {2020},
pages = {1067--1124},
doi = {10.1007/s00440-020-00996-5},
eprint = {1810.12014},
archivePrefix = {arXiv},
}
@misc{aim-uniqueness-for-stationary-fractional-stochastic-burgers-below-the-critical-threshold,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness for stationary fractional stochastic Burgers below the critical threshold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/999-uniqueness-for-stationary-fractional-stochastic-burgers-below-the-critical-threshold.md},
note = {Entry 999 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01165, "Uniqueness for stationary fractional stochastic Burgers below the critical threshold", clopen.solutions.