Metadata
Abstract
We study the inviscid Burgers equation on the circle $\mathbb{T}:= \mathbb{R}/\mathbb{Z}$ forced by the spatial derivative of a Poisson point process on $ \mathbb{R}\times \mathbb{T}$. We construct global solutions with mean $\theta $ simultaneously for all $\theta \in \mathbb{R}$, and in addition construct their associated global shocks (which are unique except on a countable set of $\theta $). We then show that as $\theta $ changes, the solution only changes through the movement of the global shock, and give precise formulas for this movement. This can be seen as an analogue of previous results by the author and Yu Gu in the viscous case with white-in-time forcing, which related the derivative of the solution in $\theta $ to the density of a particle diffusing in the Burgers flow.
References
[AD95] Hammersley’s interacting particle process and longest increasing subsequences, Probab. Theory Relat. Fields, Volume 103 (1995) no. 2, pp. 199-213 | DOI | MR | Zbl
[Bak13] The Burgers equation with Poisson random forcing, Ann. Probab., Volume 41 (2013) no. 4, pp. 2961-2989 | DOI | MR | Zbl
[BCK14] Space-time stationary solutions for the Burgers equation, J. Am. Math. Soc., Volume 27 (2014) no. 1, pp. 193-238 | DOI | MR | Zbl
[BFS25] Intertwining the Busemann process of the directed polymer model, Electron. J. Probab., Volume 30 (2025), 50, 80 pages | DOI | MR | Zbl
[BIK02] Topological shocks in Burgers turbulence, Phys. Rev. Lett., Volume 89 (2002) no. 2, 024501, 4 pages | DOI
[BK07] Burgers turbulence, Phys. Rep., Volume 447 (2007) no. 1-2, pp. 1-66 | DOI
[BK18] On global solutions of the random Hamilton–Jacobi equations and the KPZ problem, Nonlinearity, Volume 31 (2018) no. 4, p. R93-R121 | Zbl | DOI | MR
[BSS24a] Scaling limit of multi-type invariant measures via the directed landscape, Int. Math. Res. Not., Volume 2024 (2024) no. 17, pp. 12382-12432 | DOI | MR | Zbl
[BSS24b] The stationary horizon and semi-infinite geodesics in the directed landscape, Ann. Probab., Volume 52 (2024) no. 1, pp. 1-66 | DOI | MR | Zbl
[DG24] Jointly stationary solutions of periodic Burgers flow, J. Funct. Anal., Volume 287 (2024) no. 12, 110656, 43 pages | Zbl | DOI | MR
[EKMS00] Invariant measures for Burgers equation with stochastic forcing, Ann. Math. (2), Volume 151 (2000) no. 3, pp. 877-960 | DOI | MR | Zbl
[FS20] Joint distribution of Busemann functions in the exactly solvable corner growth model, Probab. Math. Phys., Volume 1 (2020) no. 1, pp. 55-100 | DOI | MR | Zbl
[GIKP05] Viscosity limit of stationary distributions for the random forced Burgers equation, Mosc. Math. J., Volume 5 (2005) no. 3, pp. 613-631 | DOI | MR | Zbl
[GRASS25] Jointly invariant measures for the Kardar–Parisi–Zhang equation, Probab. Theory Relat. Fields, Volume 192 (2025) no. 1-2, pp. 303-372 | Zbl | DOI | MR
[Ham72] A few seedlings of research, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of statistics, University of California Press (1972), pp. 345-394 | MR | Zbl
[IK03] Burgers turbulence and random Lagrangian systems, Commun. Math. Phys., Volume 232 (2003) no. 3, pp. 377-428 | DOI | MR | Zbl