Stationary solutions to the stochastic Burgers equation on the line
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866911183935635456 |
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| author | Dunlap, Alexander Graham, Cole Ryzhik, Lenya |
| author_facet | Dunlap, Alexander Graham, Cole Ryzhik, Lenya |
| contents | We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it cannot be represented as a convex combination of other invariant measures. We show that for each $a\in\mathbb{R}$, there is a unique indecomposable law of a spacetime-stationary solution with mean $a$, in a suitable function space. We also show that solutions starting from spatially-decaying perturbations of mean-$a$ periodic functions converge in law to the extremal space-time stationary solution with mean $a$ as time goes to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_07464 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Stationary solutions to the stochastic Burgers equation on the line Dunlap, Alexander Graham, Cole Ryzhik, Lenya Probability Mathematical Physics Analysis of PDEs 60H15, 35R60 We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it cannot be represented as a convex combination of other invariant measures. We show that for each $a\in\mathbb{R}$, there is a unique indecomposable law of a spacetime-stationary solution with mean $a$, in a suitable function space. We also show that solutions starting from spatially-decaying perturbations of mean-$a$ periodic functions converge in law to the extremal space-time stationary solution with mean $a$ as time goes to infinity. |
| title | Stationary solutions to the stochastic Burgers equation on the line |
| topic | Probability Mathematical Physics Analysis of PDEs 60H15, 35R60 |
| url | https://arxiv.org/abs/1910.07464 |