Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910976541982720 |
|---|---|
| author | Bringmann, Bjoern |
| author_facet | Bringmann, Bjoern |
| contents | In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument.
In this paper, we construct and study the Gibbs measure. Our approach is based on earlier work of Barashkov and Gubinelli for the $Φ^4_3$-model. Most importantly, our truncated Gibbs measures are tailored towards the dynamical aspects in the second part of the series. In addition, we develop new tools dealing with the non-locality of the Hartree interaction. We also determine the exact threshold between singularity and absolute continuity of the Gibbs measure depending on the regularity of the interaction potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_04609 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures Bringmann, Bjoern Analysis of PDEs Mathematical Physics Probability 35L15, 60H30 In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument. In this paper, we construct and study the Gibbs measure. Our approach is based on earlier work of Barashkov and Gubinelli for the $Φ^4_3$-model. Most importantly, our truncated Gibbs measures are tailored towards the dynamical aspects in the second part of the series. In addition, we develop new tools dealing with the non-locality of the Hartree interaction. We also determine the exact threshold between singularity and absolute continuity of the Gibbs measure depending on the regularity of the interaction potential. |
| title | Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures |
| topic | Analysis of PDEs Mathematical Physics Probability 35L15, 60H30 |
| url | https://arxiv.org/abs/2009.04609 |