Invariant Gibbs dynamics for two-dimensional fractional wave equations in negative Sobolev spaces

Luigi Forcella; Oana Pocovnicu

Invariant Gibbs dynamics for two-dimensional fractional wave equations in negative Sobolev spaces

Analysis of PDEs 2025-10-24 v2 Probability

Authors: Luigi Forcella , Oana Pocovnicu

Abstract

We consider a fractional nonlinear wave equations (fNLW) with a general power-type nonlinearity, on the two-dimensional torus. Our main goal is to construct invariant global-in-time Gibbs dynamics for a renormalized fNLW. We first construct the Gibbs measure associated with this equation by using the variational approach of Barashkov and Gubinelli. We then prove almost sure local well-posedness with respect to Gibbsian initial data, by exploiting the second order expansion. Finally, we extend solutions globally in time using Bourgain's invariant measure argument.

Keywords

fractional nonlinear wave equation gibbs measure invariant dynamics negative sobolev spaces local well-posedness variational approach

Cite

@article{arxiv.2306.07857,
  title  = {Invariant Gibbs dynamics for two-dimensional fractional wave equations in negative Sobolev spaces},
  author = {Luigi Forcella and Oana Pocovnicu},
  journal= {arXiv preprint arXiv:2306.07857},
  year   = {2025}
}

Comments

31 pages. The well-posedness results were improved compared to the first version. The paper was restructured. Most proofs were restructured and rendered more efficient

Invariant Gibbs dynamics for two-dimensional fractional wave equations in negative Sobolev spaces

Abstract

Keywords

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