We discuss the (1+1)-dimensional wave maps equation with values in a compact Lie group. The corresponding Gibbs measure is given by a Brownian motion on the Lie group, which plays a central role in stochastic geometry. Our main theorem is the almost sure global well-posedness and invariance of the Gibbs measure for the wave maps equation. It is the first result of this kind for any geometric wave equation. Our argument relies on a novel finite-dimensional approximation of the wave maps equation which involves the so-called Killing renormalization. The main part of this article then addresses the global convergence of our approximation and the almost invariance of the Gibbs measure under the corresponding flow. The proof of global convergence requires a carefully crafted Ansatz which includes modulated linear waves, modulated bilinear waves, and mixed modulated objects. The interactions between the different objects in our Ansatz are analyzed using an intricate combination of analytic, geometric, and probabilistic ingredients. In particular, geometric aspects of the wave maps equation are utilized via orthogonality, which has previously been used in the deterministic theory of wave maps at critical regularity. The proof of almost invariance of the Gibbs measure under our approximation relies on conservative structures, which are a new framework for the approximation of Hamiltonian equations, and delicate estimates of the energy increment.
", "doi": "10.1007/s00222-026-01414-8", "issn": "0020-9910", "publisher": "Springer Science and Business Media LLC", "publication": "Inventiones mathematicae", "publication_date": "2026-07", "series_number": "1", "volume": "245", "issue": "1", "pages": "133-490" }, { "id": "authors:6nj5f-v5884", "collection": "authors", "collection_id": "6nj5f-v5884", "cite_using_url": "https://authors.library.caltech.edu/records/6nj5f-v5884", "type": "article", "title": "Well-posedness of a gauge-covariant wave equation with space-time white noise forcing", "author": [ { "family_name": "Bringmann", "given_name": "Bjoern", "orcid": "0000-0001-7709-623X", "clpid": "Bringmann-Bjoern" }, { "family_name": "Rodnianski", "given_name": "Igor" } ], "abstract": "We first introduce a new model for a two-dimensional gauge-covariant wave equation with space-time white noise. In our main theorem, we obtain the probabilistic global well-posedness of this model in the Lorenz gauge. Furthermore, we prove the failure of a probabilistic null-form estimate, which exposes a potential obstruction towards the probabilistic well-posedness of a stochastic Maxwell–Klein–Gordon equation.
", "doi": "10.2140/pmp.2025.6.139", "issn": "2690-1005", "publisher": "Mathematical Sciences Publishers", "publication": "Probability and Mathematical Physics", "publication_date": "2025-02-05", "series_number": "1", "volume": "6", "issue": "1", "pages": "139-193" }, { "id": "authors:w91ev-pwq66", "collection": "authors", "collection_id": "w91ev-pwq66", "cite_using_url": "https://authors.library.caltech.edu/records/w91ev-pwq66", "type": "article", "title": "Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation", "author": [ { "family_name": "Bringmann", "given_name": "Bjoern", "orcid": "0000-0001-7709-623X", "clpid": "Bringmann-Bjoern" }, { "family_name": "Deng", "given_name": "Yu" }, { "family_name": "Nahmod", "given_name": "Andrea R.", "orcid": "0000-0002-3195-7249" }, { "family_name": "Yue", "given_name": "Haitian", "orcid": "0000-0002-4915-0267" } ], "abstract": "We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic Φ\u2074\u2083-model. This result is the hyperbolic counterpart to seminal works on the parabolic Φ\u2074\u2083-model by Hairer (Invent. Math. 198(2):269–504, 2014) and Hairer-Matetski (Ann. Probab. 46(3):1651–1709, 2018).The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates.The singularity of the Gibbs measure with respect to the Gaussian free field brings out a new caloric representation of the Gibbs measure and a synergy between the parabolic and hyperbolic theories embodied in the analysis of heat-wave stochastic objects. Furthermore from a purely hyperbolic standpoint our argument relies on key new ingredients that include a hidden cancellation between sextic stochastic objects and a new bilinear random tensor estimate.
", "doi": "10.1007/s00222-024-01254-4", "issn": "0020-9910", "publisher": "Springer Science and Business Media LLC", "publication": "Inventiones mathematicae", "publication_date": "2024-06", "series_number": "3", "volume": "236", "issue": "3", "pages": "1133-1411" }, { "id": "authors:1yck8-rz256", "collection": "authors", "collection_id": "1yck8-rz256", "cite_using_url": "https://authors.library.caltech.edu/records/1yck8-rz256", "type": "article", "title": "Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics", "author": [ { "family_name": "Bringmann", "given_name": "Bjoern", "orcid": "0000-0001-7709-623X", "clpid": "Bringmann-Bjoern" } ], "abstract": "In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences.
", "doi": "10.4171/jems/1317", "issn": "1435-9855", "publisher": "European Mathematical Society - EMS - Publishing House GmbH", "publication": "Journal of the European Mathematical Society", "publication_date": "2024-05-08", "series_number": "6", "volume": "26", "issue": "6", "pages": "1933-2089" }, { "id": "authors:edzy5-p4w07", "collection": "authors", "collection_id": "edzy5-p4w07", "cite_using_url": "https://authors.library.caltech.edu/records/edzy5-p4w07", "type": "article", "title": "On Gibbs measures and topological solitons of exterior equivariant wave maps", "author": [ { "family_name": "Bringmann", "given_name": "Bjoern", "orcid": "0000-0001-7709-623X", "clpid": "Bringmann-Bjoern" } ], "abstract": "We consider k-equivariant wave maps from the exterior spatial domain R³ \\ B(0,1) into the target S³. This model has infinitely many topological solitons Q_(n,k), which are indexed by their topological degree n ∈ Z. For each n ∈ Z and k ≥ 1, we prove the existence and invariance of a Gibbs measure supported on the homotopy class of Q_(n,k). As a corollary, we obtain that soliton resolution fails for random initial data. Since soliton resolution is known for initial data in the energy space, this reveals a sharp contrast between deterministic and probabilistic perspectives.
", "doi": "10.4171/rmi/1473", "issn": "0213-2230", "publisher": "European Mathematical Society - EMS - Publishing House GmbH", "publication": "Revista Matem\u00e1tica Iberoamericana", "publication_date": "2024-04-30", "series_number": "3", "volume": "40", "issue": "3", "pages": "859-900" }, { "id": "authors:pbpjh-fg957", "collection": "authors", "collection_id": "pbpjh-fg957", "cite_using_url": "https://authors.library.caltech.edu/records/pbpjh-fg957", "type": "article", "title": "The Wave Maps Equation and Brownian Paths", "author": [ { "family_name": "Bringmann", "given_name": "Bjoern", "orcid": "0000-0001-7709-623X", "clpid": "Bringmann-Bjoern" }, { "family_name": "L\u00fchrmann", "given_name": "Jonas" }, { "family_name": "Staffilani", "given_name": "Gigliola", "orcid": "0000-0003-1382-9435" } ], "abstract": "We discuss the 1+1-dimensional wave maps equation with values in a compact Riemannian manifold M. Motivated by the Gibbs measure problem, we consider Brownian paths on the manifold M as initial data. Our main theorem is the probabilistic local well-posedness of the associated initial value problem. The analysis in this setting combines analytic, geometric, and probabilistic methods.
", "doi": "10.1007/s00220-023-04885-5", "issn": "0010-3616", "publisher": "Springer Science and Business Media LLC", "publication": "Communications in Mathematical Physics", "publication_date": "2024-03", "series_number": "3", "volume": "405", "issue": "3", "pages": "60" }, { "id": "authors:0sbk0-cye47", "collection": "authors", "collection_id": "0sbk0-cye47", "cite_using_url": "https://authors.library.caltech.edu/records/0sbk0-cye47", "type": "article", "title": "An Eigensystem Approach to Anderson Localization for Multi-particle Systems", "author": [ { "family_name": "Bringmann", "given_name": "Bjoern", "orcid": "0000-0001-7709-623X", "clpid": "Bringmann-Bjoern" }, { "family_name": "Mendelson", "given_name": "Dana", "orcid": "0000-0002-3266-4851" } ], "abstract": "This paper revisits the proof of Anderson localization for multi-particle systems. We introduce a multi-particle version of the eigensystem multi-scale analysis by Elgart and Klein, which had previously been used for single-particle systems.
", "doi": "10.1007/s00023-021-01051-2", "issn": "1424-0637", "publisher": "Springer Science and Business Media LLC", "publication": "Annales Henri Poincar\u00e9", "publication_date": "2021-10", "series_number": "10", "volume": "22", "issue": "10", "pages": "3255-3290" }, { "id": "authors:wff3f-5yn25", "collection": "authors", "collection_id": "wff3f-5yn25", "cite_using_url": "https://authors.library.caltech.edu/records/wff3f-5yn25", "type": "article", "title": "The homotopy method revisited: Computing solution paths of \u2113\u2081-regularized problems", "author": [ { "family_name": "Bringmann", "given_name": "Bjoern", "orcid": "0000-0001-7709-623X", "clpid": "Bringmann-Bjoern" }, { "family_name": "Cremers", "given_name": "Daniel", "orcid": "0000-0002-3079-7984" }, { "family_name": "Krahmer", "given_name": "Felix", "orcid": "0000-0002-1959-5548" }, { "family_name": "Moeller", "given_name": "Michael", "orcid": "0000-0002-0492-6527" } ], "abstract": "\u21131-regularized linear inverse problems frequently arise in signal processing, image analysis, and statistics. The correct choice of the regularization parameter t ∈ R ≥ 0 is a delicate issue. Instead of solving the variational problem for a fixed parameter, the idea of the homotopy method is to compute a complete solution path u(t) as a function of t. In their celebrated paper A new approach to variable selection in least squares problems [IMA J. Numer. Anal. 20 (2000), no. 3, 389–403], Osborne, Presnell, and Turlach showed that the computational cost of this approach is often comparable to the cost of solving the corresponding least squares problem. Their analysis relies on the one-at-a-time condition, which requires that different indices enter or leave the support of the solution at distinct regularization parameters. In this paper, we introduce a generalized homotopy algorithm based on a nonnegative least squares problem, which does not require such a condition, and prove its termination after finitely many steps. At every point of the path, we give a full characterization of all possible directions. To illustrate our results, we discuss examples in which the standard homotopy method either fails or becomes infeasible. To the best of our knowledge, our algorithm is the first to provably compute a full piecewise linear and continuous solution path for an arbitrary combination of a measurement matrix and a data vector.
", "doi": "10.1090/mcom/3287", "issn": "1088-6842", "publisher": "American Mathematical Society (AMS)", "publication": "Mathematics of Computation", "publication_date": "2018-09", "series_number": "313", "volume": "87", "issue": "313", "pages": "2343-2364" } ]