The wave maps equation and Brownian paths

Abstract

We discuss the p1 `1q-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.

Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

“…In addition to the nonlinear wave and Schrödinger equations with odd power-type and Hartree nonlinearities discussed above, invariant Gibbs measures have also been studied in several other settings. For instance, there has been research on invariant Gibbs measures for derivative nonlinearities [BLS21,Den15,Tzv10], quadratic nonlinearities [GKO18,OOT21], exponential nonlinearities [ORTW20, ORW21,ST20b], radially-symmetric settings [BB14a,BB14b,BB14c,Den12,Tzv06], KdV and generalized KdV [Bou94, CK21, ORT16, Ric16], fractional dispersion relations [ST20a,ST21], and lattice models [AKV20].…”

“…From both a mathematical and physical perspective, it would be interesting to obtain similar results for geometric wave and Schrödinger equation with random initial data or stochastic forcing. At the time of writing, some initial progress in this direction has been made in [BLS21,KLS20], but almost all questions in this area remain wide open. We hope that the different counting and tensor estimates of this article, which were briefly discussed in Subsection 1.4.2, will also be useful in the geometric setting.…”

Well-posedness of a gauge-covariant wave equation with space-time white noise forcing

“…Despite the significant progress towards a probabilistic theory of scalar nonlinear wave equations, there has only been little progress towards a probabilistic well-posedness theory for geometric wave equations. At this point, the primary references are [BLS21] and [KLS20]. In [BLS21], the first author, Lührmann, and Staffilani proved the probabilistic local well-posedness of the p1 `1qdimensional wave maps equation with Brownian paths as initial data.…”

“…At this point, the primary references are [BLS21] and [KLS20]. In [BLS21], the first author, Lührmann, and Staffilani proved the probabilistic local well-posedness of the p1 `1qdimensional wave maps equation with Brownian paths as initial data. The most important aspect of [BLS21] is that Brownian paths, i.e., the random initial data, are natural from both geometric and probabilistic perspectives.…”

“…In [BLS21], the first author, Lührmann, and Staffilani proved the probabilistic local well-posedness of the p1 `1qdimensional wave maps equation with Brownian paths as initial data. The most important aspect of [BLS21] is that Brownian paths, i.e., the random initial data, are natural from both geometric and probabilistic perspectives. In [KLS20], Krieger, Lührmann, and Staffilani proved the small data probabilistic global well-posedness of the energy-critical Maxwell-Klein-Gordon equation (MKG) with initial data drawn from a Wiener randomization.…”