On Gibbs measures and topological solitons of exterior equivariant wave maps

TL;DR

This paper constructs Gibbs measures for equivariant wave maps with topological solitons and demonstrates that soliton resolution fails for random initial data, highlighting differences between deterministic and probabilistic approaches.

Contribution

It introduces Gibbs measures supported on topological solitons for exterior equivariant wave maps and shows soliton resolution failure for random initial data.

Findings

Gibbs measures exist and are invariant for each topological class

Soliton resolution fails for random initial data

Contrast between deterministic and probabilistic dynamics

Abstract

We consider $k$-equivariant wave maps from the exterior spatial domain $\mathbb{R}^3\backslash B(0,1)$ into the target $\mathbb{S}^3$. This model has infinitely many topological solitons $Q_{n,k}$, which are indexed by their topological degree $n\in \mathbb{Z}$. For each $n\in \mathbb{Z}$ and $k\geq 1$, 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.