On Gibbs measures and topological solitons of exterior equivariant wave maps

Abstract

We consider $k$-equivariant wave maps from the exterior spatial domain ${\mathbb{R}}^3\setminus B(0,1)$ 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 $k\ge 1$, 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.