On Gibbs measures and topological solitons of exterior equivariant wave maps

We consider k-equivariant wave maps from the exterior spatial domain R³ n 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 2 Z. For each n 2 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.