关于外部等变波映射的Gibbs测度与拓扑孤子
On Gibbs measures and topological solitons of exterior equivariant wave maps
AI总结：
本文研究了外部等变波映射的拓扑孤子，证明了支撑在孤子同伦类上的Gibbs测度的存在性与不变性，并得出孤子分解对随机初始数据失效的结论，揭示了确定性与概率论视角的鲜明对比。
摘要：
我们考虑从外部空间区域$\mathbb{R}^3\backslash B(0,1)$到目标$\mathbb{S}^3$的$k$-等变波映射。该模型具有无限多个拓扑孤子$Q_{n,k}$，它们由其拓扑度$n\in \mathbb{Z}$索引。对于每个$n\in \mathbb{Z}$和$k\geq 1$，我们证明了支撑在$Q_{n,k}$同伦类上的Gibbs测度的存在性与不变性。作为推论，我们得出孤子分解对随机初始数据失效。由于已知能量空间中初始数据的孤子分解成立，这揭示了确定性与概率论视角之间的鲜明对比。
英文摘要：
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.
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