连续自旋扩散过程中吉布斯能的损失而无恢复
Loss without recovery of Gibbsianness during diffusion of continuous spins
摘要：
我们考虑一个特定的连续自旋吉布斯分布$μ_{t=0}$的双阱势，允许铁磁有序。我们研究独立扩散下的时间演化的初始措施。对于“高温”初始测度，我们证明了时变测度$μ_{t}$对所有的$t$都是Gibbsian的.对于“低温”的初始措施，我们证明了$μ_t$保持吉布斯足够小的时间$t$，但失去其吉布斯字符足够大的$t$。与离散自旋吉布斯测度的类似情况相反，在非零外部磁场存在下，大$t$的吉布斯性质没有恢复。我们所有的结果都适用于任何维度$d\geq 2$。这个例子更普遍地表明，时间演化的连续自旋模型比离散自旋模型更容易成为非吉布斯模型。
英文摘要：
We consider a specific continuous-spin Gibbs distribution $μ_{t=0}$ for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under independent diffusions. For `high temperature' initial measures we prove that the time-evoved measure $μ_{t}$ is Gibbsian for all $t$. For `low temperature' initial measures we prove that $μ_t$ stays Gibbsian for small enough times $t$, but loses its Gibbsian character for large enough $t$. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for large $t$ in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension $d\geq 2$. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than their discrete-spin counterparts.
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