标题： 关于$Φ^4$测度的Onsager-Machlup泛函 显示英文标题

标题： On the Onsager-Machlup functional of the $Φ^4$-measure

摘要： 我们通过Onsager-Machlup(OM)泛函的视角，研究有限体积下$\Phi^4_d$$(d=1,2,3)$测度的广义密度的存在性。 后者在度量空间上的测度上被严格定义为小球概率的极限比值。 在一维情况下，我们证明$\Phi^4_1$测度的标准OM泛函与$\Phi^4$作用量一致，正如预期的那样。 在二维情况下，我们通过考虑“增强”距离来证明$P(\Phi)_2$测度的OM泛函与相应的作用量一致，这些距离是相对于高斯自由场的Wick幂定义的，类似于粗糙路径度量。 在维度$3$时，证明了OM泛函的两种自然推广是退化的。 最后，在适当的正则性条件下，通过考虑小半径-大频率联合极限，我们恢复了$\Phi^4_3$作用量。 显示英文摘要

摘要： We investigate the existence of generalised densities for the $\Phi^4_d$ $(d=1,2,3)$ measures, in finite volume, through the lens of Onsager-Machlup (OM) functionals. The latter are rigorously defined for measures on metric spaces as limiting ratios of small ball probabilities. In one dimension, we show that the standard OM functional of the $\Phi^4_1$ measure coincides with the $\Phi^4$ action as expected. In two dimensions, we show that OM functionals of the $P(\Phi)_2$ measures agree with the corresponding actions, by considering ``enhanced" distances, defined with respect to Wick powers of the Gaussian Free Field, which are analogous to rough path metrics. In dimension $3$, two natural generalisations of the OM functional are proved to be degenerate. Finally, we recover the $\Phi^4_3$ action, under appropriate regularity conditions, by considering joint small radius-large frequency limits.