论文标题
二维增量 - 果气气：偏斜的相对运动
Two-dimensional delta-Bose gas: skew-product relative motions
论文作者
论文摘要
我们证明了在二维中两体三角体气体相对运动的Feynman-kac型公式。乘法功能不是指数级的，并且该过程是eRickson [30]的偏差扩散，从$ {\ rm bes}（0，β{\ downarrow}）$ donati-martin和yor [27]作为径向部分。我们给出了两个不同的公式证明。第一个使用$ {\ rm bes}（0，β\ downarrow）$的原始偏移表征，第二个是通过预期级别的较低尺寸的贝塞尔过程。后者的证明与在拉普拉斯人中添加微型仪相互作用的长期方法对比了，因为目前的近似值来自“较低的分数维度”。此外，第二个证明是对$ {\ rm bes}（0，β\ downarrow）$的新研究，因为我们通过麦克唐纳函数的某些比率处理了漂移。事实证明的属性包括$ {\ rm bes}的SDE的强大良好性和比较。特别是，这种良好的度与Feynman-kac-type公式的偏差产物扩散的单数漂移为$ l^p _ {\ rm \ tiny loc} $仅适用于$ p \ leq 2 $的偏差，这与feynman-kac-type公式的偏差扩散形成了鲜明对比。
We prove a Feynman-Kac-type formula for the relative motion of the two-body delta-Bose gas in two dimensions. The multiplicative functional is not exponential, and the process is a skew-product diffusion uniquely extended in law, in the sense of Erickson [30], from ${\rm BES}(0,β{\downarrow})$ of Donati-Martin and Yor [27] as the radial part. We give two different proofs of the formula. The first uses the original excursion characterization of ${\rm BES}(0,β\downarrow)$, and the second is via the lower-dimensional Bessel processes at the expectation level. The latter proof contrasts the long-standing approach for delta-function interactions by adding mollifiers to the Laplacians since the present approximations are from "lower, fractional dimensions." Moreover, the second proof conducts a new study of ${\rm BES}(0,β\downarrow)$ as an SDE since we handle the drift via certain ratios of the Macdonald functions. The properties proven include the strong well-posedness and comparison of the SDE of ${\rm BES}(0,β\downarrow)$ for all initial conditions. In particular, this well-posedness contrasts the fact that the skew-product diffusion for the Feynman-Kac-type formula has a singular drift of $L^p_{\rm\tiny loc}$-integrability only for $ p\leq 2$.
