二维delta-Bose气体：斜积相对运动
Two-dimensional delta-Bose gas: skew-product relative motions
AI总结：
本文证明了二维两体delta-Bose气体相对运动的Feynman-Kac型公式，通过原有游程刻画和低维Bessel过程给出两种证明，并证明了相关SDE的强适定性。
摘要：
我们证明了二维两体delta-Bose气体相对运动的Feynman-Kac型公式。其乘性泛函不是指数型的，且该过程是一个在Erickson [30]意义下依律唯一延拓的斜积扩散，其径向部分源自Donati-Martin和Yor [27]的${\rm BES}(0,β{\downarrow})$。我们给出了该公式的两种不同证明。第一种利用了${\rm BES}(0,β\downarrow)$的原有游程刻画，第二种则是通过期望层面上的低维Bessel过程。后一种证明与长期以来通过向Laplacians添加 mollifiers 来处理delta函数相互作用的方法形成对比，因为本文的近似是来自“更低的分数维”。此外，第二种证明对作为SDE的${\rm BES}(0,β\downarrow)$进行了新研究，因为我们通过Macdonald函数的特定比值来处理漂移项。所证明的性质包括所有初始条件下${\rm BES}(0,β\downarrow)$的SDE的强适定性与比较性。特别地，这种适定性与以下事实形成对比：Feynman-Kac型公式的斜积扩散仅当$p\leq 2$时具有$L^p_{\rm\tiny loc}$可积性的奇异漂移。
英文摘要：
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$.
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