单点相互作用基态扩散中避开原点概率的次鞅：$d \in \{2,3\}$
A submartingale for the probability of avoiding the origin in one-point interaction ground-state diffusion: $d \in \{2,3\}$
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中文总结 AI 辅助
该研究针对二维和三维带单点势的奇异扩散，构造次鞅分析其击中原点的概率、首次击中时间分布及避开原点的条件动力学，以统一方法验证相关性质。
中文摘要 AI 辅助
我们研究了定义在区间$[0,T]$上的奇异扩散在靠近原点处的行为，该扩散的转移密度由$d$维薛定谔算子$L^{\gamma}$生成的半群积分核的Doob变换给出，其中$L^{\gamma}$带有一个位于原点的单点势，驱动族为$L^{\gamma}$的基态，且$d \in \{2,3\}$。我们构造了一个次鞅，其递增分量仅在扩散访问原点的时刻增长。利用该次鞅，我们证明了该扩散以正概率击中原点，并且在条件于时间$T$前击中原点的情况下，首次击中时间服从截断广义逆高斯（GIG）分布。我们进一步研究了在条件于避开原点时的动力学：在该条件律下，该扩散不是标准布朗运动，而是可以表示为正则化漂移项与连续鞅的组合。虽然这些性质在二维中已为人所知，但本文基于次鞅的方法提供了一种替代验证方式，并以统一方式处理二维和三维的情况。
英文摘要
We study the near-origin behavior on $[0,T]$ of the singular diffusion whose transition density is given by a Doob transform of the integral kernel of the semigroup generated by the $d$-dimensional Schrödinger operator $L^γ$ with a one-point potential at the origin, where the driving family is the ground state of $L^γ$ and $d\in\{2,3\}$. We construct a submartingale whose increasing component grows only at times when the diffusion visits the origin. Using this submartingale, we show that the diffusion hits the origin with positive probability and that, conditionally on hitting the origin by time $T$, the first hitting time has a truncated generalized inverse Gaussian (GIG) distribution. We further study the dynamics under conditioning to avoid the origin: under the conditional law, the diffusion is not a standard Brownian motion, but instead admits a representation in terms of a regularized drift and a continuous martingale. While these properties are known in dimension two, the present submartingale-based approach provides an alternative verification and treats dimensions two and three in a unified manner.
