AI 中文总结
本文研究带约束势时间平均低于固定水平的R^d中条件布朗运动，证明其收敛到基态扩散，得到条件概率渐近式，推广一维二次情形并解决相关猜想，方法可应用于多种可逆马尔可夫过程。
AI 中文摘要
我们研究R^d中满足如下条件的布朗运动：连续约束势的时间平均值始终低于固定水平。我们证明，在每个固定初始时间区间上，该条件化过程在全变差意义下收敛到对应合适薛定谔算子的基态扩散过程。我们还得到了该条件化事件概率的精确渐近式，包括约束的有界扰动情形。证明基于对应费曼-卡测度的局部极限定理。我们的结果将此前已知的一维二次情形推广到了任意有限维空间和广泛的约束势类，从而解决了Aurzada、Lifshits与Schickentanz提出的猜想。所提方法也适用于将布朗运动替换为合适的可逆马尔可夫过程的情形，包括多维奥恩斯坦-乌伦贝克过程、CIR过程及连续时间马尔可夫链。
英文摘要
We study Brownian motion in R^d conditioned so that the time average of a continuous confining potential remains below a fixed level. On every fixed initial time interval, we prove that the conditioned process converges in total variation to the ground-state diffusion associated with a suitable Schrödinger operator. We also obtain sharp asymptotics for the probability of the conditioning event, including bounded perturbations of the constraint. The proof is based on a local limit theorem for the corresponding Feynman-Kac measures. Our results extend the previously known one-dimensional quadratic case to arbitrary finite dimension and a broad class of confining potentials, therefore resolving a conjecture of Aurzada, Lifshits and Schickentanz. The presented approach also works when Brownian motion is replaced by suitable reversible Markov processes, including multidimensional Ornstein-Uhlenbeck processes, CIR processes and continuous-time Markov chains.
Comments34 pages
