AI 中文总结
该研究探讨二维布朗运动的惩罚问题，通过拉普拉斯变换与陶伯型定理建立相关概率测度的弱收敛性，明确了极限测度的结构性质。
AI 中文摘要
我们研究二维布朗运动的惩罚问题。从维纳测度出发，考虑通过依赖于t≥0的非负泛函F_t对路径加权得到的一族概率测度，其中F_t关于路径到时间t为止生成的σ-代数可测。在惩罚过程的适当假设下，我们建立了t→∞时这些测度的弱收敛性。极限律通过σ-有限测度W^(2)被明确确定，该测度具有涉及圆的最后击中时间的路径分解，此分解在分析中起核心作用，并给出了极限测度的鞅表示，而此时序和局部时技术不再适用。证明依赖于拉普拉斯变换方法和陶伯型定理，它们替代了游程理论工具，可精确确定极限测度及其结构性质。
英文摘要
We study a penalisation problem for two-dimensional Brownian motion. Starting from the Wiener measure, we consider a family of probability measures obtained by weighting paths by a nonnegative functional $F_t$ depending on $t \geq 0$, $F_t$ being measurable with respect to the $σ$-algebra generated by the path up to time $t$. Under suitable assumptions on the penalisation process, we establish the weak convergence of these measures when $t \rightarrow \infty$. The limiting law is identified explicitly in terms of a $σ-$finite measure $\mathbf{W}^{(2)}$, which admits a path decomposition involving the last hitting time of a circle. This decomposition plays a central role in the analysis and yields a martingale representation of the limiting measure. where ordering and local time techniques are no longer available. The proofs rely on Laplace transform methods and Tauberian theorems, which replace excursion-theoretic tools and allow a precise identification of the limiting measure and its structural properties.
