Weak convergence of the past and future of Brownian motion given the present

Proceedings Mathematical Sciences

B Rajeev¹  K B Athreya² 

[1] Stat-Math Unit, Indian Statistical Institute, Bangalore Centre, th Mile, Mysore Road, R.V. College Post, Bangalore 0 0, India$$;Department of Mathematics and Statistics, Iowa State University, Ames, Iowa 00, USA$$

关键词: Brownian motion;    weak convergence;    last entrance time;    first exit time;    coupling;    time reversal.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this paper, we show that for $t > 0$, the joint distribution of the past ${W_{t−s} : 0 leq s leq t}$ and the future ${W_{t+s} : s geq 0}$ of a $d$-dimensional standard Brownian motion $(W_s)$, conditioned on ${W_tin U}$, where $U$ is a bounded open set in $mathbb{R}^d$, converges weakly in $C[0,infty)imes C[0, infty)$ as $tightarrowinfty$. The limiting distribution is that of a pair of coupled processes $Y + B^1$, $Y + B^2$ where $Y$, $B^1$, $B^2$ are independent, $Y$ is uniformly distributed on $U$ and $B^1$, $B^2$ are standard $d$-dimensional Brownian motions. Let $sigma_t$, $d_t$ be respectively, the last entrance time before time $t$ into the set $U$ and the first exit time after $t$ from $U$. When the boundary of $U$ is regular, we use the continuous mapping theorem to show that the limiting distribution as $tightarrowinfty$ of the four dimensional vector with components $(W_{sigma_t}, t − sigma_t, W_{d_t}, d_t − t)$, conditioned on ${W_tin U}$, is the same as that of the four dimensional vector whose components are the place and time of first exit from $U$ of the processes $Y + B^1$ and $Y + B^2$ respectively.

【 授权许可】

Unknown