K B Athreya
Articles written in Proceedings – Mathematical Sciences
Volume 127 Issue 1 February 2017 pp 165-174 Research Article
Weak convergence of the past and future of Brownian motion given the present
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 $\{W_t\in U\}$, where $U$ is a bounded open set in $\mathbb{R}^d$, converges weakly in $C[0,\infty)\times C[0, \infty)$ as $t\rightarrow\infty$. 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 $t\rightarrow\infty$ of the four dimensional vector with components $(W_{\sigma_t}, t − \sigma_t, W_{d_t}, d_t − t)$, conditioned on $\{W_t\in U\}$, 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.
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