Weak convergence of the past and future of Brownian motion given the present
Proceedings of the Indian Academy of Sciences: Mathematical Sciences
In this paper, we show that for t > 0, the joint distribution of the past (Wt-s: 0 ≤ s ≤ t) and the future (Wt+s: S ≥ 0) of a d-dimensional standard Brownian motion (Ws), conditioned on (Wt ε U), where U is a bounded open set in Rd, converges weakly in C[0,∞) x C[0,∞) as t → ∞. The limiting distribution is that of a pair of coupled processes Y + B1, Y + B2 where Y, B1, B2 are independent, Y is uniformly distributed on U and B1, B2 are standard d-dimensional Brownian motions. Let αt, dt 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 → ∞ of the four dimensional vector with components (Wαt, t-αt, Wdt, dt-t), conditioned on (Wt ε 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 + B1 and Y + B2 respectively.
Athreya, K. B. and Rajeev, B., "Weak convergence of the past and future of Brownian motion given the present" (2017). Journal Articles. 2694.