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

Abstract

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 ∈ U}, where U is a bounded open set in ℝd, converges weakly in C[0,∞)×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 σt,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 \((W_{\sigma _{t}},t-\sigma _{t},W_{d_{t}},d_{t}-t)\), conditioned on {Wt∈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.