Abstract
In this paper, we show that for t > 0, the joint distribution of the past {W t−s : 0 ≤ s ≤ t} and the future {W t + s :s ≥ 0} of a d-dimensional standard Brownian motion (W s ), conditioned on {W t ∈ 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 + 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 σ 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 → ∞ of the four dimensional vector with components \((W_{\sigma _{t}},t-\sigma _{t},W_{d_{t}},d_{t}-t)\), conditioned on {W t ∈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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References
Athreya K B and Rajeev B, Brownian crossings via regeneration times, Sankhya A 75 (2) (2013) 194–210
Kallenberg O, Foundations of Modern Probability (2010) (New York: Springer)
Karatzas I and Shreve S E, Brownian Motion and Stochastic Calculus (1998) (New York: Springer Science)
Rajeev B, First order calculus and last entrance times, Seminaire de Probabilites XXX, Lecture Notes in Mathematics 1626 (1989b) (Berlin: Springer Verlag), pp. 261–287
Rajeev B, Asymptotic distribution of Brownian excursions into an interval, pre-print (2016)
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Communicating Editor: Rajeeva L Karandikar
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ATHREYA, K.B., RAJEEV, B. Weak convergence of the past and future of Brownian motion given the present. Proc Math Sci 127, 165–174 (2017). https://doi.org/10.1007/s12044-016-0314-3
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DOI: https://doi.org/10.1007/s12044-016-0314-3