The talk is based on the introductory text  http://arxiv.org/pdf/math/0310326v1

 The talk is based on the paper available at:

http://www.math.drexel.edu/~phitczen/perpdraft4.pdf

which has some references. In particular, Vervaat's paper is a good
starting point for students.

 For students (and others) interested in material for this talk, please consult:

http://stat.columbia.edu/~jcliu/paper/SimGG1Final2nd.pdf

which is a single-
server version of the talk, but under the same principle.

           On Schr¨dinger’s equation, 3-dimensional Bessel
                       o
                    bridges, and passage time problems
Abstract: The main aim of this work is finding an explicit representation of the
density ϕf of the first time T that a one-dimensional Brownian process B reaches
the moving boundary f , where
                                                      t
                                    f (t) := a +        f (s)ds
                                                    0
and
                                  T := inf{t ≥ 0|Bt = f (t)}
given that f (t) > 0, ∀t ≥ 0. We do so, by first finding the expected value of the
                                             ˜
following 3-dimesional Bessel bridge X functional
                                                s
                               E exp −                    ˜
                                                  f (u)Xu du     ,
                                              0
[the reader may consult for instance Chapter 11 in Revuz and Yor (2005) for a
general overview of this process] and exploiting its relationship with first-passage
time problems as pointed out by Kardaras (2007). It turns out that this problem
is related to Schr¨dinger’s equation with time-dependent linear potential, see Feng
                    o
(2001).
    As a by-product we solve for a family of Volterra integral equations, which were
previously only treated numerically, see Peskir (2001).
                                          References
 [1] Feng, M. (2001). Complete solution of the Schr¨dinger equation for the time-dependent linear
                                                    o
     potential, Phys. Rev . A 64, 034101.
 [2] Kardaras, K (2007). On the density of first passage times for difussions. on preparation.
 [3] Peskir, G. (2001). On integral equations arising in the first-passage problem for Brownian
     motion, J. Integral Equations Appl., 14.
 [4] Revuz, D., and M. Yor. (2005). Continuous martingales and Brownian motion, Springer-
     Verlag, New York.