  We prove the existence of semi-infinite geodesics for Brownian last-passage
percolation (BLPP). Specifically, on a single event of probability one, there
exist semi-infinite geodesics started from every space-time point and traveling
in every asymptotic direction. Properties of these geodesics include uniqueness
for a fixed initial point and direction, non-uniqueness for fixed direction but
random initial points, and coalescence of all geodesics traveling in a common,
fixed direction. Along the way, we prove that for fixed northeast and southwest
directions, there almost surely exist no bi-infinite geodesics in the given
directions. The semi-infinite geodesics are constructed from Busemann
functions. Our starting point is a result of Alberts, Rassoul-Agha and Simper
that established Busemann functions for fixed points and directions. Out of
this, we construct the global process of Busemann functions simultaneously for
all initial points and directions, and then the family of semi-infinite
Busemann geodesics. The uncountable space of the semi-discrete setting requires
extra consideration and leads to new phenomena, compared to discrete models.
