  Random walks in the quarter plane are an important object both of
combinatorics and probability theory. Of particular interest for their study,
there is an analytic approach initiated by Fayolle, Iasnogorodski and Malyshev,
and further developed by the last two authors of this note. The outcomes of
this method are explicit expressions for the generating functions of interest,
asymptotic analysis of their coefficients, etc. Although there is an important
literature on reflected Brownian motion in the quarter plane (the continuous
counterpart of quadrant random walks), an analogue of the analytic approach has
not been fully developed to that context. The aim of this note is twofold: it
is first an extended abstract of two recent articles of the authors of this
paper, which propose such an approach; we further compare various aspects of
the discrete and continuous analytic approaches.
