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Random walks in the quarter plane are an important object both ofcombinatorics 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 ofthis method are explicit expressions for the generating functions of interest,asymptotic analysis of their coefficients, etc. Although there is an importantliterature on reflected Brownian motion in the quarter plane (the continuouscounterpart of quadrant random walks), an analogue of the analytic approach hasnot been fully developed to that context. The aim of this note is twofold: itis first an extended abstract of two recent articles of the authors of thispaper, which propose such an approach; we further compare various aspects ofthe discrete and continuous analytic approaches.

Analytic approach for reflected Brownian motion in the quadrant