How to Study Reflected Brownian Motion in a Quadrant via Kernel Functional Equations? A short survey
We survey a line of works studying semimartingale reflected Brownian motion in a quadrant, covering both the non-degenerate and degenerate settings. Two main situations are emphasized: the recurrent case, where an invariant measure exists, and the transient case, where the central objects are Green's functions (potential measures). These measures typically arise from Kolmogorov forward equations. For transient or killed models one is also interested in the Martin boundary of the process and, consequently, in all positive harmonic functions, which satisfy Kolmogorov backward equations. Depending on the geometry and parameters of the model, these harmonic functions often admit probabilistic interpretations in terms of absorption, escape, or drift to infinity. All these measures and functions are studied through kernel functional equations satisfied by their Laplace transforms. Several ways of solving these equations are reviewed, each leading to different types of results. Following the analytic approach developed for quarter-plane random walks by Fayolle, Iasnogorodski and Malyshev, a key preliminary step is the analytic continuation of the relevant Laplace transforms onto the complex algebraic curve defined by the zero set of the kernel. Carleman boundary value problem techniques then yield explicit contour-integral representations for the Laplace transforms. In special parameter regimes, Tutte's invariant method provides integral-free formulas and a sharp classification of the transforms according to their algebraic-differential complexity. Singularity analysis combined with saddle-point methods carried out on the kernel algebraic curve, produces precise two-dimensional asymptotics. Finally, in the degenerate setting, the compensation approach provides an alternative constructive method, allowing one to build the densities as an infinite series through explicit iterative corrections.