| We shall study two-dimensional random walks in the positive quadrant. Each step (to a neighboring site) occurs with a certain probability, where different probabilities may be taken for the interior of the state space and the boundaries. The generating function Q(x,y) of the stationary distribution of such random walks satisfies K(x,y)=A(x,y)Q(x,0)+B(x,y)Q(0,y)+C(x,y)Q(0,0), where the functions K and A,B,C are quadratic polynomials in x and y. Finding Q(x,y) through functional equations is the universal problem for random walks in the quarter plane. Such functional equations can be solved using the theory of boundary value problems, a method pioneered by Malyshev in the 1970's. Since then, alternative methods have been developed, and the theory has advanced via its use in lattice path counting and two-server queueing models. Developing sound mathematical techniques for determining Q(x,y) and its inverse belongs to some of the most exciting problems in contemporary probability. It is at the interface of probability and analysis, and requires complex-function methods, analytic combinatorics in several variables and the asymptotics of multivariate generating functions. The novelty and urgency is underlined by the fact that three courses at international summer schools in 2009 were devoted to this topic. Within this project we focus on: (i) Classification of random walks and developing exact solution methods, (ii) Asymptotic analysis through functional equations, and (iii) Time-dependent behavior and connections with enumerative combinatorics. |
