The method of the exponential generating function for random walks in a plane quadrant. (Q1432429)

In the phase space \(N^{2}\) we consider a homogeneous random walk \(\{S_{n}\}\) and we denote by \[ h(s_{1},s_{2})=\sum_{\gamma_{1},\gamma_{2}=0}^{\infty} p_{\gamma_{1},\gamma_{2}}s_{1}^{\gamma_{1}}s_{2}^{\gamma_{2}} \] the generating function associated to the transition probabilities \(\{p_{\gamma_{1},\gamma_{2}}:(\gamma_{1},\gamma_{2})\in N^{2}\}\). The author considers the case \(h(s_{1},s_{2})=p_{20}s_{1}^{2}+ p_{02}s_{2}^{2}+p_{00}\). In the main result one derives an integral representation for the absorption probabilities \[ q_{(0,\gamma_{2})}^{(\alpha_{1},\alpha_{2})} =P\{\exists n<\infty:S_{n}=(0,\gamma_{2})\mid S_{0}=(\alpha_{1},\alpha_{2})\}. \] The proof is based on the Riemann formula for the Darboux-Picard problem for the so called telegraph equation: \[ \frac{\partial^{2}}{\partial z_{1}\partial z_{2}}g_{(0,\gamma_{2})}(z_{1},z_{2}) =g_{(0,\gamma_{2})}(z_{1},z_{2}). \] Here \(g_{(0,\gamma_{2})}\) is the exponential generating function \[ g_{(0,\gamma_{2})}(z_{1},z_{2})=\sum_{\alpha_{1},\alpha_{2}=0}^{\infty} \frac{z_{1}^{\alpha_{1}}z_{2}^{\alpha_{2}}}{\alpha_{1}!\alpha_{2}!} q_{(0,\gamma_{2})}^{(\alpha_{1},\alpha_{2})}. \]