Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 26I / b / Solution

Condition on the first jump, putting $h_{W_0}=1$. The hitting probabilities obeyThe relevant characterization is that a hitting probability is the minimal nonnegative harmonic extension of its target-boundary value. Thus among nonnegative solutions of these equations with $h_{W_0}=1$, choose the componentwise least one. Other harmonic solutions need not be probabilities. This also identifies the solution without imposing an unjustified boundary condition at infinity.