 Hitting probability

A Hitting probability describes reaching a target, possibly before a competing target. For a Markov chain and disjoint sets $A,B$, let $T_A=\inf \{n\ge 0:X_n\in A\}$ and defineIts boundary values are $h=1$ on $A$ and $h=0$ on $B$. Outside $A\cup B$, first-step analysis gives the discrete harmonic equation $h(i)={\sum }_j{P}_{ij}h(j)$. If the finite chain hits $A\cup B$ almost surely from every state, these equations determine $h$ uniquely: stop at the boundary or use the maximum principle for the difference of two solutions.

For a symmetric nearest-neighbour random walk on $\{0,1,\dots ,m\}$ absorbed at its endpoints, the Probability of hitting $m$ before zero is $h(i)=i/m$. It has the correct endpoint values and equals the average of its two neighbours; for example, starting at two on $\{0,\dots ,5\}$ gives Probability $2/5$.

On an infinite state space, boundary values and harmonicity need not imply uniqueness. The Hitting probability is the minimal nonnegative solution, obtained as the increasing limit of finite-horizon hitting probabilities. Taking $B$ empty gives the Probability of ever hitting $A$, with paths that never hit contributing zero. A Hitting probability concerns an entire sample path; it differs from the Probability of occupying the target at a specified time.