8. Let n be the subset {(0.0}.(1.0). (0.1M- 1.0),(0, - 1)} of 2.1. (That is, 0 consists of the origin and its four nest-door neighbors.) Check directly that the recipe described in example 8 for solving the \"discrete Dirichlet problem" on n is correct. Example 8. The Discrete Dirichlet Problem Let ? be a smooth, bounded region in the plane with boundary B. An important problem in electrostatics is the Dirichlet problem: Given a con- tinuous function f on B, find a function u satisfying (8) Au = 0 in Q u = f on B where Au = (82/2x)u + (82/82y)u. This problem has a discrete analogue that is itself quite interesting. Let Q Chapter 1 Measure Theory be a finite subset of Z?. A point p = (m, n) of $ is an interior point if its four next-door neighbors (m,n + 1), (m + 1,n), (m,n - 1), and (m - 1,n) are also in ?; otherwise, p is a boundary point. For instance, in the figure below, pi is an interior point and p2 a boundary point of the shaded region. For a function u on Z?, we define Adiscreteu by the formula (discrete #) (m, n) (9) u(m, n + 1) + u(m, n - 1) + u(m + 1,n) + u(m - 1,") _u(m,n) [Notice that the first term on the right is just the average of u over the next-door neighbors of the point (m, n).] The discrete analogue of the Dirichlet problem is to find a function u : Z2 -> R such that (10) Adiscrete = 0 at the interior points of , and (11) u =f on the boundary, 20, of 2, f being a given function on 20. One can solve this problem elegantly by using the random walk described in example 7: Given a point pen and a random path w starting at p, let F(w, p) be the value of f at the first point at which w hits a$2. [If c never hits the boundary, set F(w, p) = 0.] If we fix p and regard F as a function of the random path w alone, then F is a random variable in the sense of example 3. We will show in $2.8 that its expectation value is the value at p of the solution of the Dirichlet problem described in equations 10 and 11