ABSTRACT
We consider a stationary random walk of a point in many-dimensional space. It will be shown that, in the limit, the probability u(P) satisfies a partial differential equation of elliptic type. In particular, if the wandering point moves in steps of a fixed length, and in each state all the directions of motion are equally probable, then the solution of the generalized Dirichlet problem converges to a harmonic function determined by the original boundary conditions. The object of this study is the probability v(x) with which the wandering point issuing from x will eventually reach the interval x greater than or equal to b before it reaches the interval x lesser than a. The connection between the theory of probability and the boundary value problems for elliptic partial differential equations has been stressed for several years by R. Courant in Gottingen.
