ABSTRACT
We consider a stationary random walk of a point in many-dimensional space (the term “stationary” implies that the transition probability of the point moving from a position P into a set M in one step depends only on P and M but not on the time or the previous history of the random motion of the point). Let u(P) be the probability of the event that the point issuing from P makes its first exit from a given domain G ∋ P through a fixed portion of its boundary. It will be shown that, in the limit, the probability u(P) satisfies a partial differential equation of elliptic type. Finding u(P) is regarded as a special case of a more general problem (the so-called “generalized Dirichlet problem”). The same method which allows us to obtain the said limit theorem of probability theory can be used to justify obtaining an approximate solution of the classical Dirichlet problem by means of difference equations.
