Markoff Process and the Dirichlet Problem

Abstract

1. The purse of this paper is to give a general discussion of the Dirichlet problem from the standpoint of the theory of positive linear operations in a semi-ordered Banach space. It will be shown that the so-called sweeping out process of obtaining the solution of the Dirichlet problem may be observed as a kind of Markoff process) in the space of continuous functions. 2. Let be a compact Hausdorff space. The set C() of all real-valued continuous functions x(,) dened on is a Banach space with respect to the IlorII1 (1) I! = II=P-I 0() is also an (M)-spaee > wih respee to he partial ordering" and e(a)-I is he uni elemen o 3. Le D be a bounded domain in he Gauian plane. We do no ha D is simply or finitely connected. Le us corider he (M)-paces O(D) and 0(/') where D i the closure of D and /'= D-Then /-A(z) is a bounded inear operaion which mam.O(D) onto O(F), and clearly saisfieB () >=o ,Z,ll,i, es A(z,)>_O, () zl ,i, m191Ces A(z)l, (5) II A(x)II II z !1. That =A(z)is an onto-mapping means the act that, Ior may y()e C(P), there exisf an x() Cr(') such tha A(z)=/. We can take a x(')may continuou extension o y(') rom 1" to D. Such an extenion however is lmique]y determined; but it is poible z) to find in a concrete way a bunded linear operation z= B(y) which mar O(F) into C(D-) such hat AB(y)=y on C(/') and fiarther that 1) K. Yosida and S. Kakutani, Operator-theoretical treatment of Markoff process and the mean ergodic theorem, Annals of Math., 42(1941). 2. S. Kakutani, Concrete representation of abstract (M)-spaces aad the characterization of the space of continuous functions, Annals of Math., 42(1941). 3) S. Kakutani, Simultaneous extension of continuous functions considered as a posi-live operation, Jap. Journ. of Math., 19(1940).