Markoff processes and potentials III

The numbering of the sections in this part continues the numbering in the first two installments, pages 44-93 and 316-369 of the first volume of this journal.Consider again the situation treated in the introduction to the first in- stallment.Given a measure that is invariant under the transition measures.((B) fc ((dr)P(r, B), B C, one can construct a stationary Markoff process defined for all values of time, positive or negative.The process obtained by reversing the direction of time is well known to be a stationary Markoff process also.If the original transi- tion measures are absolutely continuous with respect to , P,(r, ds) p(r, s)(ds), then the reversed process has for transition measures Q,(r, ds) ==p,(s, r)((ds).Let us suppose that matters stand so, and that the Q,(r, ds) satisfy the condi- tions of regularity imposed upon the P,(r, ds).There exists then a one-to-one correspondence between measures excessive for P,(r, ds) and functions ex- cessive for Q,(r, ds), as well as a dual correspondence in which the two families of transition measures exchange roles.The results of the first two install- ments may now be stated in terms of one or the other class of excessive functions, with a consequent sharpening of certain statements.One obtains in this way, for example, the representation of an excessive function as the sum of a potential and an excessive function having certain additional properties; this particular result is unfortunately somewhat misleading, for a similar representation can be proved to hold in the setting of the second installment.One can also define naturally a capacity of sets that behaves like the Newtonian capacity and has the same interpretation by means of processes.It is rather too stringent to suppose the invariance of or the absolute con- tinuity of P,(r, ds) relative to , as we have done above, for either hypothesis rules out the potential theory of the heat equation on a bounded domain.We shall treat instead the relative theory, with ( an excessive measure and certaitt