The energy functional, balayage, and capacity.

*(English)*Zbl 0617.60073The paper contains investigations of the relationship between the energy functional and the balayage operations in the context of right processes. The results are applied in particular to the discussion of capacity and cocapacity as defined by the authors, Probab. Theory Relat. Fields 73, 415-445 (1986; Zbl 0586.60069) and discussed furthermore by the second author, Prog. Probab. Stat. 13, 195-213 (1987; see the foregoing review, Zbl 0617.60072). Special interest is in the study of capacities associated with the q-subprocesses and their behaviour as functions of q.

Assume a right process on \((E,{\mathfrak E})\) and an excessive measure m, let \(B\in {\mathfrak E}\), and define \(\Gamma^ q(B):=L^ q(m,P_ B^{q_ 1})=L^ q(R_ B^{q_ m},1)\), where \(L^ q\) denotes the energy functional, and \(P^ q_ B\) and \(R^ q_ B\) the balayage operators on functions and measures, w.r.t. the q-subprocess. Let C and \(\hat C\) denote the capacity set functions as defined in the first quoted paper. Then it is shown that \(\Gamma:=\Gamma^ 0\) defines the proper capacity extension of both C and \(\hat C\) to arbitrary sets in \({\mathfrak E}.\)

Furthermore, relations between \(\Gamma^ q\) and \(\Gamma^ r\), and between \(\Gamma^ q\) and C and \(\hat C\) are established. One main result states that if \(\Gamma^ q(B)<\infty\) for some \(q>0\), then as q tends to zero, \(\Gamma^ q(B)\) decreases to \(\Gamma(B)\), which moreover equals \(C(B)=\hat C(B)\). Another more elementary proof for this latter result has meanwhile been given by the authors [More about capacity and excessive measures. Semin. Stochastic Processes 1987, to appear], a paper that contains further results on capacities.

The results are presented in wider generality, in so far as the function 1 is replaced by some m-a.e. finite excessive function u; and the interplay between Kuznetsov measures \(Q^ u_ m\), the w.r.t. u Doob- transformed semigroup, the functional L, and the associated balayage operations is investigated. This yields several relations, which are of interest in themselves as well.

Assume a right process on \((E,{\mathfrak E})\) and an excessive measure m, let \(B\in {\mathfrak E}\), and define \(\Gamma^ q(B):=L^ q(m,P_ B^{q_ 1})=L^ q(R_ B^{q_ m},1)\), where \(L^ q\) denotes the energy functional, and \(P^ q_ B\) and \(R^ q_ B\) the balayage operators on functions and measures, w.r.t. the q-subprocess. Let C and \(\hat C\) denote the capacity set functions as defined in the first quoted paper. Then it is shown that \(\Gamma:=\Gamma^ 0\) defines the proper capacity extension of both C and \(\hat C\) to arbitrary sets in \({\mathfrak E}.\)

Furthermore, relations between \(\Gamma^ q\) and \(\Gamma^ r\), and between \(\Gamma^ q\) and C and \(\hat C\) are established. One main result states that if \(\Gamma^ q(B)<\infty\) for some \(q>0\), then as q tends to zero, \(\Gamma^ q(B)\) decreases to \(\Gamma(B)\), which moreover equals \(C(B)=\hat C(B)\). Another more elementary proof for this latter result has meanwhile been given by the authors [More about capacity and excessive measures. Semin. Stochastic Processes 1987, to appear], a paper that contains further results on capacities.

The results are presented in wider generality, in so far as the function 1 is replaced by some m-a.e. finite excessive function u; and the interplay between Kuznetsov measures \(Q^ u_ m\), the w.r.t. u Doob- transformed semigroup, the functional L, and the associated balayage operations is investigated. This yields several relations, which are of interest in themselves as well.

##### MSC:

60J45 | Probabilistic potential theory |