GENERALIZED DYSON SERIES AND FEYNMAN'S OPERATIONAL CALCULUS 5

and both are very different in spirit from the present paper. In partic-

ular, the connections with path integration as well as the complicated

combinatorics associated with the disentanglement that leads to our gen-

eralized Dyson series do not appear in either of [41] or [361.

It is reasonable to refer to the functional

(0.5) F(x) := exp ( / 9(s,x(s))dn(s))

(0,t)

as the Feynman-Kac functional with Lebesgue-Stieltjes measure n. I t is

natural to ask if the corresponding operator, considered as a function of

time, satisfies a differential equation analogous to the heat or Schro-

dinger equations. This is the case, as is shown by Lapidus in [33, 34]

where a "Feynman-Kac formula with a Lebesgue-Stieltjes measure" is

established and related results are given. (See Kac's papers [23; 24,

pp. 62-65] for the classical Feynman-Kac formula.) For an exponential

functional of the type (0.5), for instance, the study conducted in [33,

34] reveals the distinct roles played by the continuous part and the dis-

crete part of n. I t also makes explicit connections with the theory of

the product integral [5].

We now describe briefly the organization of this paper. In the

remainder of the present section, we introduce notation and give two

preliminary lemmas.

In Section 1, we discuss the prototypical example n - y + ^T men-

tioned above; our most detailed analytic proofs are given in this case.

Generalized Dyson series for the full class of functionals treated

in this paper are obtained in Section 2. Some readers might wish to con-

sult this section only briefly on a firs t reading.

Section 3 may be particularly helpful to the reader as i t deals with

a variety of concrete examples of perturbation expansions. The emphasis

in Sections 2 and 3 is largely on the combinatorics.

In Section 4, we give theorems insuring stabilit y with respect to the

potentials and with respect to the measures. We also give some applica-

tions of the stabilit y theorems for measures.

We present, in Section 5, a graphical representation of our gen-

eralized Dyson series in terms of generalized Feynman diagrams.