Introduction

Mark Kac [32, 33, 13], introduced a method for calculating the distribution of the integral

for a function v defined on the state space E of a Markov process , and a time T that may be fixed or random. In [32] and [33] Kac considered the case when X is a Brownian motion (BM), but his method leading to the Feynman-Kac formula in that setting has since been developed and applied much more generally. See [62, 11, 14, 34, 64] for textbook treatments of the F-K formula for BM, Section III.19 of [57] for a modern treatment of the F-K formula for a Feller-Dynkin process, and Section 5 of [37] for a survey with further references. In this paper we review an aspect of Kac's method not mentioned in these treatments. This is his formula for moments of for suitable T, first derived in [33] for BM on the line, then generalized in [13] to a Markov process with abstract state space. The reader is not assumed to be acquainted with the modern theory of Markov processes beyond what can be found, for example, in Chapter III of [57].

To state the basic form of Kac's formula in some generality, let be the family of probability measures governing a Markov process set up on a suitable probability space ; is the law of X under the initial condition . We assume that there is a -algebra on E such that (i) is -measurable for each , and (ii) is a -measurable mapping of into E, where is the Borel -algebra on . It is convenient to assume that accommodates a random variable ( ) which (under for all ) is independent of X, and has the exponential distribution with parameter . Other random times T involving extra randomization may also be assumed to be defined on the same basic setup.

Call T a Markov killing time of X if under each the killed process is Markovian with (sub-Markovian) semigroup :

 

In addition we assume that is -measurable for all t>0 and all positive -measurable f. In formula (2) (and elsewhere in the paper), 1(B) is the indicator of the event B and serves double duty as the expectation operator for the probability measure . Define the Green's operator or potential kernel G associated with T by

 

for non-negative -measurable f. For example, is a Markov killing time with , where is the semigroup of X, in which case is the resolvent or -potential operator associated with . Other Markov killing times are , and T the first entrance or last exit time of a suitable subset B of the state space of X. A finite fixed time T is typically not a Markov killing time unless X is set up as a space-time process, so T becomes a hitting time. Other Markov killing times can be constructed (i) by killing the process at state-dependent rate for some killing rate function k defined on E, (ii) by killing according to a multiplicative functional, and (iii) by combinations of these kinds of operations. See [8]. As shown by the example of last exit times, a Markov killing time of X is not necessarily a stopping time. See [47, 60] for further examples in this vein.

Kac's moment formula [33, 13] Let T be a Markov killing time for X, let be an arbitrary initial distribution on E, and let v be a non-negative measurable function on E. Then the moment of under is given by

 

where , G is the potential kernel for the killed process as in (3), and stands for the function that is identically 1.

In terms of operators, where is the operator of multiplication by v. For n=1, formula (4) just restates the definition (3) of the Green's operator G. For n=2 the formula reads

 

Note the special case in (4): , , so (4) becomes

 

The first appearance of formula (4) seems to be (3.5) in Kac [33]. There is a space-time BM derived from a one-dimensional BM B, and T is a fixed time. Darling-Kac [13] (page 445, line 4) give the Laplace transformed version of the same formula for B a two-dimensional BM, which amounts to the present formula (4) for X = B and . The formula (4) for general X and v, and , is implicit in the discussion on page 446 of [13], and is used there for an asymptotic calculation of moments which identifies the limit distribution of as for a large class of Markov processes. See [7, 3] for more recent developments in this vein. To illustrate with three more examples from the literature, Exercise 4.11.10 of Itô -McKean [30] is (6) for X a one-dimensional diffusion and T the first exit time from an interval; Nagylaki [48] gives the more general formula (4) in the same setting; Propositions 8.6 and 8.7 of Iosifescu [29] are (4) for X a Markov chain, , and v either the indicator of the set of all transient states, or the indicator of a single transient state.

As noted by Kac [33] in the Brownian setting, summing the moment formula (4) weighted by 1/n! yields the

Feynman-Kac formula [21, 32] For ,

 

where is the minimal positive solution f of

 

Informally, we may write

 

The meaning of has just been precisely defined for , but this expression also makes sense for signed v under appropriate conditions.

Note that replacing in (7) by for gives an expression for the moment generating function of , which may however diverge for all . If the m.g.f. does converge for some , it of course determines the distribution of . But even if not, Kac's moment formula still allows evaluation of whatever moments of are finite.

Khas'minskii [38] found (4) and (7) for a general X and T the exit time of a domain. He also noted the following immediate consequence of (4) which has found numerous applications in the theory of Schrödinger semigroups [2, 63]. See also [5, 9, 49] for various refinements and further references.

Khas'minskii's condition If is bounded then for all x the moment generating function converges for .

If the infinitesimal generator of the semigroup is a differential operator (such as for Brownian motion), then integral equation (8) can be recast as a differential equation subject to suitable boundary conditions depending on the nature of T. For details in various settings see [14, 34, 64] and Section 13.4 of [16]. Ciesielski-Taylor [12] used (7) to derive the distribution of for X a BM in for , and v the indicator function of a solid sphere in , in which case represents the total time spent by B in the sphere. See also [57] Section III.20 for a different treatment.

The rest of this paper is organized as follows. Kac's moment formula as stated above is proved in Section 2. Some variations and corollaries are presented in Section 3. In Section 4 we explain how these results relate to the more customary statement of the F-K formula that the semi-group of the process obtained by killing X at rate v(x) has infinitesimal generator . In Section 5 the general results are specialized to the context of a Markov chain with finite state space, where the F-K formula can be understood with almost no calculation by direct probabilistic argument. In Section 6 we point out how the F-K formula for occupation times of Markov chains applies to local times of more general Markov processes. Such formulae were the basis of Ray's [55] derivation of the Ray-Knight description of the local time field of a one-dimensional diffusion evaluated at a Markov killing time T, and of calculations by Williams [65, 66] for Markov chains.