SOME PROPERTIES OF THE FEYNMAN-KAC FUNCTIONAL

The Feynman-Kac formula and its connections with classical analysis were 
initiated in the now celebrated paper [6] of M. Kac. It soon became obvious that 
the formula provides a powerful tool for solving partial differential equations by 
running the Brownian motion process. K.L. Chung and K.M. Rao in [4] used it 
to characterize solutions of the Schrodinger equation. In this paper we study 
some properties of the Feynman-Kac functional using the Brownian motion process. In particular, we are going to use it in connection with the gauge function 
in order to obtain an energy formula similar to one obtained by G. Dal Maso and 
U. Mosco in [5].


Introduction
In [5], G. Dal Maso and U. Mosco studied a relaxed Dirichlet problem in an open region f of R d d > 2 which can formally be written as: Au + u 0 in (1.1) where A is the Laplace operator and # is an arbitrary non-negative Borel measure not charging polar sets in Rd.The measure # may take the value + o.Special cases of (1.1) are Dirichlet problems of the type -Au-0inf-, u-0onE, (1.2) (E denotes the closure of E), as well as the stationary SchrSdinger equation" where q is a non-negative potential.The main result in [5] is an approximation of equation (1.1) by a sequence of Dirichlet problems of the form (1.2).In order to carry out this procedure, one needs to study the behavior of an arbitrary local weak solution u of (1.1) such that u E oc() V Loc( #) and having finite local g-energy given by: / Vu]2dx+ j'u2d#, a'ca. (1.3) The methods in [5] are variational in nature.In this paper, we study the Feynman-Kac func- tional, directly using the Brownian motion process.Using the gauge function, we obtain an ener- gy formula in part 5 similar to formula (1.3).Section 3 deals with the characterization of the null set of gq.For example, in Proposition 3.3, it is shown that the set {gq-0} is a polar set.Section 4 deals with continuity properties of the gauge function gq.Specifically, Theorem 4.2   shows that if gq is nonvanishing and continuous in , then q is in local Kato class in .Finally, Section 5 deals with the energy as introduced in [7].In addition, it is shown that if s is an exces- sive function, then its corresponding gauge function satisfies the equation Ag s gs"

Notations and Preliminaries
Throughout this paper, X {Xt; >_ 0} denotes the Brownian Motion process in Rd, d >_ 2; denotes a domain in Rd.Let q >_ 0 be measurable.Let q(t)-exp[-fq(Xs)ds], gq(X)- 0 EXe[ q(V)], where v-exit time from f.Using strong Markov property, we see that gq <_ EX[gq(XrB)], where r B -exit time from a ball with a center x.Also, it is seen that if q is bounded, then gq is continuous; hence, in general, q is upper semi-continuous.It follows that 9q is subharmonic in f.
Throughout this paper we deal with a topology on R d that is finer than the Euclidean metric topology.Namely, the fine opoloy on R d is the smallest topology on R d for which all superhar- monic functions are continuous in the extended sense.It is easily seen that the fine topology is larger than the Euclidean metric topology on Rd.So we speak of fine interior, finely continuous, etc.
Another concept which is used in the paper and which is related to the behavior of the Brownian motion process is that of a regular point of a set.Namely, given a set D denote by T the exit time for the Brownian motion from D. (Which is the same as the hitting time of the complement of D.) Then, a point a E OD is called regular for D c if pa{T 0) 1.In other words, starting at a regular point the Brownian motion hits the complement of a set in question immediately.

The Null Set of gq
In this section we study the set on which function gq vanishes.Proposition 3.1: q < c a.e. on the set {gq > 0}.
Prf: Suppose q is bounded.Then, assuming the domain f is bounded, f q(Xo)O 1 -eq(V) q(Xs)e s ds.

(3.1)
Some Properties of the Feynman-Kac Functional 3 Therefore if q is bounded, which is a consequence of (3.1) and Markov property.
For general q, let a n q An.Using (3.2), one gets, Let n tend to infinity in the above inequality.We get where qgq is defined to be zero if gq 0. Thus, qgq < oo a.e.
A more precise result is the following: Proposition 3.2: Suppose gq()-0 and that is regular for the set {gq "explosion point" for q, i.e., Vt > O, P q(Xs)ds oo

o
Proof: Let > O, F {gq >_ g} and let T be the hitting time of F.Then, >0}.Then, is an 0 gq() >_ EX[eq(T)gq(XT) T < r] implying eq(T)-0 on the set {T < r}.As 0, the hitting times decrease to the hitting time to {9q > 0} which is zero P-a.s. by assumption.This completes the proof.Pmark a.l" f gq(g)-0 and { is not regular for {gq of {gq 0}.Thus, > 0}, then must be in the fine interior {gq 0} {finely open set} U {the set of explosion points of q).l{emark a.2: Suppose gq O. Then {gq > O} is a finely open non-empty set.So, some point of {gq-0} must be a regular point for {gq > 0}.However, such a point is an explosion point for q.Thus, we can say if q has no explosion points, then gq cannot vanish unless gq =_ O.
Let us show that some point for which gq 0 is regular for {gq > 0}.Let gq() 0 and T hitting time to {gq > 0}.Since gq(Xt) is continuous in for > 0, we see that gq(XT)-0 if T < oe and X T is regular for {gq > 0}, almost surely.Hence, there are points for which gq-O, and which are regular for {gq > 0}.
ltemark 3.3: The proof of formula (3.3) is as follows.
Since gq(Xt) > O, eq(t) 0 for every t, which means that x is an explosion point.
Example: Let q(x) Ix -3, 2 </ < 3 in R3.Then, using scaling for Brownian motion we see that for any > 0, pO Xs -ds < a P Xs/e2 -ds < a o I / t e2 p e2-Xsl -ds < a 0 The last inequality holds for every > 0. It follows that i.e., 0 is an explosion point for q and, of course, gq(O)-O.
f Xs -Ods oe, Vt > O, P-a.s., 0 Note that q is locally integrable.If Some Properties of the Feynman-Kac Functional 5 {i} is a dense countable set, let qi-x-i[-" Then qi is locally integrable.For suitable constants i, q rliq E Ltoc, and Vt > 0 and Vi, pi[ f q(Xs)d s c] 1.Thus, it is possible that 9q 0 on a dense set with q integrable, o 4. Continuity Properties of gq The following remark will be helpful in the proof of Proposition 4.1.
$ Also, G(., t) is continuous for each t.Thus, if gq is continuous, G(., t) tends to gq uniformly on compacts by Dini's theorem.We conclude that 9q is continuous if and only if E(')[gq(Xt): tends to gq uniformly on compacts as tends to zero.
If h is any of 91,92 or g and r any of ql + q2, ql or q2, we have by Markov property Proof:

t<,]
The last term above clearly tends to zero uniformly on compacts because Px{r < t} does this as t0.From the last sentence of the above remark, the continuity of h is equivalent to E()[h(Xt)(1 e(t)): < r] (4.1) to tend to zero uniformly as t--0.If h-gi and r-qi, i-1,2, this is the case because gi are continuous.We have with h-g and r-ql + q2, that E()[g(Xt)(1 er(t))" < r] E(')[g(Xt)(1 eql (t) -eql (t) er(t)): < T].
Since g _< gl, E()[g(Xt)(1 eql (t)): < 7] _< E(")[gl(Xt)(1 -eql (t)): < 7-], using the conclusion stated in the last sentence of the previous remark, it follows that the left- hand side of this inequality tends to zero uniformly as t0, because the right-hand side does it, since gl is continuous.Likewise, E (")[g(Xt)(eql(t)-er(t)):t where again the left-hand side of this inequality tends to zero uniformly as t0, because the right- hand side does it, since g2 is continuous.This concludes the proof showing that gql + q2 is also continuous.
Proposition 4.2: If gq is continuous, so is g for all > O.
As in the proof of Proposition 4.1, we need only to show that if g,x g,x q EX[g(Xt)(1 e (t))" t < r] q tends to zero uniformly on compacts.or equal to (4.2) By HSlder's inequality, the expression in (4.2) is less-than {EX[g-(Xt)(1-% (t))A-l"t <: T]} A.
q q Thus, it follows from (4.3) that the expression in (4.2)is less than or equal to EX[gq(Xt)(1 eq(t)): < r]   and the result follows.
Theorem 4.1: Suppose Gq is locally bounded.Then gq continuous.
is continuous if and only if Gq is Proof."Step 1. Suppose Gq is bounded.Then gq is continuous if and only if Gq is contin- uous.To see this, write Kqf EX[ ] eq(t)f(Xt)dt The first equation shows that gq is continuous if and only if K qq is continuous, and from the second equation, we see that under the condition that Gq is bounded, the continuity of Gq is equivalent to that of K qq.
Step 2. Now suppose D is a relatively compact subset of Q.Then, G Dq <_ Gq and Gq bounded on D implies that GDq is bounded.Suppose gq is continuous.We have gq-EX[eq(rD)gq(XrD)].
gq is finely continuous so that set A-{gq-0} is finely closed.hitting time of A, we have r , Z.R. POP-STOJANOVI, and K. MURALI RAO -f