EIGENVALUE EXPANSIONS FOR BROWNIAN MOTION WITH AN APPLICATION TO OCCUPATION TIMES

: Let B be a Borel subset of R d with (cid:12)nite volume. We give an eigenvalue expansion for the transition densities of Brownian motion killed on exiting B . Let A 1 be the time spent by Brownian motion in a closed cone with vertex 0 until time one. We show that lim u ! 0 log P 0 ( A 1 < u ) = log u = 1 =(cid:24) where (cid:24) is de(cid:12)ned in terms of the (cid:12)rst eigenvalue of the Laplacian in a compact domain. Eigenvalues of the Laplacian in open and closed sets are compared. Abstract . Let B be a Borel subset of R d with (cid:12)nite volume. We give an eigenvalue expansion for the transition densities of Brownian motion killed on exiting B . Let A 1 be the time spent by Brownian motion in a closed cone with vertex 0 until time one. We show that lim u ! 0 log P 0 ( A 1 < u ) = log u = 1 =(cid:24) where (cid:24) is de(cid:12)ned in terms of the (cid:12)rst eigenvalue of the Laplacian in a compact domain. Eigenvalues of the Laplacian in open and closed sets are compared.

volume. Multidimensional Brownian motion will be denoted X t . For a Borel set B ∈ R d , let τ (B) = τ B = inf{t > 0 : X t / ∈ B} and T (B) = T B = inf{t > 0 : X t ∈ B}.
Let p B (t, x, y) be the transition densities for Brownian motion killed on exiting B, let G B (x, y) be the corresponding Green function, and set P B t f(x) = B p B (t, x, y)f(y) dy. For definitions and further information, see Bass [B], Sections III.3, III.4. We use f, g to denote f(x)g(x) dx. The sphere in R d with center y and radius r will be denoted S(y, r), while the corresponding open ball will be denoted B(y, r).
Theorem 1.1. Suppose B ⊂ R d is a Borel set whose Lebesgue measure is finite and positive and let µ denote the restriction of the Lebesgue measure to B. There exist reals 0 < λ 1 ≤ λ 2 ≤ · · · < ∞ and a complete orthonormal system ϕ i for L 2 (B) such that (i) the sequence {λ i } has no subsequential limit point other than ∞, (ii) for each t we have P B t ϕ i = e −λ i t ϕ i , µ-a.e., (iii) if f ∈ L 2 (B), then the convergence is absolute, and the convergence takes place in L ∞ (B), (iv) we have the expansion for µ 2 -almost every pair (x, y), the convergence is absolute, and the convergence takes place in L ∞ (B × B), and (v) if for some t > 0 we have p B (t, x, y) > 0 for µ 2 -almost every pair (x, y), then λ 1 < λ 2 and ϕ 1 > 0, µ-almost everywhere.
Theorem 1.2. Let J n be Borel subsets of R d with finite volume decreasing to a compact set K. The eigenvalues λ(J n ) converge to λ(K) as n → ∞.
It is standard to adapt the proof of Theorem 1.1 to see that a result analogous to Theorem 1.1 holds for a compact subset K of a sphere S(y, r) such that S(y, r) \ K has a non-empty interior relative to S(y, r). In this case, the transition probabilities p K (t, x, y) are those of the Brownian motion on S(y, r). Suppose that K is a subset of S(0, 1) and let λ(K) be the first eigenvalue (i.e., λ 1 ). Then let v = d/2 − 1 and ξ(K) = (v 2 + 2λ(K)) 1/2 + v λ(K) .
Let C be a closed cone in R d with non-empty interior and vertex (0, . . . , 0) and let J be the closure of S(0, 1) \ C. We will write i.e., A t is the amount of time spent by X inside C before time t.
Theorem 1.3. Assume that the d-dimensional Lebesgue measure of the boundary of C is zero and that both C and its complement have non-empty interiors. Then It seems that the only case when an explicit formula for P 0 (A 1 < u) is known is when C is a half-space, where the distribution is known as the "arc-sine law." See Bingham and Doney [BD] for related results. Meyre and Werner [MR] proved that P 0 (A 1 < u) is comparable to u 1/ξ(J) , but they had to assume that C is convex. Meyre [M] considered more general cones but made a strong assumption of regularity on the boundary. Our paper is inspired by a question posed by W. Werner. We are grateful for discussions with P. Baxendale, E.B. Davies, R. Howard, P. March, M. van den Berg, W. Werner, and Z. Zhao.
2. Eigenvalue expansions. In this section we prove Theorem 1.1, 1.2, and also give a result (Proposition 2.1) on hitting times which is of independent interest.
Proof of Theorem 1.1. For each t, p B (t, x, y) ≤ p(t, x, y), the density of unkilled Brownian motion, which is bounded by a constant depending only on t. Since B has finite volume, B B p B (t, x, y) 2 dx dy < ∞. By Riesz and Sz.-Nagy [RN], page 179, P B t is a completely continuous operator, that is, the image of the unit ball in L 2 (B) under P B t is a set whose closure in L 2 is compact. By the Hilbert-Schmidt expansion theorem (Riesz and Sz.-Nagy [RN] or Bass [B], Section III.4), P B t has an eigenvalue expansion as in (iii) and (iv). The fact that the eigenvalues of P B t are of the form e −λ i t , that the λ i and ϕ i do not depend on t, and that (i) holds may be proved as in Bass [B] or Port and Stone [PS], using the complete continuity of P B t in L 2 in place of the equicontinuity of By the semigroup property and Parseval's identity applied to f(y) = p B (t/2, x, y), Integrating over B, for all t > 0. Since (2.2) holds for all t, this and (2.1) imply that the convergence in (iii) and (iv) is absolute and takes place in L ∞ (B) and L ∞ (B × B), respectively. Assertion (v) is an immediate consequence of the Krein-Rutman theorem [KR].
The hypothesis of finite volume is sufficient, but not necessary and sufficient; see, e.g., [vdB]. On the other hand, some restriction on B is required for the conclusion of Theorem 1.1 to hold, as the case B = R d shows.
Proposition 2.1. Suppose K ⊂ R d is a compact set.
(i) There exist λ, c 1 ∈ (0, ∞) such that P x (τ K ≥ t) ≤ c 1 e −λt for all x ∈ K and all t > 0; (ii) There exists c 2 ∈ (0, ∞) and a Borel set L contained in K such that P x (τ K > t) ≥ c 2 e −λt for all x ∈ L and all t > 0, where λ is the same constant as in (i). (iii) Suppose there exists a ball B = B(x 0 , r) contained in the interior of K such that for almost every x that is not regular for K c , Then there exists c 3 ∈ (0, ∞) and an open set M contained in K such that P x (τ K > t) ≥ c 3 e −λt for all x ∈ M and all t > 0, where λ is the same constant as in (i) and (ii).
Proof of Proposition 2.1. Let us apply Theorem 1.1(iii) with the function f = 1 K to get for a.e. x, Let |K| denote the Lebesgue measure of K. Note that By (2.1) and (2.2) of the proof of Theorem 1.1 it follows that From these estimates, there exists c 5 such that for t large, with c 7 independent of t and x. Property (i) follows easily from this with λ = λ 1 .
We turn next to the proof of (iii). There are at most finitely many i such that λ i = λ 1 because {λ i } has only ∞ as a subsequential limit point. Φ cannot be equal to 0, a.e., because that would contradict the linear independence of the ϕ i . Since P K t 1 K ≥ 0 for all t and Ψ(t, x) = o(Φ(x)) for almost every x on the set where Φ = 0, we must have Φ ≥ 0, a.e. Hence Φ must be positive on a set of positive measure. Note We conclude that {x ∈ K : x is not regular for K c , Φ(x) > 0} has positive measure.
Suppose x is not regular for K c and P x (T B < τ K ) > 0. By the strong Markov property and the support theorem for Brownian motion, So for all y ∈ B, G K (x, y) > 0 for almost all x that are not regular for K c . Therefore for y ∈ B, the ball with the same center as B but half the radius, there exists δ > 0 such that 3), there exist c 8 > 0 and t 0 such that for t ≥ t 0 − 1, If t ≥ t 0 and x ∈ M, since p B (1, x, y) is bounded below for x, y ∈ M. Property (iii) follows for t ≥ t 0 with the same λ as in (i), namely, λ = λ 1 . To remove the restriction involving t 0 , note that if t < t 0 , then Finally, we show (ii). By the first paragraph of the proof of (iii), Φ is positive on a set of positive measure. Hence there exists c 11 > 0 and a Borel set L contained in K such that L has positive measure and Φ > c 11 on L. The conclusion (ii) follows from this similarly to the above.
Proof of Theorem 1.2. Applying (2.3) with t − n −1 and letting n → ∞, we get a.e., by (2.1), (2.2), and dominated convergence. Hence, for all x and t, Since the events {τ J n ≥ t} are decreasing in n, the same is true for the probabilities a.e., from which it follows that λ(J n ) is an increasing function of n.
Suppose that λ * = lim n→∞ λ(J n ) is strictly smaller than λ(K). So there exists ε > 0 such that λ(J n ) < λ(K) − ε for all n. We will show that this assumption leads to a contradiction.
Let ϕ n 1 be the eigenfunction corresponding to the first eigenvalue in J n . Recall from the proof of Theorem 1.1 that and so we can find a constant c 1 such that for all x ∈ J n and all n sufficiently large. Since ϕ n 1 (x) ≤ c 1 but the L 2 (J n ) norm of ϕ n 1 is equal to 1, for large n we can find c 2 , c 3 > 0 and a set M = M(n) ⊂ K such that the Lebesgue measure of M is greater than c 3 and ϕ n 1 (x) > c 2 for every x ∈ M.
Let |K| denote the Lebesgue measure of K and let µ be the probability measure on K obtained by renormalizing the Lebesgue measure restricted to K. Then, for n large, If n is sufficiently large, for all t, On the other hand, if n is sufficiently large, Since the events {τ K < t < τ J n } decrease to ∅, the right hand side tends to 0 as n → ∞, a contradiction.
3. Time spent in a cone. This section is devoted to the proof of Theorem 1.3. In this section we want to consider eigenvalue expansions for Brownian motion on S(y, r) killed on exiting a subset B of S(y, r). The proofs of Section 2 are easily adapted to this situation; we leave it to the reader to supply the details, and we apply the results of Section 2 without further mention.
(i) There is an open set M ⊂ S(0, 1) and a constant c 1 > 0 such that for every x with x/|x| ∈ M we have for t ≥ |x| 2 , (ii) There exists c 2 < ∞ such that for all x = 0, Proof. Lemma 3.1 follows from Proposition 2.1 in the same way as Proposition 2.3 follows from Proposition 2.2 in Meyre [M].
Proof of Theorem 1.3.
Step 1. First we will show that there is c 1 > 0 such that for all In other words, with probability greater than c 1 the cone is hit at a place at least half as far from the origin as the starting point. The constant c 1 may depend on C δ * and C but does not otherwise depend on C.
The cone C δ * has non-empty interior so it is easy to see that for every point y ∈ S(0, 2|x|), Brownian motion starting from y will hit C δ * before hitting S(0, |x|) ∪ S(0, 3|x|) with probability greater than c 2 > 0. If R T > |x|/2 then either X T ∈ C or X T ∈ S(0, 2|x|). By applying the strong Markov property at T we conclude that which proves (3.1).
Step 2. Suppose that 0 < a < r. Assume that C is a cone with C δ * ⊂ C ⊂ C and for all x ∈ S(0, a). Let C = {y ∈ C : |y| ≥ a/2}. We will prove that for all x ∈ S(0, a) where c 3 may depend on C and δ * but does not otherwise depend on C, a or r. Let T 0 = 0, S k = inf{t > T k : X t ∈ C ∪ S(0, a/2)}, k ≥ 0, Typically, S k and T k are finite for small k and infinite for large k in dimensions higher than 2. For the event {T k < T C } to happen, the process would have to return k times to S(0, a) and after each return it would have to hit S(0, a/2) before hitting C. A repeated application of the strong Markov property and (3.1) yield for x ∈ S(0, a), (1 − c 1 ) k ρ ≤ c 3 ρ and the proof of (3.2) is complete.

A standard estimate gives
for small s and so We also have P x (T C > T S(0,s α ) ) ≤ P x (T C > s 3α ) + P x (T S(0,s α ) < s 3α ).
It is easy to see that P x (T S(0,s α ) < s 3α ) ≤ exp(−s −α/2 ) for small s, so it follows that Recall that δ * > 0 is small and let ξ * = ξ(J δ * ). According to Lemma 3.1 (ii) there exists c 4 such that for all δ ∈ (0, δ * ), C = C δ , all t > 0 and all x ∈ S(0, a), We obtain from (3.2), for all x ∈ S(0, a). We apply this formula with t = s 3α and combine it with (3.3) to obtain for small s. Now we let δ = s α/2 (we consider only small s). Note that for small s, the distance between C δ and ∂C is greater than s 1/2−α . A standard estimate for Brownian motion shows that for x ∈ C δ and small s, P x (T ∂C < s 1−α ) ≤ exp(−s −α/2 ) and so Another standard estimate gives for small a, P 0 (T S(0,a) > 1/4) ≤ exp(−a −1 ).
We combine our estimates to see that for small s, P 0 (A 1 < s 1−α ) ≤ P 0 (T S(0,a) > 1/4) If we substitute u = s 1−α , we obtain It follows that lim inf We proved in Theorem 1.2 that lim δ→0 ξ(J δ ) = ξ(J ). Hence, by choosing sufficiently small δ * > 0 we can assume that ξ * = ξ(J δ * ) is arbitrarily close to ξ(J ). This and the fact that α may be chosen arbitrarily close to 0 show that This proves the lower bound in Theorem 1.3.
Step 4. Next we prove the opposite inequality. Find a set M as in Lemma 3.1 (i) and let c 7 be equal to the c 1 in that same lemma. Let M 1 = {x : x/|x| ∈ M} and a = u 1/2 . The probability p = P 0 (A T S(0,a) < u, X(T S(0,a) ) ∈ M 1 ) does not depend on u, by scaling, and it is strictly positive.
Let C o denote the interior of C. Recall that we have assumed that the boundary of C has zero d-dimensional Lebesgue measure. Hence, This, the strong Markov property applied at T S(0,a) and Lemma 3.1 (i) imply that It follows that lim sup u→0 log P 0 (A 1 < u)/ log u ≤ 1/ξ(J ).
The proof of Theorem 1.3 is complete. applies to many sets that have empty interior (see Example 4.1 below) and "Laplacian" has to be defined using, for example, Brownian transition probabilities. It is natural to ask about the relationship between eigenvalues in open and closed sets. It is easy to prove that the spectrum for a compact sets is the same as the spectrum for its interior if the common boundary of these sets is smooth. This section is devoted to a discussion of what happens for non-smooth sets. We will be concerned only with the first eigenvalue and denote it λ. We start with a simple example of a highly irregular set showing that the first eigenvalue for a compact set and its interior can be different.

Eigenvalues of the
be an ordering of all points with rational coordinates in Q, except (0, 0). For each i, Then Brownian motion starting from (0, 0) does not hit K c immediately a.s. and so the spectrum given in Theorem 1.1 is non-trivial. At the same time, the interior of K is empty.
For the rest of this section we will consider only compact sets K such that K is the closure of the interior D of K. It is clear from the above example that typically λ(D) = λ(K) if we do not make this assumption.
For a compact set K, the set of all x ∈ ∂K such that P x (T K c = 0) = 0 will be denoted I(K). In other words, I(K) is the set of points which are not regular for K c .
The following result has been inspired by discussions of eigenvalue continuity with R. Howard. We will only sketch its proof.
Proposition 4.2. Suppose that K is a compact set and K is the closure of its interior D. Then λ(D) = λ(K) if and only if I(K) is polar.
Proof. Let D n = B(D, 1/n) and K n = B(K, 1/n). The sets D n are open and K n are compact. It is easy to check that D n ⊂ K n ⊂ D n−1 . Hence λ(D n ) ≤ λ(K n ) ≤ λ(D n−1 ). The monotonicity of λ follows, for example, from the first paragraph of the proof of Theorem 1.2. We see that lim n→∞ λ(D n ) = lim n→∞ λ(K n ).
By Theorem 1.2, λ(K) = lim n→∞ λ(K n ). Thus it will suffice to show that lim n→∞ λ(D n ) = λ(D) if and only if I(K) is polar.
That the polarity of I(K) implies lim n→∞ λ(D n ) = λ(D) was proved by Le Gall [LG]. Now we will sketch how to prove the opposite implication. Assume that I(K) is not polar. We modify the proof of Theorem 1 in Gesztesy and Zhao [GZ] as follows.
We will discuss only the case when the boundary of D has zero d-dimensional Lebesgue measure. This is the case covered by the second part of the proof in [GZ]. The other case can be adapted to our purposes in an analogous manner.
Suppose that x ∈ D and consider a Brownian bridge X starting at x at time 0 and returning to x at time 1. The path properties of a Brownian bridge are the same as those of Brownian motion away from the starting point and end point. Since I(K) is non-polar, there is a positive probability that X will hit I(K). The strong Markov property applied at the hitting time of I(K) may be used to show that X does not enter K c just after hitting I(K) with positive probability. Then we can use time reversal to see that X may hit I(K) before hitting K c with positive probability. Another application of the strong Markov property shows that {X t , 0 ≤ t ≤ 1} may hit I(K) but not hit K c with positive probability. In terms of unconditioned Brownian motion, this implies that for x ∈ D, Hence This is the analogue of (40) in [GZ]. The same argument as in [GZ] then implies for some t > 0 and ε(x) > 0. This is the analogue of (43) in [GZ].
Here η > 0 may be chosen independently of n because δ(x) does not depend on n. This completes the argument.
Can one describe in geometric terms all sets K with I(K) polar? We do not have a complete characterization but two partial results should shed some light on the problem. The first result is due to R. Howard who has an analytic proof. We supply our own proof which is probabilistic and seems to illustrate well the special role of the main assumption of the proposition. We would like to point out that if K ⊂ R d , d ≥ 3, and ∂K is locally the graph of a continuous function then the boundary of K may contain infinitely many points x which are irregular for K c , i.e., such that P x (T K c = 0) = 0. A standard example of such a point is the "Lebesgue thorn." If d = 2 then there are no such points.
Proof. Suppose that f : Suppose that x ∈ ∂D is regular for D c . Then P xa.s. there exist times t n , n ≥ 1, such that t n > 0, t n → 0 and X(t n ) ∈ D c for every n. Let Y (t) = X(t) − t · (0, . . . , 0, 1), i.e., Y is a Brownian motion with drift. We have Y (t n ) = X(t n )−t n ·(0, . . . , 0, 1) ∈ K c since f is a continuous function and X(t n ) ∈ D c . We see that Brownian motion with constant drift starting from x hits K c immediately with probability 1. The distributions of Brownian motion and Brownian motion with constant drift are mutually absolutely continuous on the finite time interval [0, 1] so we conclude that Brownian motion starting from x hits K c immediately with probability 1. We have shown that if x is regular for D c then it is also regular for K c .
The set of all points in ∂D which are irregular for D c is polar (Blumenthal and Getoor [BG]) so I(K) is polar.
It is easy to adapt the proof to the case when the boundary can be represented only locally as the graph of a function.
The next example goes in the opposite direction to Proposition 4.3. If the boundary of K is represented locally by a bounded function, I(K) need not be polar.
We will identify R 2 with {(x 1 , x 2 , x 3 ) ∈ R 3 : x 3 = 0} and similarly for subsets of R 2 . Let and let M 1 be the orthogonal projection of M on R 2 . Fix a base point z = (0, 0, 1/2) ∈ Q. Let p = P z (T M < T ∂Q ).
It is clear that p is strictly positive. Choose a sequence of distinct points y k = (y 1 k , y 2 k ) ∈ R 2 , k ≥ 1, such that each point of M 1 is an accumulation point of the sequence {y k } but there are no accumulation points outside the closure of M 1 . Let F k (r) = {(x 1 , x 2 , x 3 ) ∈ Q : x 3 ≤ 0, |x 1 − y 1 k | ≤ r, |x 2 − y 2 k | ≤ r}.
For a fixed k, P z (T F k (r) < T ∂Q ) goes to zero as r → 0. We choose r k > 0 so small that P z (T F k (r k ) < T ∂Q ) < p/2 k+1 for every k > 0. Moreover we choose r k so small that the sets F k (r k ), k ≥ 1, are disjoint.
We let D = Q \ M ∪ k F k (r k ) .
We will now verify that properties (i)-(iv) hold for D.
(i) Let G be the projection of M ∪ k F k on R 2 . Then let f be equal to 0 on G and equal to −1 otherwise. It is easy to see that (i) is satisfied by this function f and domain D.
(ii) The boundary of D is a subset of ∂Q ∪ M ∪ k ∂F k . Hence, ∂D is a subset of the countable union of sets whose three-dimensional Lebesgue measure is zero.
(iii) Every open set is a subset of the interior of its closure. Recall that every point of M 1 is a cluster point of the sequence {y k }. Hence every point of M is a cluster point of some points in the interiors of F k 's. This implies that no point of M may belong to the interior of K. It is evident that no other point of D c may belong to the interior of K.
By applying the strong Markov property at T M we conclude that there is a non-polar subset M of M \ ∂Q such that for every x ∈ M we have P x (T K c > 0) ≥ P x (T ∂Q < T K c ) > 0.
The referee for this paper suggested the following two problems.
Problem 1. If K is compact and is equal to the closure of its interior, estimate |λ(D) − λ(K)| in terms of the capacity of I(K).

Problem 2.
Is it possible to define the Dirichlet Laplacian in terms of "generalized" derivatives at most points of a compact set and extend the theory of Sobolev spaces to such sets?