A RAY KNIGHT THEOREM FOR SYMMETRIC MARKOV PROCESSES

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1 The Annals of Probability, Vol. 8, No. 4, A RAY KNIGHT THEOREM FOR SYMMETRIC MARKOV PROCESSES By Nathalie Eisenbaum, Haya Kaspi, Michael B. Marcus, 1 Jay Rosen 1 and Zhan Shi Université Paris VI, Technion, City College, CUNY, College of Staten Island, CUNY, Université Paris VI et X be a strongly symmetric recurrent Markovprocess with state space S and let t denote the local time of X at S. For a fied element in the state space S, let } τt =inf s s >t The -potential density, u y, of the process X killed at T = inf s X s = is symmetric and positive definite. et η =η S be a mean-zero Gaussian process with covariance E η η η y =u y The main result of this paper is the following generalization of the classical second Ray Knight theorem: for any b R and t>, } ( ) τt + 1 η + b S = 1 } η + t + b S in law A version of this theorem is also given when X is transient. 1. Introduction. The goal of this paper is to generalize the second Ray Knight theorem to all strongly symmetric Markovprocesses with finite 1-potential densities. In [13], Chapter XI, Section, the authors use the terminology second Ray Knight theorem to describe the law of the Brownian local time process τt, where τt =inf s s >tis the rightcontinuous inverse of the mapping t t. In its standard formulation, this theorem says that for any fied t>, the process τt, under the measure P, is a Markovprocess in and identifies this process as a squared Bessel process of dimension starting at t. For a general Markov process X, the methods of [6] can be used to show that the analogous local time process will not be a Markovprocess in the state variable unless X has continuous paths. There is an alternate formulation of the second Ray Knight theorem. It states that for any t>, under the measure P P, τt + 1 B } = ( 1 B + t ) } (1.1) in law Received June 1999; revised March. 1 Supported in part by grants from the NSF and PSC-CUNY. AMS 1991 subject classification. 6J55. Key words and phrases. ocal time, Markovprocesses, Ray Knight theorem. 1781

2 178 EISENBAUM, KASPI, MARCUS, ROSEN AND SHI where B / is a real-valued Brownian motion, with measure P, independent of the original Brownian motion. The equivalence of this with the standard formulation mentioned above is a consequence of the additivity property of squared Bessel processes, (see, e.g., [13], Chapter XI, Theorem 1.). (1.1) is the version of the second Ray Knight theorem which we generalize to strongly symmetric Markovprocesses by replacing B by a different meanzero Gaussian process. et X = F t X t P t R +, denote a strongly symmetric standard Markovprocess with state space S, which is a locally compact separable metric space, and with reference measure m. The full definition of these terms is given in [9]. Strong symmetry means that for each α> and S, the potential measures (1.) U α A = e αt E 1 A X t dt are absolutely continuous with respect to the reference measure m, and we can find densities u α y, for the potential measures, which are symmetric in y. Throughout this paper we assume that the densities u α y are finite for some, hence all α>. This is the necessary and sufficient condition for the eistence of local times; see, for eample, Theorem 3. of [9]. et t denote the local time of X at S. Heuristically, t is the amount of time that the process spends at, up to time t. We can define t (1.3) t = lim f ε X s ds ε where f ε is an approimate delta function at. Specifically, we take the support of f ε to be B ε, the ball of radius ε centered at and f ε y dmy =1. Convergence here is locally uniform in t, almost surely. Hence t inherits from t f ε X s ds both continuity in t and the additivity property: t+s = t + s θ t. et denote a fied element in the state space S and let (1.4) τt =inf s s >t τt is the right-continuous inverse of the mapping t t et Y denote the process X killed when it first hits. Y is a strongly symmetric Markovprocess. We use u α y to denote its α-potential densities. These densities, and in particular the -potential density u y are finite and positive definite. See emma 5.1. et η =η S be a mean-zero Gaussian process with covariance (1.5) E η η η y =u y (We define P η to be the measure for η and denote the corresponding epectation operator by E η.) The main result of this paper is the following generalization of the classical second Ray Knight theorem (1.1).

3 RAY KNIGHT THEOREM 1783 Theorem 1.1. et X = F t X t P be a strongly symmetric recurrent Markov process and let τt S be the local time process for X as defined above. For any b R and t>, under the measure P P η τt + 1 η + b S } = 1 ( η + t + b ) } (1.6) S in law. When X is a real-valued symmetric évy process, the Gaussian process in Theorem 1.1 has stationary increments, with η = In particular when X is a real-valued symmetric stable process of inde β 1 the Gaussian process is a fractional Brownian motion of inde β 1 This is shown in Section 6. Theorem 1.1 is reminiscent of Dynkin s isomorphism theorem [3], which is used in [9], [1], [4] and [5] to obtain many interesting facts about the local time process t S by eploiting properties of associated Gaussian processes. However, Dynkin s theorem refers to the total local time λ S, for the process killed at an independent eponential time λ, orto S for transient processes. Because of this Dynkin s theorem lends itself primarily to the study of properties of t S which are uniform in t. Theorem 1.1 allows one to obtain results about t S which may vary with t. Theorem 1.1 is used in [1] to show that the most visited site up to time t, of a symmetric stable process, is transient. This result is etended to a larger class of évy processes in [8]. In [11], Theorem 1.1 is used to give a relatively simple proof of the necessary and sufficient condition for the joint continuity of the local time of symmetric évy processes. (Actually, the proofs in [11] are also valid for all strongly symmetric Markov processes.) In Theorem 1.1 the Markovprocess X is taken to be recurrent. If it is transient, then with probability 1, < Consequently, (1.6) can not hold for transient processes. To see this, note that it follows immediately from (1.6) that τt = t for all t, which is impossible if t> We now present a generalization of (1.6) so that it also holds when X is transient. When X is transient, its -potential, u is finite and, under P is an eponential random variable with mean u. When X is recurrent, its -potential, u is infinite. In the net theorem we define ρ to be an eponential random variable with mean u, when u < and set ρ to be identifically equal to when u =. et τ t =inf s s t, the left-continuous inverse of t t et η be as defined in (1.5) and take η and ρ to be independent. Theorem 1.. et X = F t X t P be a strongly symmetric Markov process and h = P T <. For any b R 1 and t, under the measure P P η, we have } τ t + 1 η + h b S (1.7) ( = η + h ) } t ρ+b S in law. 1

4 1784 EISENBAUM, KASPI, MARCUS, ROSEN AND SHI In particular, when X is transient, (1.8) } τ + 1 η + h b S ( = η + h ) } ρ + b S 1 in law. An important tool used to prove these results is a new interpretation of a familiar formula due to M. Kac for the moment generating function of the total accumulated local time process. This is given in Section 3. In Section we present some preliminaries concerning Gaussian processes. In Section 4 we prove an isomorphism theorem closely related to Theorems 1.1 and 1., which facilitates their proofs. Section 5 is devoted to the proofs of Theorems 1.1 and 1.. In Section 7 we use the Kac-type formula to give an easy proof of the first Ray Knight theorem. To apply Theorems 1.1 and 1. one must obtain the -potential density u y, of the process X killed at T = inf s X s =, in order to identify the Gaussian process η, defined in (1.5). In Section 6 we do this when X is a real-valued symmetric évy process. It turns out that when X is recurrent, η has stationary increments. In Remark 6.1, we show that when X is a symmetric stable process with inde β 1 ηis a fractional Brownian motion with inde β 1.. Preliminaries on Gaussian processes. In Corollary.1 and emma. we establish some equivalence relationships between the sums of squares of two independent Gaussian processes which play an important role in the proof of Theorems 1.1 and 1.. They are consequences of the following routine calculation which we provide for the convenience of the reader. emma.1. et ζ =ζ 1 ζ n be a mean zero, n-dimensional Gaussian random variable with covariance matri. Assume that is invertible. et λ =λ 1 λ n be an n-dimensional vector and an n n diagonal matri with λ j as its jth diagonal entry. et u =u 1 u n be an n-dimensional vector. We can choose λ i > i= 1n sufficiently small so that 1 is invertible and (.1) ( n ) E ep λ i ζ i + u i / i=1 = ( 1 γ u ep + γ γt ) deti 1/ where = 1 1 and γ =λ 1 u 1 λ n u n

5 RAY KNIGHT THEOREM 1785 Proof. (.) and (.3) We write ( n E ep = ep λ i ζ i + u i / i=1 ( γ u ) ) ( n ) E ep λ i ζi / + u iζ i i=1 ( n ) E ep λ i ζi / + u iζ i i=1 1 = ep (γ ζ ζ 1 ζ t ) dζ det 1/ = det 1/ det 1/ Ẽeγ ξ where ξ is an n-dimensional Gaussian random variable with mean zero and covariance matri and Ẽ is epectation with respect to the probability measure of ξ. Clearly, ( γ γ Ẽe γξ t ) (.4) = ep Putting these together gives us (.1). We have the following immediate corollary of emma.1. Corollary.1. et η =η S be a mean-zero Gaussian process and f a real-valued function on S. It follows from emma 1 that for a + b = c + d, η + f a + η + f b S (.5) =η + f c + η + f d S in law, where η is an independent copy of η. et G =G S be a mean-zero Gaussian process with covariance u y. We use the standard decomposition to write G = ρ + h G, where h = u /u and ρ = G h G. Thus ρ and h G are independent Gaussian processes. emma.. et G be an independent copy of G. Then G + G } ρ S = + ρ + h T } (.6) S in law, where T is an eponential random variable with mean EG, independent of G and G.

6 1786 EISENBAUM, KASPI, MARCUS, ROSEN AND SHI Proof. et E ρ denote epectation with respect to ρ S and E G denote epectation with respect to G. Clearly, E G = E ρ E G. et γ =γ 1 γ m and τ = τ 1 τ m be two vectors. We write γ τ = m i=1 γ i τi. et λ =λ λ n and λ =λ 1 λ n. et G =G G n and let h = h h n, where =. Note that for all λ λ n sufficiently small we have (.7) E G e λ G / E Ge λ G / = E G E G (e λ h G + G /( E ρ e λ ρ+hg / ) ) where ρ =ρ 1 ρ n. Also, by Corollary.1, (.8) ( Eρ e λ ρ+hg / ) ( ) = Eρ e λ ρ / E ρ e λ ρ+h G + G / Combining these we see that the left-hand side of (.7) equals (.9) ( ) E ρ e λ ρ / E G E G E ρ e λ ρ+h G + G / ast, set EG = 1/q, and note that λ ( / ρ+h G E G E G E ρ e ) + G (.1) = q λ ( /e ρ+h s+t) s+tq E π ρ e 1 ds dt st = Ee λ ρ+h T / Substituting this in (.9) completes the proof of this lemma. 3. A Kac-type formula. We give a version of Kac s formula for the moment generating function of the total accumulated local time process which enables us to easily obtain the classical first and second Ray Knight theorems and generalize the second Ray Knight theorem. In what follows we use the notation A 1 for the matri obtained by replacing each of the n entries in the first column of the n n matri A by the number 1. emma 3.1. et X be a strongly symmetric Markov process with -potential density u y. et denote the total accumulated local time of X. et be the matri with elements i j = u i j ij = 1n and assume that is invertible. et y = 1. et be the matri with elements i j = λ i δ i j.for all λ 1 λ n sufficiently small we have (3.1) ( ) n E y ep λ i i = det ( I 1) deti i=1

7 RAY KNIGHT THEOREM 1787 (3.) Proof. By Kac s moment formula, ( ( n ) ) E y k λ j j = k! j=1 n j 1 j k =1 uy j1 λ j1 u j1 j λ j u j j3 u jk jk 1 λ jk 1 u jk 1 jk λ jk for all k. The proof of this is standard; see, for eample, [9], (4.7) or [7]. et β = uy 1 λ 1 uy n λ n and 1 be the transpose of an n-dimensional vector with all of its elements equal to 1. Observe that n jk =1 u jk 1 jk λ jk is an n 1 matri with entries 1 jk 1 j k 1 = 1n Note also that 1 isann 1 matri and (3.3) n j k 1 =1 u jk jk 1 λ jk 1 1 jk 1 = 1 jk Iterating this relationship we get ( ( n ) ) k E y λ j j = k!β k 1 1 (3.4) j=1 = k! k 1 1 where we use the facts that 1 = y and β k 1 is an n-dimensional vector, which is the same as the first row of k. It follows from this that ( n ) E y } ep λ i i = k 1 (3.5) i=1 1 k= =I Equation (3.1) now follows from Cramer s theorem. This completes the proof of this lemma. Remark 3.1. The -potential u y in emma 3.1 is positive definite; see, for eample, [9], Theorem 3.3. et G =G S be a mean-zero Gaussian process with covariance u y. Note that deti 1 is the moment generating function of G n / + G n /, where G n =G 1 G n, and similarly for G n (see emma.1). Suppose that we can find a stochastic process H S such that deti 1 1 is the moment generating function of H 1 H n, and suppose this can be done for all finite joint distributions. Then we would have that under the measure P y P H P G P G, (3.6) + H S } = G / + G / S} in law.

8 1788 EISENBAUM, KASPI, MARCUS, ROSEN AND SHI 4. A related isomorphism theorem. In this section we prove an isomorphism theorem closely related to Theorems 1.1 and 1. which facilitates their proof. Essentially, for the proofs of Theorems 1.1 and 1., we will only have to verify that the conditions of this section apply. et Z is a strongly symmetric Markovprocess with state space S and -potential density v y which can be written in the following form: (4.1) v y =v y+1/q where, q> and v y is positive definite and satisfies v y for all y S. et η S be the mean-zero Gaussian process with covariance v y. We have the following isomorphism theorem. Theorem 4.1. et Z be a strongly symmetric Markov process with -potential density v y satisfying 41, and let S denote the total accumulated local time of Z. Then under the measure P P n, +η + b / S } = (4.) η + T + b / S } in law for all b R, where T is an eponential random variable with mean 1/q. Proof. Set (4.3) φ = η + G Where G N 1/q so that φ E is the mean-zero Gaussian process with covariance function v y. et be the covariance matri of φ 1 φ n where 1 = and n are arbitrary. Note that i j = v i j and in particular j 1 = v j = 1/q. Consider det ( I 1). Subtract the first row of the matri in this determinant from each of the other rows. Using this row operation it is clear that (4.4) deti 1 =deti where is the covariance matri of η η n and is the natural restriction of, that is, i j = λ i+1 δ i j 1 i j n 1. That is, the reciprical of (4.4) is the moment generating function of 1/η 1 η n +1/ η 1 η n, where η is an independent copy of η. (Note that we can include η 1 since it is equal to ). Taking into account the comments made in Remark 3.1 we see that we have obtained (4.5) + η / + η / S} = φ / + φ / S} in law under the measure P P φ P φ. To obtain (4.) we note that it follows from emma. that (4.6) φ / + φ / S = η / +η + T / S in law,

9 RAY KNIGHT THEOREM 1789 since by assumption h = v /v 1 Substituting this in (4.5) we get (4.7) + η / + η / S} = η / +η + T / S } in law, which implies that (4.8) + η / S} = η + T / S } in law. Since T is independent of the Gaussian processes it follows from Corollary.1 that ( η + T ) / + η + b / S } = η / + ( η + T + b ) } / S in law. Thus we see that adding η + b / to each side of (4.8) gives us (4.). 5. Proofs of Theorems 1.1 and 1.. et X be a strongly symmetric Markovprocess. Two Markovprocesses determined by X play an important role in these proofs. The first is the process Y =Y t t R +, which is X killed when it first hits. This process is introduced prior to the statement of Theorem 1.1. To define the second, let T = Tq be an eponential random variable with mean 1/q. We define Z =Z t t R + by (5.1) Xt if t<τt, Z t = otherwise, where τt is defined in (1.4). Recall that u y denotes the -potential density of Y; consequently u y, for all y S. emma 5.1. Both Y and Z are strongly symmetric Markov processes. If X is recurrent and ũ y denotes the -potential density of Z, then (5.) ũ y =u y+1/q Furthermore, both u y and ũ y are finite, symmetric and positive definite. Proof. The fact that Y is strongly symmetric follows easily from Hunt s switching identity; see, for eample, [9], Section 3, where the reader will also find a proof that u y is positive definite. [Although the results of that reference are stated for the process killed on eisting a compact set, the proofs hold just as well for u y The finiteness of u y follows from the finiteness of u 1 y and the well-known fact that T is eponential under P, hence T would be infinite a.s. if u =E T were infinite. The strong symmetry of Z follows from more general results in [1] or [14]. Finally,

10 179 EISENBAUM, KASPI, MARCUS, ROSEN AND SHI to prove (5.) we use the fact that τt =T +τt θ T and the strong Markov property at T to see that (5.3) ũ y =E y τt =E y T +E y τt = E y T +E y τt =E y T +E y T = u y+1/q where the third equality uses the symmetry of ũ y in the form E y τt =ũy=ũy =Ey τt This completes the proof of emma 5.1. Proof of Theorem 1.1. Given X, with local time process t S, we consider the associated Markovprocess Z defined in (5.1) with -potential ũ y. By emma 5.1, Z satisfies the conditions of Theorem 4.1. Also, the total accumulated local time process for Z is precisely τt S. Hence by Theorem 4.1, under the measure P P η, τt +η + b / S } = ( η + T + b ) } (5.4) / S in law for all b R. (1.6) follows from this by taking the aplace transform. Proof of Theorem 1.. Suppose now that the Markovprocess X is transient. et u y denote its -potential, which is finite. Under P, the total accumulated local time of X has an eponential distribution with mean u. et ρ to be an independent eponential random variable with mean u. Consider the Markovprocesses Y and Z defined in (5.1), for the transient process X. emma 5.. is given by The potential density of the Markov process Z with respect to m 1 (5.5) ũ y =u y+h h y q where h = P T < =u /u and 1/q = ET ρ. Both ũ y and u y are symmetric and positive definite and u y, for all y S. Proof. The fact that the potential of Z is a absolutely continuous with respect to m, is symmetric and positive definite is the content of emma 5.1. et ũ y be its density with respect to m. Then ũ y =u y+p T < ũy = u y+p T < ũy = u y+p T < P y T < ũ

11 RAY KNIGHT THEOREM 1791 where the second equality follows from the symmetry of ũ. We now recall that ũ = E τt, and that τt is equal to the minimum of the two eponential random variables, T and. This completes the proof of emma 5.. Note that emma 5. is actually a generalization of emma 5.1 since when X is recurrent h = P T < 1. Proof of Theorem 1.(Continued). et A t be a continuous additive functional of X and let R A ω =inf ta t ω >. We call A t a local time of X at S if P R A = =1, and P y R A = = for all y. IfA t and B t are two local times for X at S, then A t = cb t for some constant c>. It is easy to check that t defined in (1.3) is a local time of X at with potential ( ) (5.6) E y e αt d t = u α y α> We call t the normalized local time of X at. However, note that for any real-valued function g > S g t is also a local time for X at, which we call a nonnormalized local time of X at [unless g 1]. Dividing (5.5) by h h y we get (5.7) ũ y h h y = u y h h y + 1 q It is easy to see that the function h is ecessive for X Y and Z. et X h be an h-path transform of X taken with respect to h and let P /h denote the measure corresponding to X h. Denote by Y h and Z h the corresponding h-path transforms of Y and Z. Note that u y/h h y u y/h h y and ũ y/h h y are the potential densities of X h Y h and Z h, respectively, all with respect to h md. Clearly, they are all symmetric. et y t be the normalized local time at y of X h with respect to h md, so that E /h e αt d y t = uα y (5.8) h h y Consider η /h S. This is a mean-zero Gaussian process with covariance u y/h h y. By (5.7), Z h satisfies the conditions of Theorem 4.1. Also, the total accumulated local time process for Z h is precisely τt S. Hence by Theorem 4.1, under P /h P η, τt + η /h + b } S (5.9) η /h = + V + b } S in law,

12 179 EISENBAUM, KASPI, MARCUS, ROSEN AND SHI where V is an eponential random variable with parameter q and T is an eponential random variable with parameter q. Equivalently, under P /h P η, τt + η + h b } S (5.1) η + h = V + b } S in law, where t = h t. Then = t S is a nonnormalized local time for X h. Its potential with respect to the process X h is (5.11) E /h e αt h y d y t = uα yh y h h y = uα yh y h which is the α-potential of X h with respect to m. This is precisely the potential of with respect to the process X h, where = t S is the normalized local time for X. Since X h is X killed at the last eit from, conditioned to hit after time, we see that t = t until the last eit of X from. et τt = inf s s t. It follows that τt = τ T Therefore (5.1) is equivalent to the statement that under P P η, τ T + η + h b } S η + h = T ρ+b } S in law where ρ is an eponentially distributed random variable with mean u independent of T. Since this is the aplace transform of (1.7), the proof of Theorem 1. is complete. 6. u y for symmetric évy processes. To apply Theorems 1.1 and 1. one must obtain the -potential density u y of the process X killed at T = inf s X s =, in order to identify the Gaussian process η, defined in (1.5). In this section we do this when X is a real-valued symmetric évy processes. et X be a real-valued symmetric évy process defined by (6.1) Ee iλx t = e tψλ We assume that X has a local time, which is equivalent to the condition that 1/1 + ψλ 1. It follows simply from (6.1) that the α-potential density of X, u α y = 1 cosλ y (6.) dλ π α + ψλ Since, in this case, u α y depends only on y, we also write it as u α y.

13 RAY KNIGHT THEOREM 1793 In our presentation we distinguish between whether X is transient or recurrent. Consider the integral (6.3) 1 1 ψλ dλ The évy process X is transient when this integral is finite and is recurrent when this integral is infinite. (6.4) Theorem 6.1. When X is recurrent, u y =φ+φy φ y where (6.5) φ = 1 π 1 cos λ ψλ dλ R Consequently, the Gaussian process η defined in 15 is such that η = and (6.6) Eη ηy = π 1 cosλ y ψλ dλ When X is transient we can define the stationary Gaussian process G R as the process with covariance u y =u y, given in 6. et ρ be the projection of G in the subspace of orthogonal to G. That is, we write G = ρ + h G, where h = P T < = u /u and ρ = G h G. Thus ρ and h G are independent Gaussian process. In this case, (6.7) (6.8) Proof. (6.9) u y =Eρ ρ y = u y It follows from (6.) that u uy u u α u α = 1 1 cosλ dλ π α + ψλ = φ α Consequently, (6.1) lim α uα u α = 1 1 cosλ dλ π ψλ = φ

14 1794 EISENBAUM, KASPI, MARCUS, ROSEN AND SHI Using the Markovproperty at time T shows that ( ) u α y =E e αt d y t ( T ) ( ) (6.11) = E e αt d y t + E e αt d y t T = u α y+e e αt u α y It follows from (6.11) that u α y uα (6.1) = u α y u α u α y u α E e αt = φ α +φ α ye e αt φα y The assumption that X is recurrent means that P T < = 1. Consequently, lim α E e αt =1. Also, clearly, u α y tends to u y as α decreases to zero. Therefore we can take the limits in (6.1) and use the fact that u α y = to obtain (6.4). Since (6.13) Eη ηy = u +u y y u y we get (6.6). Now suppose that X is transient. Since the -potential of X eists in this case we can simply take the limit α in (6.11) to get (6.14) u y =u y p T < u α y Using P T < = u /u we get (6.7). Remark 6.1. When X is a symmetric stable process with inde β 1, ψλ=λ β. Using this in (6.5) and making the change of variables λ y = s we see that (6.15) where (6.16) Eη ηy = c β y β 1 c β = π 1 cos s s β 7. First Ray Knight theorem. We show that the first Ray Knight theorem is an immediate consequence of our version of Kac s moment formula. Theorem 7.1 (First Ray Knight theorem). et B be Brownian motion on R +, starting from y> and let t t R + R + denote its local time process. et T denote the first hitting time of. Then, under the measure P y P 1 P, (7.1) T y } = B 1 + B y} in law, ds

15 RAY KNIGHT THEOREM 1795 where B i i= 1 are independent real-valued Brownian motions with measures P i, independent of the original Brownian motion B. Proof. We apply emma 3.1 to the process Y which is Brownian motion, starting at y>and killed the first time it hits zero. In this case T is the total accumulated local time of Y. The -potential of Y is u y = y when y and u y =and when y <. This is well known and is also included in Theorem 6.1. We take y = 1 > > n > and i j = u i j in emma 3.1. In the net paragraph we show that deti 1 =1, which implies that H in Remark 3.1 is identically zero. Consequently, (7.1) follows from (3.6) since the Gaussian process G in (3.6), with covariance y for y >, is equal in law to B. et D be the matri obtained by subtracting the first row of I 1 from each of the other rows. Clearly, D j j = 1 j= 1n D j 1 = j= n; and D j k = k= j+1n. This shows that the matri D = D i j n i j= is a lower triangular matri; that is, all its entries above the diagonal are equal to zero. Thus deti 1 =detd =D 1 1 det D =1. This completes the proof of this theorem. Acknowledgments. We are grateful to Jean Bertoin and Marc Yor for interesting and helpful discussions. REFERENCES [1] Bass, R., Eisenbaum, N. and Shi, Z. (). The most visited sites of symmetric stable processes. Probab. Theory Related Fields [] Bertoin, J. (1996). évy Processes. Cambridge Univ. Press. [3] Dynkin E. B. (1983). ocal times and quantum fields. In Seminar on Stochastic Processes 198 (E. Çinlar, K.. Chung and R. K. Getoor, eds.) Birkhäuser, Boston. [4] Eisenbaum, N. (1994). Dynkin s isomorphism theorem and the Ray Knight theorems. Probab. Theory Related Fields [5] Eisenbaum, N. (1997). Théorèmes limites pour les temps locau d un processus stable symétrique. Séminaire de Probabilités XXXI. ecture Notes in Math Springer, Berlin. (Errata: Séminaire de Probabilités XXXIII. ecture Notes in Math ). [6] Eisenbaum, N. and Kaspi, H. (1993). A necessary and sufficient condition for the Markov property of the local time process. Ann. Probab [7] Fitzsimmons, P. and Pitman, J. H. (1999). Kac s moment formula and the Feyman Kac formula for additive functionals of a Markov process. Stochastic Process Appl [8] Marcus, M. B. (). The most visited sites of certain évy processes. J. Theoret. Probab. To appear. [9] Marcus, M. B. and Rosen, J. (199). Sample path properties of the local times of strongly symmetric Markovprocesses via Gaussian processes. Ann. Probab [1] Marcus, M. B. and Rosen, J. (1996). Gaussian chaos and sample path properties of additive functionals of symmetric Markovprocesses Ann. Probab [11] Marcus, M. B. and Rosen J. (). Gaussian processes and the local times of symmetric évy processes. In évy Processes and Their Applications (O. Barnsdorff Nielsen, T. Mikosch and S. Resnick, eds.) Birkhauser, Boston. To appear.

16 1796 EISENBAUM, KASPI, MARCUS, ROSEN AND SHI [1] Mitro, J. B. (1979). Dual Markovfunctionals: applications of a useful auiliary process. Z. Wahrsch. Verw. Gebiete [13] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin. [14] Walsh, J. B. (197). Markovprocesses and their functionals in duality. Z. Wahrsch. Verw. Gebiete N. Eisenbaum aboratoire de Probabilités Université Paris VI 4 place Jussieu 755 Paris Cede 5 France nae@ccr.jussieu.fr H. Kaspi Industrial Engineering and Management Technion Haifa 3 Israel iehaya@t.technion.ac.il M. B. Marcus Department of Mathematics City College, CUNY New York, New York mbmarcus@earthlink.net Z. Shi aboratoire de Probabilités Université Paris VI 4 place Jussieu 755 Paris Cede 5 France zhan@proba.jussieu.fr J. Rosen Department of Mathematics College of Staten Island, CUNY Staten Island, New York jrosen3@idt.net