Markovian loop soups: permanental processes and isomorphism theorems

We construct loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.


Introduction
A Markovian loop soup is a particular Poisson point process L on paths associated to a Markov process X. It is determined by its intensity measure µ which we refer to as loop measure. Loop measure for Brownian motion was introduced by Symanzik in his seminal paper [24] on Euclidean quantum field theory, where it is referred to as 'blob measure', and is a basic building block in his construction of quantum fields. Brownian loop soup was introduced by Lawler and Werner [20], in their work on SLE and conformally invariant processes in the plane. Le Jan extended the notion of loop soups to other Markov processes [12], and this has been generalized further in [14,15]. In all this work the loop measure is constructed using bridge measures for X. This requires that X have transition densities. The main point of this paper is to show how to construct loop measures and hence loop soups for Markov processes which have potential densities but not transition densities.
Our motivation in studying Markovian loop soups is to better understand the wonderful and mysterious Isomorphism Theorem of Dynkin, [7,8], which connects the family of total local times L = {L x ∞ , x ∈ S} of a symmetric Markov process X in S with the Gaussian process G = {G x , x ∈ S} of covariance u(x, y). (When X is symmetric, u(x, y) is positive definite.) Actually, in the Isomorphism Theorem it is the family of squares of G, that is x , x ∈ S}, which is connected with L. This theorem is not an isomorphism in the usual sense, but the connection between L and G 2 is sufficiently tight that it has been used to derive many new properties of the local times, as described in [22]. This is why we consider the Isomorphism Theorem to be wonderful. We call it mysterious because it is hard to see intuitively why there should be any connection between Markov local times and Gaussian processes.
As noted by Le Jan,[13,Theorem 9], loop soups offer a deep understanding of this connection. Recall that each realization of L is a countable collection of paths ω. Set We call L x the loop soup local time at x. A simple application of the Palm formula for Poisson point processes provides a connection, an Isomorphism Theorem, between L = {L x ∞ , x ∈ S} and L = { L x , x ∈ S}. Since L is defined in terms of local times of X this should not be surprising. What may be surprising is that when X is symmetric then L = { L x , x ∈ S} has the distribution of G 2 = {G 2 x , x ∈ S}! Furthermore, the definition of (1.1) of L does not require the symmetry of X, so we obtain an Isomorphism Theorem for non-symmetric X.
In 1997, D. Vere-Jones, [26], introduced the α-permanental process θ := {θ x , x ∈ S} with kernel u(x, y), which is a real valued positive stochastic process with joint distributions that satisfy u(x j , x π(j) ), (1.2) for any x 1 , . . . , x n ∈ S, where c(π) is the number of cycles in the permutation π of [1, n]. In addition, by [26, p. 128], the joint moment generating function of (θ x 1 , . . . , θ xn ) has a non-zero radius of convergence. Consequently, an αpermanental process is determined by its moments. It is not hard to show that in the symmetric case G 2 /2 = {G 2 x /2, x ∈ S} is a 1/2-permanental process with kernel u(x, y), the covariance of G.
In [9], Eisenbaum and Kaspi were able to show the existence of an αpermanental process with kernel u(x, y) whenever u(x, y) is the potential density of a transient Markov process X, and use this to obtain an Isomorphism Theorem for non-symmetric X, where the role played in the symmetric case by the Gaussian squares G 2 is now played by a permanental process. In this paper we will see that the loop soup local time L is an α-permanental process with kernel u(x, y).
The advantage of using loop soups to construct permanental processes and obtain Isomorphism Theorems is two-fold. First, as mentioned, loop soups provide an intuitive understanding of the connection between permanental processes and local times. Second, this approach is capable of great generalization. Recent work, [14,15], uses loop soups for Markov processes with potential densities u(x, y) which may be infinite on the diagonal. In this case there are no local times and no permanental processes. Rather, loop soups are used to prove the existence of permanental fields (indexed by measures rather than points in S) with which to establish Isomorphism Theorems: for continuous additive functionals in [14], and for intersection local times in [15]. We know of no way other than using loop soups to prove the existence of permanental fields associated with not necessarily symmetric X, and the Isomorphism Theorems contain constructs which seem inaccessible without the loop soup. For example, in the symmetric case the Isomorphism Theorems contain random variables which are not in the associated Gaussian sigma field.
Here is an outline of this paper. The loop measure is constructed and studied in Section 2. In the short sub-section 2.1 we show that when transition densities exist, our definition of loop measure agrees with the standard definition using bridge measures. In Section 3 we introduce the loop soup and quickly show that the loop soup local timeL is an α-permanental process with kernel u(x, y). In the short Section 4 we use the Palm formula to prove our Isomorphism Theorem. Further properties of the loop measure are derived in Sections 5-7. These include invariance under loop rotation, and the behavior of the loop measure under restriction and space-time transformations. Here again the novelty is in deriving these properties in great generality and without the assumption of transition densities.
where d is a metric for the topology of S, and f ǫ is a continuous function supported on [0, ǫ], and define L y t (ω) = lim inf n→∞ t 0 f n −1 ,y (X s ) ds. (This is used to show measurability in y).
Under our assumption that u(x, y) is continuous, it follows as in the proof of [22,Lemma 3.4.3], that uniformly in x, u(x, y) as a function of y is locally bounded and continuous. This implies that for any β > 0, the same is true for and it follows from the resolvent equation that for each x, u β (x, y) is a density for U β (x, dy) with respect to m(dy). It then follows from the resolvent equation that for any α, β > 0 and all x, y (2.3) Using (2.2) and the resolvent equation for additive functionals we see that We now show that m(K) < ∞ for each compact K ⊆ S. To see this note first that from (2.4) and our assumption that u(x, y) > 0 for all x, y ∈ S, that also u 1 (x, y) > 0 for all x, y ∈ S. It then follows from the last paragraph that y → u 1 (x, y) is bounded below for y ∈ K by a constant C = C(x) > 0. Consequently We may take the canonical representation of X in which Ω is the set of right continuous paths ω in S ∆ = S ∪∆ with ∆ / ∈ S, and is such that ω(t) = ∆ for all t ≥ ζ = inf{t > 0 |ω(t) = ∆}. Then X t (ω) = ω(t). We define a σ-finite measure µ on (Ω, F) by for all F-measurable functions F on Ω. Here k t is the killing operator defined by k t ω(s) = ω(s) if s < t and k t ω(s) = ∆ if s ≥ t. We call µ the loop measure of X because, when X has continuous paths, µ is concentrated on the set of continuous loops. See also Lemma 2.4 below. Even if X is not assumed to have continuous paths we can verify that µ is concentrated on To see this, note first of all that since 1 {ζ=∞} • k t = 0 for each t, it is clear from (2.6) that µ(ζ = ∞) = 0. Then, since L x t is constant for t ≥ ζ, while on But by right-continuity of paths, the set of times for which X t− (ω) either fails to exist or exists but is different from X t (ω) is at most countable, for each ω ∈ Ω, [4, IV, Theorem 88D], while L x t is continuous in t so that d t L x t has no atoms. Hence (2.7) where the last equality used the fact that d t L x t is supported on {X t = x}. As usual, if F is a function, we often write µ(F ) for F dµ. u(y k , y π(1) ) · · · u(y π(k−2) , y π(k−1) )u(y π(k−1) , y k ), (2.9) where P k−1 denotes the set of permutations of [1, k − 1]. When k = 1 this means µ (L y 1 ∞ ) = u(y 1 , y 1 ).
Let Q x,y denote the measure defined on (Ω, F) by We remark that under the measures P x/h = 1 u(x,y) Q x,y , the paths of X are conditioned to hit y and die on their last exit from y. P x/h is the h-transform of P x , with h(x) = u(x, y)/u(y, y) = P x (T y < ∞).
In the proof of the Isomorphism Theorem we will need the following Lemma.
Comparing (2.26) with y = x and (2.9) we see that (2.24) holds for all polynomial H. But it is easily seen from (2.9) and (2.26) that the random variables L z ∞ are exponentially integrable both under Q x,x and µ (L x ∞ ·), hence finite dimensional distributions are determined by their moments.
Since ζ • k t = ζ ∧ t, we note for future reference that In the sequel we will use the fact that (t, x) → L x t (ω) is an occupation density with respect to m: for all t ≥ 0 and all non-negative Borel functions f , almost surely. It suffices to prove this for f ≥ 0 which are continuous and compactly supported. This case follows from the proof of [22, Theorem 3.7.1], with one change. That theorem assumed the joint continuity of L x t in order to show that the right hand side of (2.28), which we denote by A t , is a CAF. But this can be seen directly. A t is obviously monotone increasing in t and constant for t ≥ ζ. Also, using (2.5) s. Hence the a.s. continuity of A t follows from the dominated convergence theorem after applying Fubini to the fact for each for each x ∈ S, a.s. in ω. Hence by Fubini this holds a.s. in ω for a.e. x ∈ S. From the right hand side of (2.28) we then see that a.s. in ω, A s+t (ω) = A s (ω) + A t • θ s (ω), which completes the proof that A t is a CAF, and hence the proof of (2.28).

Transition densities
For this subsection only, we assume that P t (x, dy) ≪ dm(y) for each t > 0 and x ∈ S; in other words, P t (x, dy) has transition densities with respect to m. Under this assumption we give an alternate description of the loop measure. This is the description found in the literature. Using this description we give a simple proof of the fact that the loop measure is invariant under loop rotation. A proof of this fact without assuming transition densities is given in Section 5. The material in this sub-section will not be used in the following sections of the paper.
Under our assumption that P t (x, dy) has transition densities with respect to m, it follows from [6] that we can find jointly measurable transition densities p t (x, y) with respect to m which satisfy the Chapman-Kolmogorov equation Assume that p t (x, y) < ∞, for all 0 < t < ∞ and x, y ∈ S. It then follows as in [10] that for all 0 < t < ∞ and x, y ∈ S, there exists a finite measure Q x,y for all F s ∈ F s with s < t. In particular, for any 0 < t 1 ≤ · · · ≤ t k−1 ≤ t k < t and bounded Borel measurable functions for all F measurable functions F on Ω.
Proof of Lemma 2.3 Let us temporarily use the notation µ(F ) to denote the right hand side of (2.33). It suffices to show that µ Note that this last condition implies that Similarly, using the Markov property For y, x ∈ S, we define the measure Γ y,x (·) on [0, ∞) with cdf: We claim that for each y, we can find a set S y ⊂ S with m(S y ) = 0 such that for all t and x ∈ S c y . Then by (2.38), for any t k and all x ∈ S c Since the right hand side of (2.36) involves a dm(x) integral, by Fubini we can replace the term E y k ∞ t k 1 t d t L x t−t k which appears there with the left hand side of (2.40). Thus we will obtain Comparing with (2.35) then shows that µ(F ) = µ(F ). It only remains to verify our claim concerning (2.39). Note that since the left hand side of (2.39) is continuous in t and the right hand side is monotone, it suffices to find a set S y which works for all rational t, hence for each fixed t. By the occupation density formula (2.28) Since this holds for all bounded measurable f , our claim for fixed t is established.
For later use we note that applying the Chapman-Kolmogorov equation The next result justifies our calling µ the loop measure even for a process with discontinuous paths. This result will be proved in full generality in section 5. Define the loop rotation ρ u by Here, for two positive numbers a, b we define a mod b = a− mb for the unique . . , f k be bounded Borel measurable functions on S ∪ ∆ with f j (∆) = 0, j = 1, . . . , k. Fix some t and u. Since Integrating both sides with respect to dm(x) and applying the Chapman-Kolmogorov equation (2.30) we obtain where in the last line we used Comparing with (2.43) we obtain our Lemma.

The loop soup
Let Ω be the path space for X described after (2.5). For any α > 0, let L α be a Poisson point process on Ω with intensity measure αµ. Note that L α is a random variable; each realization of L α is countable subset of Ω. To be more specific, let Then for any disjoint measurable subsets A 1 , . . . , A n of Ω, the random variables N (A 1 ), . . . , N (A n ), are independent, and N (A) is a Poisson random variable with parameter αµ(A), i.e.
The Poisson point process L α is called the 'loop soup' of the Markov process X. The term 'loop soup' is used in [20], [19] and [17,Chapter 9]. In [12] L α is referred to, less colorfully albeit more descriptively, as Poissonian ensembles of Markov loops. See also [25] and [21]. We define the 'loop soup local time', L x , of X, by The next theorem is given for associated Gaussian squares in [13,Theorem 9].
Theorem 3.1 Let X be a transient Borel right process with state space S, as described in the beginning of this section, and let u(x, y), x, y ∈ S denote its potential density. Let { L x , x ∈ S} be the loop soup local time of X. Then { L x , x ∈ S}, is an α-permanental process with kernel u(x, y).
Proof By the master formula for Poisson processes, [16, (3.6)], Differentiating each side of (3.4) with respect to z 1 , . . . , z n and then setting z 1 , . . . , z n equal to zero, we get where the second sum is over all partitions B 1 , . . . , B l of [1, n]. Using (2.9) it is easily seen that this is u(x j , x π(j) ). (3.6) Let θ = {θ x , x ∈ S} be an α-permanental process with kernel u(x, y), x, y ∈ S, as considered in Theorem 3.1. Clearly, by our loop soup construction, θ is infinitely divisible. In [9, Corollary 3.4], Eisenbaum and Kaspi show that the Lévy measure of {θ x , x ∈ S} is given by the law of for any y ∈ S. However it follows from Theorem 3.1 that the loop measure αµ is the Lévy measure of {θ x , x ∈ S}. Therefore for any y ∈ S. This fact is also an immediate consequence of Lemma 2.2.

The Isomorphism Theorem via loop soup
For our Isomorphism Theorem we will need a special case of the Palm formula for Poisson processes L with intensity measure n on a measurable space S, see [2,Lemma 2.3]. This says that for any positive function f on S and any measurable functional G of L Proof We apply the Palm formula with intensity measure αµ, and Note that Then by (4.1) so that our Theorem follows from Lemma 2.2.

Invariance under loop rotation
In subsection 2.1, assuming the existence of transition densities, we gave a simple proof of the fact that the loop measure is invariant under loop rotation. In this section we give a proof of this fact without assuming transition densities. This proof is considerably more complicated.
Because the lifetime ζ is rotation invariant (ζ(ρ v ω) = ζ(ω) so long as ζ(ω) < ∞), the rotation invariance of the loop measure µ is equivalent to that of the measure ν defined by ν(F ) := µ(ζF ). By (2.27) and (2.22) we have The measure ν is more convenient for the calculations that follow, because of the following formula, where Γ x,y is defined in (2.37): as measures on the product space S k × (t k , ∞). Furthermore, with t 1 = 0, (5.2) holds for all 0 < t 2 < t 3 < · · · < t k .
Proof of Lemma 5.1: Using (5.1) we see that Using the Markov property and (2.4) we see that Here P β t (x, dy) = e −βt P t (x, dy). Using (2.38) and then the Markov property as in the previous display We claim that for a.e. t 1 , as measures in y 1 , To see this, it suffices to integrate both sides with respect to e −αt 1 dt 1 , use (2.3) with α replaced by α + β, and the fact that S has a countable base. (It is important to note that we allow y k = y 1 ).
Combining (5.3)-(5.6) we obtain for a.e. t 1 This agrees with what we obtain from the right hand side of (5.2): This completes the proof of our Lemma when t 1 > 0. When t 1 = 0, it follows from (5.3)-(5.4), and then (2.4) and (2.38) that This agrees with (5.7) for t 1 = 0, and the rest of the proof follows as in (5.8).
As a byproduct of our proof we now show that for any continuous compactly supported f 1 . To see this, note that by (5.3)- By (5.6), for a.e. t 1 this equals S S P β t 1 (y 1 , dz) u β (z, y 1 ) f 1 (y 1 ) dm(y 1 ). (5.12) But as noted in the paragraph containing (2.2), u β (z, y 1 ) is bounded, uniformly in z for y 1 in the compact support of f 1 (y 1 ). Hence (5.12) is bounded by Thus we have shown that for some dense D ⊆ R 1 and the right hand side is finite by (2.5). (5.10) then follows using right continuity. We will also need the following.
Let us define the process X to be the periodic extension of X; that is, It will be convenient to write The key observation is that for all α > 0. This follows from Hence for any continuous compactly supported f The rotation invariance of µ or ν is equivalent to the statement that for all 0 < t 1 < · · · < t k and r > 0 and all f j ≥ 0 continuous with compact support. This will follow once we show that the joint distribution of (X, ζ) is invariant under time shifts. That is, ((X t+v ) t≥0 , ζ) has the same distribution (under ν) as ((X t ) t≥0 , ζ) for all v > 0. To prove this we will first show that for all k and all α 1 , . . . , α k , for all g of the form g(ζ) = (1 − e −αζ )e −βζ , and where By (5.20) the left hand side of (5.23) is finite for all α 1 ≥ α, while the right hand side is finite since By uniqueness of Laplace transforms, it then follows that for Lebesgue a.e. k-tuple (t 1 , . . . , t k ), and in particular, for any r > 0, It follows that for any r > 0, But it is easily seen that F k (t 1 +r, . . . , t k +r) = F k (t 1 , . . . , t k ) so that, canceling the common constant factor e − k j=1 α j r , we obtain and thus comparing with (5.23) we have that for each r > 0 It follows that This holds for any k, in particular for k = 1, so that using (5.21) we have Thus by Fubini we can find a set T ⊆ R + with T c of Lebesgue measure 0 such that for all t 1 ∈ T we have (5.32), and (5.31) holds for a.e. t 2 , . . . , t k .
Using the boundedness and continuity of the f j and the right continuity of X t it follows from the Dominated Convergence Theorem that (5.31) holds for all (t 1 , t 2 , . . . , t k ) ∈ T × R k−1 + . Let now f 1,n be a sequence of continuous functions with compact support with the property that f 1,n ↑ 1. By the above, (5.31) with f 1 replaced by f 1,n holds for all (t 1 , t 2 , . . . , t k ) ∈ T n × R k−1 + for an appropriate T n ⊆ R + with T c n of Lebesgue measure 0. In particular T * = ∩ n T n = ∅, and we can apply the Monotone Convergence Theorem with t 1 ∈ T * to conclude, spelling out g(ζ), that for all t 2 , . . . , t k . Applying once again the Monotone Convergence Theorem for α → ∞ we obtain for all t 2 , . . . , t k . Fix a compact K ⊆ S. If we replace f 2 by a sequence f 2,n ↑ 1 K and then set t 2 = 0, we can conclude from (5.34) and (5.25) that the finite measures 1 K (X 0 ) · ν and 1 K (X 0 ) · ρ r * ν agree on the σ-algebra generated byX t , t ≥ 0 and ζ. Since this holds for any compact K ⊆ S, so do ν and ρ r * ν. Here and below we use the notation f * ν(A) = ν(f −1 (A)). It remains to prove (5.23). Using (5.19) and using (5.2) We now make the change of variables r = t 1 , s j = t j − t j−1 (j = 2, . . . , k), s 1 = t − t k + t 1 (with accompanying limits of integration 0 < r < s 1 , s j > 0) and then integrate out r. Writingŝ j := s 2 + · · · + s j ands := k j=1 s j , the expression in (5.36) is thereby transformed to e −α σ(j)ŝj 1 − e −α σ(j)s P s j (y j−1 , dy j ) f σ(j) (y j )Γ y k ,y 1 (ds 1 ) ds 2 · · · ds k .
We now turn to the right hand side of (5.23) and try to rewrite it in terms which are similar to our last expression for the J k (σ)'s. Using k j=1 α j t j = k j=1 α σ(j) t σ(j) we have Let us now fix σ ∈ P k and consider the corresponding term in (5.39) Changing variables (r 1 = t 1 , r j = t j − t 1 for j = 2, . . . , k) and integrating out r 1 , this can be rewritten as  (5.41) Summing first over all permutations σ ∈ P k with σ(1) = i and then over i we obtain (5.42) Using (5.19) we can express this as Using Lemma 5.1 we then see that with the convention that t 1 = 0. Once more making the change of variables Γ y k ,y 1 (ds 1 ) ds 2 · · · ds k .

The restriction property
Let B ⊆ S be open and set Clearly, t → X t is right continuous. With and we show in [22,Section 4.5] that X = (Ω, G t , G, X t , θ t , P x ) is a Borel right process with state space B and potential densities with respect to the measure m(dx) restricted to B. Here we have used the convention that u(∆, y) = 0 and that X t (ω) = ∆ when t = +∞. It follows as before that uniformly in x, u(x, y) is locally bounded and continuous in y.
Let {L x t , (x, t) ∈ S × R + } be the family of local times for X used in the construction of µ.
It is easy to see that L x t is a CAF for X and It follows that { L x t , (x, t) ∈ B × R + } are local times for X. We can then define the loop measure µ for X by the formula (In our definition (2.6) of µ we assumed that X had continuous potential densities. We do not know if u(x, y) is continuous in x. However, the continuity of u(x, y) was only used to guarantee a nice family of local times for X, and by the above this is inherited by { L x t , (x, t) ∈ B × R + }). Note that B c = S − B does not contain ∆. Proof of Theorem 6.1: It suffices to prove this for F of the form But this is clearly which is precisely what we obtain from µ k j=1 f j (X t j ) by proceeding as in (6.9).
7 Transformations of the loop measure 7.1 Mappings of the state space LetS be another locally compact topological space with a countable base, and let f : S →S be a topological isomorphism. Then forms a Borel transition semigroup onS. LetΩ be the set of right continuous paths ω inS ∆ =S ∪ ∆ with ∆ / ∈S, and such that ω(t) = ∆ for all t ≥ ζ = inf{t > 0 |ω(t) = ∆}. Then withX t (ω) = ω(t) it follows from [23, Section 13] thatX = (Ω,F t ,X t , θ t ,P x ) is a Borel right process. Furthermore, ThusX has continuous potential densitiesū(x, y) = u(f −1 (x), f −1 (y)) with respect to the measure f * m.

Unit Weights
We say that a random variable T ≥ 0 is a unit weight if Of course, since ζ is invariant under loop rotation, 1/ζ is an example of a unit weight. (7.20) will provide another example, which is be used in the proof of Theorem 7.2 to determine how the loop measure transforms under a time change.
Let I ρ (X) be the collection of measurable functions on Ω which are invariant under loop rotation.
Lemma 7.1 If T is a unit weight then for all F ∈ I ρ (X) (7.8) Proof: By invariance of µ we have that for each u > 0 and F ∈ I ρ (X) Since ζ is invariant under loop rotation, this implies that for any u > 0 This shows that

Time change by the inverse of a CAF
where ν A is a Borel measure on S. We suppose that P x (A t = ∞, t < ζ) = 0 for all x ∈ S and t > 0. (This is the case, for instance, if ν A (K) < ∞ for each compact K ⊂ S.) By the argument at the beginning of the proof of Lemma 2.1, (7.13) defines a CAF of X. Let S A denote the fine support of A; that is, the set of x ∈ S such that P x (R = 0) = 1 where R = inf{t > 0 | A t > 0}, see [23,Section 64]. Because m is a reference measure and v(x) := E x (exp(−R)) is a 1-excessive function, Let τ t be the right continuous inverse of A t , and let Y t = X τt . Then Y = (Ω, G t , Y t , θ t , P x ) is a Borel right process with state space S A and lifetime A ζ , see [23,Theorem 65.9] for details, noting that [23, (60.4)] applied to exp(−A t ) allows us to assume that A is a perfect CAF. Here θ t (ω) = θ τt(ω) (ω). Using the change of variables formula, [5, Chapter 6, (55.1)], we see that so that Y t has continuous potential densities u(x, y) with respect to the measure ν A (dy) on S A . (In the last step we used the fact that for any measurable function h s , we have ∞ 0 h s dA s = ∞ 0 h s dL y s ν A (dy). It suffices to verify this for functions of the form h s = 1 [0,t] (s), in which case it is obvious). Furthermore, since S A is the fine support of A, L x τt is continuous in t for each x ∈ S A , see [11, p. 1659], and of course E y L x τ∞ = u(y, x). It follows that {L x τt , (x, t) ∈ S A × R + } is the family of local times for Y . See [23, Theorem 65.6] for additivity.
It will be convenient to use the canonical notation X ′ = (Ω, , which is the same as X t (ω), but we use the notation X ′ t to emphasize that it is associated with the measures P ′x which we now define. If we set g(ω)(t) = ω(τ t (ω)) we have Y t = X t • g and put Using [23, (62.20)], compare (2.23), we see that if t 1 < · · · < t n , Let Q ′x,x be the measure in (2.22) associated with X ′ . Using (2.23) and the fact that X ′ also has potential densities u(x, y) we have that if Hence for all measurable F g * Q x,x (F ) = Q ′x,x (F ) , ∀x ∈ S A . (7.17) Before considering general ν A 's, we first study the special case where the measure ν A is equivalent to m. Thus ν A (dx) = h(x)m(dx) where h is a measurable function on S with 0 < h(x) < ∞ for all x. It follows from (2.28) that A t = t 0 h (X s ) ds, (7.18) and thus S A = S. Let µ, µ ′ be the loop measures for X, X ′ respectively. Proof of Theorem 7.2: Define the unit weight T (ω) = h(ω(0)) A ζ (ω) . (7.20) By (7.8) we have µ (F ) = S Q x,x (T F ) m(dx) for all F ∈ I ρ (X). It is easy to see that F ∈ I ρ (X ′ ) implies that F • g ∈ I ρ (X). Noting that A ζ = ζ • g, and using (7.17) The last equality used (2.27) and the fact that ν A (dx) = h(x)m(dx).
We next show how to combine Theorems 7.1 and 7.2. Let S ′ be another locally compact topological space with a countable base, and let f : S → S ′ be a topological isomorphism. With h as above, let m S ′ be the measure on S ′ defined by m S ′ (dy) := f * (h m S ) (dy). (7.22) It follows from the discussion in sub-section 7.1 and the present sub-section that if we setX ′ t := f (X τt ) = f (Y t ) and {P ′x , x ∈ S ′ } the measures induced by {P x , x ∈ S}, thenX ′ = (Ω,F t ,X ′ t , θ t ,P ′x ) is a Borel right process with continuous potential densities u ′ (x, y) = u(f −1 (x), f −1 (y)) (7.23) with respect to the measure f * (h m S ) = m S ′ on S ′ . Set f ♯ (ω)(t) = f (ω(τ t )) and let µ,μ ′ be the loop measures for X,X ′ respectively. Combining Theorems 7.1 and 7.2 we obtain Corollary 7.1 f ♯ * µ (F ) =μ ′ (F ) , ∀F ∈ I ρ X ′ . (7.24) Remark 7.1 Let D, D ′ be two simply connected domains in the complex plane and let f be a conformal map from D onto D ′ . Let X be Brownian motion in D. Since the potential density for X with respect to λ D , Lebesgue measure on D, is not continuous, (it has a logarithmic singularity on the diagonal), X does not fit into the framework of this paper. Nevertheless, we argue by analogy. Let h(x) = |f ′ (x)| 2 . ThenX ′ is a Brownian motion in D ′ , and f * (h λ D )(dy) = λ D ′ (dy). It follows formally that (7.24) would yield [18,Proposition 5.27], the conformal invariance of Brownian loop measures.
We now turn to a general CAF as in (7.13), Our results are not as complete as (7.19), but see the Remark following the proof of Theorem 7.3.
For any B ⊆ S let L B (X) be the σ-algebra generated by the total local times {L x ∞ , x ∈ B} of X, and let µ ′ be the loop measure for X ′ . Recall that L ′x ∞ = L x ∞ • g so that Consider F ∈ L S A (X). Since A ζ ∈ L S A (X) and A ζ > 0, P x a.s. for all x ∈ S A , by replacing F in (7.26) by F/A ζ and then integrating with respect to dν A (x) we can deduce immediately that , ∀F ∈ L S A (X) . (7.28) Although S A may not be locally compact, X ′ inherits from X all the properties needed to define µ ′ as in (2.6), and it then follows as in (2.22) that µ ′ (F ) = S A Q ′x,x 1 ζ F dν A (x). (7.29) By (7.17) this shows that Noting that A ζ = ζ • g, (7.28) and (7.27) then imply our Theorem. · · · dL xn s n−j+1 dL x 1 s n−j+2 dL x 2 s n−j+3 · · · dL x j−1 sn , that is, we measure n-tuples of times in which x 1 , . . . , x n are visited in cyclic order. If n = 2 and x 1 = x 2 , then L x 1 ,x 2 t = L x 1 t L x 2 t , but in general L x 1 ,...,xn t is not a product of the corresponding local times. Let M(X) denote the σalgebra generated by the multiple local times. When Supp (ν A ) = S we can show that (7.25) holds for all F ∈ M(X) = M(X ′ ). When S is finite, it is known that M(X) = I ρ (X), [13, p. 24]. For diffusions, see [21], especially Corollary 2.9, and for more general processes see [3].
We leave to the interested reader the task of combining Theorem 7.3 with spatial transformations as in Corollary 7.1.