Isomorphisms of $\beta$-Dyson's Brownian motion with Brownian local time

We show that the Brydges-Fr\"ohlich-Spencer-Dynkin and the Le Jan's isomorphisms between the Gaussian free fields and the occupation times of symmetric Markov processes generalize to the $\beta$-Dyson's Brownian motion. For $\beta\in\{1,2,4\}$ this is a consequence of the Gaussian case, however the relation holds for general $\beta$. We further raise the question whether there is an analogue of $\beta$-Dyson's Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix-valued Gaussian free fields. In the case $n=2$ we give a simple construction.


Introduction
There is a class of results, known as isomorphism theorems, relating the squares of Gaussian free fields (GFFs) to occupation times of symmetric Markov processes. They originate from the works in mathematical physics [33,2]. For a review, see [26,34]. Here in particular we will be interested in the Brydges-Fröhlich-Spencer-Dynkin isomorphism [2,8,9] and in the Le Jan's isomorphism [21,22]. The BFS-Dynkin isomorphism involves Markovian paths with fixed ends. Le Jan's isomorphism involves a Poisson point process of Markovian loops, with an intensity parameter α " 1{2 in the case of real scalar GFFs. For vector-valued GFFs with d components, the intensity parameter is α " d{2. We show that both Le Jan's and BFS-Dynkin isomorphisms have a generalization to β-Dyson's Brownian motion, and provide identities relating the latter to local times of one-dimensional Brownian motions. By doing so, we go beyond the Gaussian setting.
For β P t1, 2, 4u, a β-Dyson's Brownian motion is the diffusion of eigenvalues in a Brownian motion on the space of real symmetric pβ " 1q, complex Hermitian pβ " 2q, respectively quaternionic Hermitian pβ " 4q matrices. Yet, the β-Dyson's Brownian motion is defined for every β ě 0. The one-dimensional marginals of β-Dyson's Brownian motion are Gaussian beta ensembles GβE. The generalization of Le Jan's and BFS-Dynkin isomorphisms works for every β ě 0, and for β P t1, 2, 4u it follows from the Gaussian case. The intensity parameter α appearing in the Le Jan's type isomorphism is given by 2α " dpβ, nq " n`npn´1q where n is the number of "eigenvalues". In particular, α takes not only half-integer values, as in the Gaussian case, but a whole half-line of values. The BFS-Dynkin type isomorphism involves polynomials defined by a recurrence with a structure similar to that of the Schwinger-Dyson equation for GβE. These polynomials also give the symmetric moments of the β-Dyson's Brownian motion. We further ask the question whether an analogue of GβE and β-Dyson's Brownian motion could exist on electrical networks and interpolate and extrapolate the distributions of the eigenvalues in matrix-valued GFFs. Our motivation for this is that such analogues could be related to Poisson point process of random walk loops, in particular to those of non half-integer intensity parameter. If the underlying graph is a tree, the construction of such analogues is straightforward, by taking β-Dyson's Brownian motions along each branch of the tree. However, if the graph contains cycles, this is not immediate, and one does not expect a Markov property for the obtained fields. However, in the simplest case n " 2, we provide a construction working on any graph.
Our article is organized as follows. In Section 2 we recall the BFS-Dynkin and the Le Jan's isomorphisms in the particular case of 1D Brownian motion. In Section 3 we recall the definition of Gaussian beta ensembles and the corresponding Schwinger-Dyson equation. Section 4 deals with β-Dyson's Brownian motion and the corresponding isomorphisms. Section 5 deals with general electrical networks. We give our construction for n " 2 and ask our questions for n ě 3.

Isomorphism theorems for 1D Brownian motion
Let pB t q tě0 be the standard Brownian motion on R. L x will denote the Brownian local times: We will denote by ppt, x, yq the heat kernel on R, and by p R`p t, x, yq the heat kernel on R`with condition 0 in 0: ppt, x, yq " 1 ?
We will denote by P t,x,y p¨q the Brownian bridge probability from x to y in time t, and by P t,x,y

R`p¨q
(for x, y ą 0) the probability measures where one conditions P t,x,y p¨q on that the bridge does not hit 0. Let pG R`p x, yqq x,yě0 be the Green's function of 1 2 d 2 dx 2 on R`with 0 condition in 0, and for K ą 0, pG K px, yqq x,yě0 the Green's function of 1 2 d 2 dx 2´K on R: Let pµ x,y R`q x,yą0 , resp. pµ x,y K q x,yPR be the following measures on finite-duration paths: (2.1) µ x,y R`p¨q :" ż`8 0 P t,x,y R`p¨q p R`p t, x, yqdt, µ x,y K p¨q :" ż`8 0 P t,x,y p¨qppt, x, yqe´K t dt.
The total mass of µ x,y R`, resp. µ x,y K , is G R`p x, yq, resp. G K px, yq. The image of µ x,y R`, resp. µ x,y K , by time reversal is µ y,x R`, resp. µ y,x K . Let T x denote the first hitting time of a level x by the Brownian motion pB t q tě0 . We will denote by γ a generic path on R. Let pμ x,y p¨qq xăyPR , resp. pμ x,y K p¨qq xăyPR be the following measures on paths from x to y: µ x,y pF pγqq " E B 0 "y rF ppB Tx´t q 0ďtďTx qs,μ x,y K pF pγqq " The measureμ x,y has total mass 1 (probability measure), whereas the total mass ofμ x,y K is For 0 ă x ď y ă z, the measure µ x,z R`c an be obtained as the image of the product measure µ x,y R`bμ y,z under the concatenation of two paths. Similarly, for x ď y ă z P R, the measure µ x,z K is the image of µ x,y K bμ y,z K under the concatenation of two paths. Let pW pxqq xPR denote a two-sided Brownian motion, i.e. pW pxqq xě0 and pW p´xqq xě0 being two independent standard Brownian motions starting from 0 (W p0q " 0). Note that here x is rather a one-dimensional space variable then a time variable. The derivative dW pxq is a white noise on R. Let pφ R`p xqq xě0 denote the process p ?
2W pxqq xě0 . The covariance function of φ Rì s G R`. Let pφ K pxqq xPR be the stationary Ornstein-Uhlenbeck process with invariant measure N p0, 1{ ? 2Kq. It is a solution to the SDE The covariance function of φ K is G K . What follows is the BFS-Dynkin isomorphism (Theorem 2.2 in [2], Theorems 6.1 and 6.2 in [8], Theorem 1 in [9]) in the particular case of a 1D Brownian motion. In general, the BFS-Dynkin isomorphism relates the squares of Gaussian free fields to local times of symmetric Markov processes.
Theorem 2.1 (Brydges-Fröhlich-Spencer [2], Dynkin [8,9]). Let F be a bounded measurable functional on CpR`q, resp. on CpRq. Let k ě 1 and x 1 , x 2 , . . . , x 2k in p0,`8q, resp. in R. Then where the sum runs over the p2kq!{p2 k k!q partitions in pairs, the γ i -s are Brownian paths and the Lpγ i q-s are the corresponding occupation fields x Þ Ñ L x pγ i q.

Remark 2.2.
Since for x ă y, the measure µ x,y R`, resp. µ x,y K , can be decomposed as µ x,x R`bμ x,y , resp. µ x,x K bμ x,y K , Theorem 2.1 can be rewritten using only the measures of type µ x,x R`a ndμ x,y , resp. µ x,x K andμ x,y K .
To a wide class of symmetric Markov processes one can associate in a natural way an infinite, σ-finite measure on loops [20,19,18,21,22,23,12]. It originated from the works in mathematical physics [31,32,33,2]. Here we recall it in the setting of a 1D Brownian motion, which has been studied in [24]. The range of a loop will be just a segment on the line, but it will carry a non-trivial Brownian local time process which will be of interest for us.
Given a Brownian loop γ, T pγq will denote its duration. The measures on (rooted) loops are Usually one considers unrooted loops, but this will not be important here. The 1D Brownian loop soups are the Poisson point processes, denoted L α R`, resp. L α K , of intensity αµ loop R`, resp. αµ loop K , where α ą 0 is an intensity parameter. LpL α R`q , resp. LpL α K q, will denote the occupation field of L α R`, resp. L α K : The following statement deals with the law of LpL α R`q , resp. LpL α K q. See Proposition 4.6, Property 4.11 and Corollary 5.5 in [24]. For the analogous statements in discrete space setting, see Corollary 5, Proposition 6, Theorem 13 in [21] and Corollary 1, Section 4.1, Proposition 16, Section 4.2, Theorem 2, Section 5.1 in [22]. In general, one gets α-permanental fields (see also [23,12]). For α " 1 2 in particular, one gets square Gaussians. We recall that given a matrix M " pM ij q 1ďi,jďk , its α-permanent is Perm α pM q :" ÿ σ permutation of t1,2,...,ku Theorem 2.3 (Le Jan [21,22], Lupu [24]). For every α ą 0 and x P R`, resp. x P R, the r.v. L x pL α R`q , resp. L x pL α K q, follows the distribution Gammapα, G R`p x, xq´1q, resp. Gammapα, G K px, xq´1q. Moreover, the process α Þ Ñ L x pL α R`q , resp. L x pL α K q, is a pure jump Gamma subordinator with Lévy measure with initial condition L 0 pL α R`q " 0. That is to say it is a square Bessel process of dimension 2α, reflected at level 0 for α ă 1. For x P R, x Þ Ñ L x pL α K q is a stationary solution to the SDE . In particular, for α " 1 2 , one has the following identities in law between stochastic processes:
For q ě 1, p q pλq will denote the q-th power sum polynomial p q pλq :" n ÿ j"1 λ q j .
The recurrence (3.3) and the initial condition p 0 pλq " n determine all the moments xp ν pλqy β,n .
Next are some elementary properties of GβE, which follow from the form of the density (3.1).
Proposition 3.2. The following holds.
Proof. One can factorize the density (3.1) as This immediately implies (3) and (1). The property (2) is implied by (4), (3) and (1). The property (4) can be obtained by computing a Laplace transform. Fix K ą 0. We have that By performing the change of variablesλ " pK`1q 1 2 λ, we get that the expression above equals Thus, So we get the Laplace transform of a Gammappdpβ, nq´1q{2, 1q r.v.
Next is an embryonic version of the BFS-Dynkin isomorphism (Theorem (2.1)) for the GβE. One should imagine that the state space is reduced to one vertex, and a particle on it gets killed at an exponential time. (1) Let a ě 0. Let h : R n Ñ R be a measurable function such that x|hpλq|y β,n ă`8.
(1) clearly implies (2). It is enough to check (3.4) for F of form F ptq " e´K t , with K ą 0. Then where on the second line we used the change of variablesλ " pK`1q 1 2 λ, and on the third line the homogeneity. Further, pK`1q´1 2`n`n pn´1q β 2`a˘" Ere´K θ s.

β-Dyson's Brownian motions and the occupation fields of 1D Brownian loop soups.
For references on β-Dyson's Brownian motion, see [10,4,30,5,6], [27,Chapter 9] and [1,Section 4.3]. Let β ě 0 and n ě 2. The β-Dyson's Brownian motion is the process pλpxq " pλ 1 pxq, . . . , λ n pxqqq xě0 with λ 1 pxq ě¨¨¨ě λ n pxq, satisfying the SDE with initial condition λp0q " 0. The derivatives pdW j pxqq 1ďjďn are independent white noises. Since we will be interested in isomorphisms with Brownian local times, the variable x corresponds here to a one-dimensional spatial variable rather than a time variable. For every 2x, is distributed, up to a reordering of the λ j pxq-s, as a GβE (3.1). For β equal to 1, 2 resp. 4, pλpxqq xě0 is the diffusion of eigenvalues in a Brownian motion on the space of real symmetric, complex Hermitian, resp. quaternionic Hermitian matrices. For β ě 1, there is no collision between the λ j pxq-s, and for β P r0, 1q two consecutive λ j pxq-s can collide, but there is no collision of three or more particles [6]. Note that for β ą 0 and j P 2, n , pλ j pxq´λ j´1 pxqq{2 behaves near level 0 like a Bessel process of dimension β`1 reflected at level 0, and since β`1 ą 1, the complication with the principal value and the local time at zero does not occur; see [35,Chapter 10]. In particular, each pλ j pxqq xě0 is a semimartingale. For β " 0, pλpxq{ ? 2q xě0 is just a reordered family of n i.i.d. standard Brownian motions.
Remark 4.1. We restrict to β ě 0 because the case β ă 0 has not been considered in the literature. The problem is the extension of the process after a collision of λ j pxq-s. The collision of three or more particles, including all the n together for β ă´2 pn´3q npn´1q , is no longer excluded. However, we believe that the β-Dyson's Brownian motion can be defined for all β ą´2 n . This is indeed the case if n " 2. One can use the reflected Bessel processes for that. Let pρpxqq xě0 be the Bessel process of dimension β`1, reflected at level 0, satisfying away from 0 the SDE with ρp0q " 0. The reflected version is precisely defined for β ą´1 "´2 2 ; see [29, Section XI.1] and [17, Section 3]. Let p Ă W pxqq xě0 be a standard Brownian motion starting from 0, independent from pW pxqq xě0 Then, for n " 2, one can construct the β-Dyson's Brownian motion as Next are some simple properties of the β-Dyson's Brownian motion.
Proposition 4.2. The following holds.
(1) The process`1 ? n p 1 pλpxqq˘x ě0 has the same law as φ R`. (2) The process p 1 2 p 2 pλpxqqq xě0 is a square Bessel process of dimension dpβ, nq starting from 0.
Proof. With Itô's formula, we get dp 1 pλpxqq " ? 2 n ÿ j"1 dW j pxq, where the points x P R`for which λ j pxq " λ j´1 pxq for some j P 2, n can be neglected. This gives (1), (2) and (4) since the processes are both standard Brownian motions. Again, one can neglect the points x P R`where p 2 pλpxqq1 n p 1 pλpxqq 2 " 0, which only occur for n " 2.
For (3), we have that The Brownian motion p 1 pW q " 1 ? 2 p 1 pλq is independent from the family of Brownian motions`W j´1 n p 1 pW q˘1 ďjďn . Further, the measurability of`λ j´1 n p 1 pλq˘1 ďjďn with respect to`W j´1 n p 1 pW q˘1 ďjďn follows from the pathwise uniqueness of the solution to (4.1); see [ Corollary 4.3. The process`1 2 p 2 pλpxqq˘x ě0 has the same law as the occupation field pL x pL α R`q q xě0 of a 1D Brownian loop soup L α R`, with the correspondence R`b e two independent 1D Brownian loop soups, α still given by (4.4). Then, one has the following identity in law between pairs of processes:

4.2.
Symmetric moments of β-Dyson's Brownian motion. We will denote by x¨y Rβ ,n the expectation with respect to the β-Dyson's Brownian motion (4.1). This section will be devoted to deriving a recursive way to express the symmetric moments for ν be a finite family of positive integers with |ν| even and x 1 ď x 2 ď¨¨¨ď x mpνq P R`. This generalizes the Schwinger-Dyson equation (3.3). Note that if |ν| is odd then the moment equals 0.
We will also use in the sequel the following notation. For k ě k 1 P N, k, k 1 will denote the interval of integers k, k 1 " tk, k`1, . . . , k 1 u. We start by some lemmas.
Proof. This is a straightforward computation. Proof. The process (4.6) is a local martingale. Its quadratic variation is given by ż x 0 p ν pλpyqq 2 p 2q pλpyqqdy.
So the quadratic variation is locally bounded in L 1 . It follows that (4.6) is a true martingale.
Let ν be a finite family of positive integers. and let x 1 ď x 2 ď¨¨¨ď x mpνq P R`. For k P 1, mpνq and x ě x k´1 , let f k pxq denote the function The main idea for expressing a symmetric moment (4.5) is that for x ě x k´1 , the derivative f 1 k pxq is a linear combination of symmetric moments of degree |ν|´2, with coefficients depending on β and n. The precise expressions for these coefficients can be deduced from Lemmas 4.4 and 4.5. Further, the moment (4.5) equals f mpνq px mpνq q, for every k P 2, mpνq , f k px k´1 q " f k´1 px k´1 q, and where xp ν pλqy β,n is the moment of the GβE, given by Proposition 3.1. So given the above initial conditions, and knowing the derivatives f 1 k pxq one gets the moment (4.5). It turns out that this moment is a multivariate polynomial in px k q 1ďkďmpνq . Next we describe the recursion for this polynomial.
Let pY kk q kě1 denote a family of formal commuting polynomials variables. We will consider finite families of positive integers ν " pν 1 , ν 2 , . . . , ν mpνq q with |ν| even. The order of the ν k will matter. That is to say we distinguish between ν and pν σp1q , ν σp2q , . . . , ν σpmpνqq q for σ a permutation of 1, mpνq . We want to construct a family of formal polynomials Q ν,β,n with parameters ν,β and n, where Q ν,β,n has for variables pY kk q 1ďkďmpνq . To simplify the notations, we will drop the subscripts β, n and just write Q ν . The polynomials Q ν will appear in the expression of the symmetric moments (4.5). We will denote by cpν, β, nq the solutions to the recurrence (3.3), which for β P p´2{n,`8q are the moments xp ν pλqy β,n . By convention, cpp0q, β, nq " n and cpH, β, nq " 1. For k ě 1 and Q a polynomial, Q kÐ will denote the polynomial in the variables pY k 1 k 1 q 1ďk 1 ďk , obtained from Q by replacing each variable Y k 1 k 1 with k 1 ě k`1 by the variable Y kk . Note that Q mpνqÐ ν " Q ν and that Q 1Ð ν is an univariate polynomial in Y 11 . For Y a formal polynomial variable, deg Y will denote the partial degree in Y.
Definition 4.7. The family of polynomials pQ ν q |ν| even is defined by the following.
Note that since the polynomials Q ν,β,n are formal, one is not restricted by a specific range for β. One could take any β P C or even consider β as a formal parameter. The specific range for β will only matter when relating Q ν,β,n to the symmetric moments of the β-Dyson's Brownian motion.
Proposition 4.8. Definition 4.7 uniquely defines a family of polynomials pQ ν q |ν| even . Moreover, the following properties hold.
(1) For every A monomial of Q ν and every k P 2, mpνq , In particular, Q ν is a homogeneous polynomial of degree |ν|{2.
(2) For every k P 1, mpνq and every permutation σ of k, mpνq , Proof. The fact that the polynomials Q ν are well defined can be proved by induction on |ν|{2.
We are ready now to express the symmetric moments (4.5). Proposition 4.9. Let β ě 0. Let ν be a finite family of positive integers, with |ν| even. Let Q ν " Q ν,β,n be the polynomial given by Definition 4.7. Let x 1 ď x 2 ď¨¨¨ď x mpνq P R`. Then, Proof. The proof is done by induction on |ν|{2.
The case |ν|{2 " 1 corresponds to ν " p1, 1q or ν " p2q. These are treated by Proposition 4.2, and taking into account that the one-dimensional marginals of square Bessel processes follow Gamma distributions. Now consider the induction step. Assume |ν|{2 ě 2. Recall the function f k pxq (4.7) for k P 1, mpνq . We have that (4.10) f 1 px 1 q " cpν, β, nqp2x 1 q |ν|{2 " Q 1Ð ν pY 11 " 2x 1 q, where for the second equality we applied the condition (1) in Definition 4.7. If mpνq " 1, there is nothing more to check. In the case mpνq ě 2, we need only to check that for every k P 2, mpνq and every x ą x k´1 , Indeed, given (4.10), by applying (4.11) to k " 2, we further get f 2 px 2 q " P 2Ð ν pY 11 " 2x 1 , Y 22 " 2x 2 q, and by successively applying (4.11) to k " 3, . . . , k " mpνq, we at the end get which is exactly what we want. To show (4.11), we proceed as follows. Let pF x q xě0 be the filtration of the Brownian motions ppW j pxqq 1ďjďn q xě0 . Then, for x ą x k´1 , where x¨|F x k´1 y Rβ ,n denotes the conditional expectation. To express A mpνq ź we apply Itô's formula to mpνq ź The local martingale part is, according to Lemma 4.6, a true martingale, and thus gives a 0 conditional expectation. The bounded variation part is a linear combination of terms of form pνpλpxqqdx, with |ν| "´m pνq ÿ k 1 "k ν k 1¯´2, the exact expressions following from Lemma 4.4 and Lemma 4.5. By comparing these expressions with the recurrence (4.8), and using the induction hypothesis at the step |ν|{2´1, we get (4.11). At this stage we omit detailing the tedious but completely elementary computations.

4.3.
More general formal polynomials. In previous Section 4.2, we defined recursively a family of formal polynomials Q ν " Q ν,β,n (Definition 4.7), which encode the symmetric moments of the β-Dyson's Brownian motion (Proposition 4.9). However, these polynomials are insufficient both for the generalization of the BFS-Dynkin isomorphism (forthcoming Proposition 4.14) and for expressing the symmetric moments of the stationary version of the β-Dyson's Brownian motion (forthcoming Proposition 4.22). Therefore we introduce an other family of formal polynomials P ν " P ν,β,n , with P ν constructed out of Q ν in a straightforward way which we describe next. On top of the formal commuting polynomial variables pY kk q kě1 appearing in the polynomials Q ν , we also consider the family of the formal commuting variables p q Y k´1 k q kě2 , also commuting with the first one. A polynomial P ν will have for variables pY kk q 1ďkďmpνq and p q Y k´1 k q 2ďkďmpνq . 13 Definition 4.10. Given ν a finite family of positive integers with |ν| even, let P ν be the polynomial in the variables pY kk q 1ďkďmpνq , p q Y k´1 k q 2ďkďmpνq defined by the following.
(1) P ν ppY kk q 1ďkďmpνq , p q Y k´1 k " 1q 2ďkďmpνq q " Q ν ppY kk q 1ďkďmpνq q. (2) For every A monomial of P ν and every k P 2, mpνq , The property (4.9) ensures that P ν " P ν,β,n is well defined. As for Q ν,β,n , P ν,β,n is defined for every β P C. Proposition 4.9 and Definition 4.10 immediately imply the following.
Proposition 4.12. Let m P Nzt0u. Let M " pM kk 1 q 1ďk,k 1 ďm be the formal symmetric matrix with entries given by The following holds.
For other examples of P ν , see the Appendix. As a side remark, we observe next that the value β "´2 n plays a special role for the polynomials Q ν,β,n and P ν,β,n . In particular, P ν,β"´2 n ,n gives the moments of the stochastic processes pφ R`p xqq xě0 and pφ K pxqq xPR introduced in Section 2, which are Gaussian. This is also related to the fact that in the limit β Ñ´2 n , the GβE converges in law to n identical Gaussians (3.2). Proposition 4.13. Let n ě 1. Let K ą 0. Let ν be a finite family of positive integers with |ν| even. Let x 1 ď¨¨¨ď x mpνq be mpνq points in p0,`8q, resp. in R. Then Q ν,β"´2 n ,n ppY kk " 2x k q 1ďkďmpνq q " That is to say, the variables Y kk are replaced by G R`p x k , x k q, resp. G K px k , x k q, and the variables q Proof. First, one can check that (4.14) c´ν, β "´2 n , n¯" n mpνq´|ν|{2 |ν|! 2 |ν|{2 p|ν|{2q! .

This follows from Proposition 3.2. The key point is that
d´β "´2 n , n¯" 1.
Further, let p r Q ν q |ν| even be the following formal polynomials: To conclude, we need only to check that r Q ν " Q ν,β"´2 n ,n for all ν with |ν| even. Indeed, this immediately implies that P ν,β"´2 n ,n " n mpνq´|ν|{2 ÿ where the M kk 1 are given by (4.13), and thus n´m pνq`|ν|{2 P ν,β"´2 n ,n corresponds to the Wick's rule. So by evaluating in Y kk " G R`p x k , x k q and q Y k´1 k " G R`p x k´1 , x k q{G R`p x k´1 , x k´1 q, resp. Y kk " G K px k , x k q and q Y k´1 k " G K px k´1 , x k q{G K px k´1 , x k´1 q, one gets the moments of φ R`, resp. φ K .
The identity r Q ν " Q ν,β"´2 n ,n can be checked by induction over |ν|{2 by following Definition 4.7. From (4.14) follows that the r Q ν satisfy the condition (1) in Definition 4.7. One can further check the recurrence (4.8), and this amounts to counting the pairs in k´1 ν p k, mpνq q.

BFS-Dynkin isomorphism for β-Dyson's Brownian motion.
We will denote by Υ a generic finite family of continuous paths on R, Υ " pγ 1 , . . . , γ J q, and JpΥq will denote the size J of the family. We will consider finite Brownian measures on Υ where JpΥq is not fixed but may take several values under the measure. Given x P R, L x pΥq will denote the sum of Brownian local times at x: LpΥq will denote the occupation field x Þ Ñ L x pΥq. Given ν a finite family of positive integers with |ν| even and 0 ă x 1 ă x 2 ă¨¨¨ă x mpνq , µ ν,x 1 ,...,x mpνq R`p dΥq (also depending on β and n) will be the measure on finite families of continuous paths obtained by substituting in the polynomial P ν " P ν,β,n for each variable Y kk the measure µ x k ,x k R`, and for each variable q Y k´1 k the measureμ x k´1 ,x k R`; see Section 2. Since we will deal with the functional LpΥq under µ ν,x 1 ,...,x mpνq R`p dΥq, the order of the Brownian measures in a product will not matter. For instance, for ν " p2, 1, 1q (see Appendix), Note that depending on values of n and β, a measure µ ν,x 1 ,...,x mpνq R`m ay be signed. Next is a version of BFS-Dynkin isomorphism (Theorem (2.1)) for β-Dyson's Brownian motion.
Proposition 4.14. Let ν be a finite family of positive integers, with |ν| even and let 0 ă x 1 ă x 2 ă¨¨¨ă x mpνq . Let F be a bounded measurable functional on CpR`q. Then Remark 4.15. In the limiting case when x k " x k´1 for some k P 2, mpνq , q Y k´1 k in P ν has to be replaced by the constant 1 instead of a measure on Brownian paths. Let us first outline our strategy for proving Proposition 4.14. By density arguments it is enough to show (4.16) for functionals F of form where χ is a continuous non-negative function with compact support in p0,`8q. For such F , the value returned by the right-hand side of (4.16) is well understood and is related to the local times of Brownian motions with a killing rate given by χ. In order to deal with the left-hand side of (4.16), one interprets  Then xD χ p`8qy Rβ ,n " 1. Moreover, Then pM χ pxqq xě0 is a martingale with respect to the filtration pF x q xě0 and for all x ě 0, Proof. (4.17) and (4.18) follow from the properties of square Bessel processes. See Theorem (1.7), Section XI.1 in [29]. pM χ pxqq xě0 is obviously a (true) martingale, as can be seen with the quadratic variation. Further, Lemma 4.18. Let be pλpxq " pλ 1 pxq, . . . ,λ n pxqqq xě0 withλ 1 pxq ě¨¨¨ěλ n pxq, satisfying the SDE with initial conditionλp0q " 0. Further consider a change of measure with density D χ p`8q (4.17) on the filtered probability space with filtration pF x q xě0 . Then λ after the change of measure andλ before the change of measure have the same law.
Let ψ χ denote the following diffeomorphism of R`: Let ψ´1 χ be the inverse diffeomorphism. Proof. The process´1 u χÓ pxqλ pxq¯x ě0 satisfies d´1 u χÓ pxqλ j pxq¯" By further performing the change of variable given by ψ χ , one gets (4.1).

Proof. From Lemma 4.19 and Proposition 4.9 it follows that
Further, let A be a monomial of P ν . One has to check that This amounts to counting the power for each u χÓ px k q on both sides. On the left-hand side, each u χÓ px k q appears with power ν k . The power of u χÓ px k q on the right-hand side is By (4.12), this is again ν k . Finally, by (4.20), Proof of Proposition 4.14. It is enough to show (4.16) for functionals F of form where χ is a continuous non-negative function with compact support in p0,`8q. For such a χ, whereλ is given by (4.19), withλp0q " 0. The symmetric moments ofλ are given by Lemma 4.20. To conclude, we use that ż γ exp´´ż R`L z pγqχpzqdz¯µ x,x R`p dγq " G R`,χ px, xq, 4.5. The stationary case. In this section we consider the stationary β-Dyson's Brownian motion on the whole line and state the analogues of Propositions 4.2, 4.9 and 4.14 for it. The proofs are omitted, as they are similar to the previous ones. As previously, n ě 2 and β ě 0. Let K ą 0. We consider the process pλpxq " pλ 1 pxq, . . . , λ n pxqqq xPR with λ 1 pxq ě¨¨¨ě λ n pxq, satisfying the SDE (4.21) dλ j pxq " ? 2dW j pxq´?2K λ j pxq`β ? 2K ÿ j 1 ‰j dx λ j pxq´λ j 1 pxq , the dW j , 1 ď j ď n, being n i.i.d. white noises on R, and λ being stationary, with p2Kq  (1) The process`1 ? n p 1 pλpxqq˘x PR has the same law as φ K . (2) Consider a 1D Brownian loop soup L α K , with α given by (4.4). The process p 1 2 p 2 pλpxqqq xPR has the same law as the occupation field pL x pL α K qq xPR . (3) The processes pp 1 pλpxqqq xPR and`λpxq´1 n p 1 pλpxqq˘x PR are independent.
(4) Let L α´1 2 K and r L 1 2 K be two independent 1D Brownian loop soups, α given by (4.4). Then, one has the following identity in law between pairs of processes: We will denote by x¨y K β,n the expectation with respect to the stationary β-Dyson's Brownian motion. Given ν a finite family of positive integers with |ν| even and x 1 ă x 2 ă¨¨¨ă x mpνq P R, µ ν,x 1 ,...,x mpνq K pdΥq (also depending on β and n) will be the measure on finite families of continuous paths obtained by substituting in the polynomial P ν " P ν,β,n for each variable Y kk the measure µ x k ,x k K , and for each variable q Y k´1 k the measureμ Proposition 4.22. Let ν a finite family of positive integers with |ν| even. Let x 1 ď x 2 ď¨¨¨ď x mpνq P R. Then, Further, let F be a bounded measurable functional on CpRq. For x 1 ă x 2 ă¨¨¨ă x mpνq P R, 5. The case of general electrical networks: a construction for n " 2 and further questions 5.1. Formal polynomials for n " 2. In this section n " 2, and β is arbitrary, considered as a formal parameter. Note that dpβ, n " 2q " β`2. In Section 4.2 we introduced the formal commuting polynomial variables pY kk q kě1 . Here we further consider the commuting variables pY kk 1 q 1ďkăk 1 , and by convention set Y kk 1 " Y k 1 k for k 1 ă k. Givenν " pν 1 , . . . ,ν m q with ν k P N (value 0 allowed), Pν ,β will be the following multivariate polynomial in the variables pY kk 1 q 1ďkďk 1 ďm : where f is a map f : 1,ν 1`¨¨¨`νm Ñ 1, m , such that for every k P 1, m , |f´1pkq| "ν k . Recall the expression of the α-permanents (2.3). It is clear that Pν ,β does not depend on the particular choice of f . In caseν 1 "¨¨¨"ν m " 0, by convention we set Pν ,β " 1. Given ν a finite family of positive integers with |ν| even, let k ν : 1, |ν| Þ Ñ 1, mpνq be the map given by (4.15). Let I ν be the following set of subsets of 1, |ν| : I ν :" tI Ď 1, |ν| | @k P 1, mpνq , |k´1 ν pkqzI| is even u, where |¨| denotes the cardinal. Note that necessarily, for every I P I ν , the cardinal |I| is even. Let p P ν,β be the following multivariate polynomial in the variables pY kk 1 q 1ďkďk 1 ďmpνq : Y kν pa i qkν pb i q¯P p 1 2 |k´1 ν pkqzI|q 1ďkďmpνq ,β .

Indeed, in the expansion of´Ă
W px k q`ρpx k q¯ν k`´Ă W px k q´ρpx k q¯ν k only enter the even powers of ρpx k q, which is how I ν appears. Then one uses that the square Bessel process pρpxqq xě0 is a pβ`1q{2-permanental field with kernel pG R`p x, yqq x,yPR`. Because of the particular form of G R`, we have that for x 1 ď x 2 ď¨¨¨ď x mpνq P R`, ,n"2 " r P ν,β ppY kk " 2x k q 1ďkďmpνq , p q Y k´1 k " 1q 2ďkďmpνq q.
By combining with Corollary 4.11, we get that the following multivariate polynomials in the variables pY kk q 1ďkďmpνq are equal for β ě 0: r P ν,β pp q Y k´1 k " 1q 2ďkďmpνq q " P ν,β,n"2 pp q Y k´1 k " 1q 2ďkďmpνq q.
Since the coefficients of both are polynomials in β, the equality above holds for general β. To conclude the equality r P ν,β " P ν,β,n"2 , we have to deal with the variables p q Y k´1 k q 2ďkďmpνq . For this we use that both in case of P ν,β,n"2 and in case of r P ν,β , each monomial satisfies (4.12). For r P ν,β this follows from (5.1).

5.2.
A construction on discrete electrical networks for n " 2. Let G " pV, Eq be an undirected connected graph, with V finite. We do not allow multiple edges or self-loops. The edges tx, yu P E are endowed with conductances Cpx, yq " Cpy, xq ą 0. There is also a nonuniformly zero killing measure pKpxqq xPV , with Kpxq ě 0. We see G as an electrical network. Let ∆ G denote the discrete Laplacian p∆ G f qpxq " ÿ y"x Cpx, yqpf pyq´f pxqq.
Let pG G,K px, yqq x,yPV be the massive Green's function G G,K " p´∆ G`K q´1. The (massive) real scalar Gaussian free field (GFF) is the centered random Gaussian field on V with covariance G G,K , or equivalently with density Let X t be the continuous time Markov jump process to nearest neighbors with jump rates given by the conductances. The process X t is also killed by K. Let ζ P p0,`8s be the first time X t gets killed by K. Let p G,K pt, x, yq be the transition probabilities of pX t q 0ďtăζ . Then p G,K pt, x, yq " p G,K pt, y, xq and G G,K px, yq " ż`8 0 p G,K pt, x, yqdt.
Let P t,x,y G,K be the bridge probability measure from x to y, where one conditions on t ă ζ. For x, y P V , let µ x,y G,K be the following measure on paths: µ x,y G,K p¨q :" ż`8 0 P t,x,y G,K p¨qp G,K pt, x, yqdt.
It is the analogue of (2.1). The total mass of µ x,y G,K is G G,K px, yq, and the image of µ x,y G,K by time reversal is µ y,x G,K . Similarly, one defines the measure on (rooted) loops by µ loop G,K pdγq :" where T pγq denotes the duration of the loop γ. It is the analogue of (2.2). The measure µ loop G,K has an infinite total mass because it puts an infinite mass on trivial "loops" that stay in one vertex. For α ą 0, one considers Poisson point processes L α G,K of intensity αµ loop G,K . These are (continuous time) random walk loop soups. For details, see [19,18,21,22].
For a continuous time path γ on G of duration T pγq and x P V , we denote L x pγq :" ż T pγq 0 1 γpsq"x ds.
One has equality in law between pL x pL 1 2 G,K qq xPV and p 1 2 φ G,K pxq 2 q xPV , where φ G,K is the GFF distributed according to (5.2) [21,22]. This is the analogue of (2.4). For general α ą 0, the occupation field pL x pL α G,K qq xPV is the α-permanental field with kernel G G,K [21,22,23]. In this sense it is analogous to squared Bessel processes. If pχpxqq xPV P R V is such that´∆ G`K´χ is positive definite, then (5.3) E " exp´ÿ xPV χpxqL x pL α G,K q¯ı "ˆd etp´∆ G`K q detp´∆ G`K´χ q˙α .