Interacting diffusions on positive definite matrices

We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to $K$-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda chain.


INTRODUCTION
In recent years, there has been much progress in the development of integrable models in probability, particularly interacting particle systems related to representation theory and integrable systems. A well-known example is the coupled system of SDE's (1.1) where are independent one-dimensional Brownian motions. This process is closely related to the Toda lattice and has been extensively studied [5,6,23,34,39,44,45].
It is natural to consider non-commutative generalisations of such processes. In this paper, we consider some interacting systems of Brownian particles in the space P of positive × real symmetric matrices. One of the main examples we consider is a generalisation of the system (1.1), a diffusion process in P with infinitesimal generator where Δ denotes the Laplacian, and denotes the partial matrix derivative, on P. In the case = 1 with = ln , it is equivalent to the system (1.1). We will show that this process is related to a quantisation of the non-Abelian Toda lattice in P.
In another direction, Matsumoto and Yor [30] obtained an analogue of Pitman's 2 − theorem for exponential functions of Brownian motion, which is closely related to Dufresne's identity [16]. These results were recently extended to the matrix setting by Rider and Valkó [42] (see also Bougerol [7] for related results in the complex case). We discuss this example in some detail, and give a new proof (and slight generalisation) of Rider and Valkó's result. We also consider an example of a pair of interacting Brownian particles in P with a 'reflecting wall'.
The outline of the paper is as follows. In the next section we present some preliminary background material. This is followed, in Sections 3-6, by a series of examples with small numbers of particles. In Section 7, we discuss the example (1.2) and its relation to a quantisation of the non-Abelian Toda lattice. In Section 8, we outline how this example is related to some Bäcklund transformations for the classical non-Abelian Toda lattice. In Section 9, we briefly discuss a related class of processes in which the underlying motion of particles is not governed by the Laplacian, but rather a related diffusion process which was introduced and studied by Norris, Rogers and Williams [33]. In Section 10, we briefly outline 1 how the framework developed in this paper extends to the complex setting, with particular emphasis on some extensions of the (Hermitian) Matrix Dufresne identity of [42].

PRELIMINARIES
We mostly follow the nomenclature of Terras [47], to which we refer the reader for more background. Let P denote the space of positive × real symmetric matrices. For ∈ ( ) and ∈ P, write [ ] = . This defines an action of ( ) on P. For ∈ P, we will use the notation etr ( ) = exp( tr ( )), and denote by 1/2 the unique positive square root of .
2.1. Differential operators. The partial derivative on P is defined, writing = ( ), by We define the Laplacian on P by Δ = tr 2 , where = . The Laplacian is a ( )invariant differential operator on P, meaning If = −1 , then = − ′ , where ′ = ( ) . It follows that = (−1) . In particular, the Laplacian is invariant under this change of variable.
For ∈ C and ∈ P, define This function was introduced by Herz [22], and is related to by We note that − ( ) = ( ) ( ), which implies The asymptotic behaviour of ( ) for large arguments has been studied via Laplace's method in the paper [12], see also [20,Appendix B]. If we fix ∈ P and let = 2 2 /2, then it holds that, as → ∞, and denote the eigenvalues of . In particular, taking = and 2 /2 = , say, we deduce from the same application of Laplace's method, the following lemma, which we record here for later reference. Then 1/2 ( ) → , in probability, as → ∞.

2.7.
Standard probability distributions. The Wishart distribution on P with parameters Σ ∈ P and > − 1 has density If ≥ is an integer and is a × random matrix with independent standard normal entries, then Σ 1/2 Σ 1/2 is Wishart distributed with parameters Σ and . The inverse Wishart distribution on P, with parameters Σ ∈ P and > − 1, is the law of the inverse of a Wishart matrix with parameters Σ and , and has density The matrix GIG (generalised inverse Gaussian) distribution on P with parameters ∈ R and , ∈ P is defined by For sufficiently smooth, this implies that Examples include and, more generally, for ∈ C, We note that the kernel defined by (2.19) satisfies

2.9.
Brownian motion and diffusion. We define Brownian motion in P with drift ∈ R to be the diffusion process in P with generator More generally, if is a positive eigenfunction of the Laplacian on P with eigenvalue , then we may consider the corresponding Doob transform If ( ) = | | for some ∈ R, then = 2 and Δ ( ) ≡ Δ ( ) . We shall refer to the diffusion process with infinitesimal generator Δ ( ) as a Brownian motion in P with drift .
A Brownian motion in P, with drift , may be constructed as follows. Let , ≥ 0 be a standard Brownian motion in the Lie algebra ( , R) of real × matrices, that is, each matrix entry evolves as a standard Brownian motion on the real line. Set = / √ 2 + . Define a Markov process , ≥ 0 in ( ) via the Stratonovich SDE: = . When = 0, this is called a right-invariant Brownian motion in ( ); thus, we shall refer to as a right-invariant Brownian motion in ( ) with drift . Then = is a Brownian motion in P with drift . Note that satisfies the Stratonovich SDE = ( + ) .
By orthogonal invariance of the underlying Brownian motion in ( , R), one may replace the and factors in this equation by 1/2 to obtain a closed SDE for the evolution of . We will also consider more general diffusions on P , with generators of the form where the are locally Lipschitz functions on P . For such generators, we may take the domain to be 2 (P ), the set of continuously twice differentiable, compactly supported, functions on P . If is a probability measure on P and the martingale problem associated with ( , ) is well posed, then we may construct a realisation of the corresponding Markov process by solving the (Stratonovich) SDE's: where 1 , . . . , are independent standard Brownian motions in ( , R), = ( + )/ √ 2, and (0) is chosen according to .

BROWNIAN PARTICLES WITH ONE-SIDED INTERACTION
Consider the differential operator on P 2 defined by Let ( , ) = etr (− −1 ) and consider the integral operator defined, for suitable test functions on P 2 , by Then, on a suitable domain, the following intertwining relation holds: Indeed, let us write = ( , ), = ( , ) and note the following identities: It follows, using the fact that Δ is self-adjoint with respect to , that as required. Now suppose is a positive eigenfunction of Δ with eigenvalue such that Then˜ is also a positive eigenfunction of Δ with eigenvalue : For example, if = , for some ∈ R satisfying (2.13), then˜ = Γ ( ) (see, for example, [47,Exercise 1.2.4]). Similarly, if ∈ R satisfies (2.13), and = ℎ or = , for some ∈ ( ), then it also holds that˜ = Γ ( ) . More generally,˜ is a constant multiple of whenever is a simultaneous eigenfunction of the Laplacian and the integral operator with kernel ( , ); note that these two operators commute, since Δ = Δ .
Then (3.2) extends to: The intertwining relation (3.3) has a probabilistic interpretation, as follows. Set Let be a probability measure on P and define a probability measure on P × P by Suppose that is such that the martingale problems associated with (Δ (˜ ) , ) and ( ( ) , ) are well-posed, and that ( , ) is a diffusion process with infinitesimal generator ( ) and initial law . Then it follows from the theory of Markov functions (see Appendix A) that, with respect to its own filtration, is a Brownian motion with drift˜ and initial distribution ; moreover, the conditional law of , given , ≤ , only depends on and is given by ( , ) ( ). This statement is analogous to the Burke output theorem for the / /1 queue, although in this context the 'output' (a Brownian motion with drift ) need not have the same law as the 'input' (a Brownian motion with drift ). Note however that these Brownian motions are equivalent whenever˜ is a constant multiple of , so whenever this holds the output does have the same law as the input. This is always the case when = 1, as was observed in the paper [39].
We note that the intertwining relation (3.3) also implies that where ( ( ) ) * is the formal adjoint of ( ) . One can replace by any invariant kernel ′ and the above remains valid with ).
In this case, we require˜ = to be finite, where  and initial law 0 ( )Π − ( , ). Then, with respect to its own filtration, is a Brownian motion with drift started at 0 . Moreover, the conditional law of , given , ≤ , only depends on and is given by Π − ( , ).
The above example extends naturally to a system of particles with one-sided interactions, as follows. Let 2 , . . . , ∈ R, and a positive eigenfunction of Δ such that For example, if ( ) = | | 1 then this condition is satisfied provided < 1 for all 1 < ≤ , in which case we have˜ .
This implies that * = 0 and, moreover, if is such that the relevant martingale problems are well posed and the system is started in equilibrium, then is a Brownian motion, in its own filtration, with drift˜ . This certainly holds in the case ( ) = | | 1 , with < 1 for all 1 < ≤ . Note that this can also be seen as a direct consequence of Theorem 2.

CONNECTION WITH BESSEL FUNCTIONS
The previous example, with two particles, extends naturally to . Note that this is a combination of the and ′ of the previous section.
Writing = ( 1 , 2 ), define Consider the integral operator, defined for suitable on P 2 × P by Then the following intertwining relation holds: Indeed, let us write = ( , ), = ( , ) and note that The claim follows, using the fact that Δ is self-adjoint with respect to . Suppose that is a positive eigenfunction of Δ with eigenvalue such that Then is a positive eigenfunction of with eigenvalue . We remark that, if = , then . Then (4.1) extends to: This intertwining relation has a probabilistic interpretation, as follows. Let be a probability measure on P 2 and define a probability measure on P 2 × P by Suppose that is such that the martingale problems associated with ( ( ) , ) and ( ( ) , ) are well-posed, and that ( , ) is a diffusion process with infinitesimal generator ( ) and initial law . Then, with respect to its own filtration, is a diffusion with generator ( ) and initial distribution ; moreover, the conditional law of , given , ≤ , only depends on and is given by The above example is a special case of a more general construction which will be discussed in Section 7.

MATRIX DUFRESNE IDENTITY AND
To see this, let = ( , ) = ( , ) and first note that, by invariance, Let us write 1 ( , ) = ( , ), 2 ( , ) = ( , ). By (2.1) and (2.2), It follows that and the corresponding integral operator Then, on a suitable domain, the following intertwining relation holds: Indeed, let us write = ( , ), = ( , ) and first note that Now, using the fact that Δ is self-adjoint with respect to , together with the identity It follows that ( ) ( ) = ( ) ( ), as required. Note that the intertwining relation (5.1) implies where is the linear operator defined, for suitable : P → C by The intertwining relation (5.2) is essentially equivalent to [42,Corollary 6]. Now suppose is a positive eigenfunction of Δ with eigenvalue such that = < ∞. Then it follows from (5.2) that is a positive eigenfunction of with eigenvalue . Note that if we write ( ) = ( ) ( ), then this implies This suggests that, for suitable , the function admits a natural probabilistic interpretation, via the Feynman-Kac formula, and this is indeed the case.

Proposition 3. Let be a positive eigenfunction of Δ such that
< ∞, and the martingale problem associated with Δ ( ) is well posed for any initial condition in P. Let be a Brownian motion in P with drift started at , and denote by E the corresponding expectation. Assume that, for any ∈ P, where > 0 is a constant. Then ( ) = ( ) and, moreover, is the unique bounded solution to (5.3) satisfying the boundary condition (5.5).
Proof. It follows from the Feynman-Kac formula that satisfies To prove uniqueness, up to a constant factor, suppose ( ) is another bounded solution which vanishes as → 0. Note that, by (5.4), it must hold that → 0 almost surely as → ∞. Thus, tr is a bounded martingale which converges to 0 almost surely, as → ∞, hence must be identically zero almost surely, which implies = 0, as required.
If ( ) = | | − /2 , then = − is the matrix -Bessel function defined by (2.17). If we denote the eigenvalues of by ( ), arranged in decreasing order, then, as shown in [42], it holds almost surely that, for any initial condition ∈ P, In particular, if > ( − 1)/2, then (5.4) holds. In this example, the process is ( , R)invariant, so we may write and it follows, using bounded convergence, that lim →0 ( ) = 1. On the other hand, again using bounded convergence, we have Putting this together and applying Proposition 3 yields the following conclusion, in agreement with [42,Theorem 2]. When = 1, this is Dufresne's identity [16]. More generally, suppose = ℎ , where ∈ R . Define new variables by It is well known that the spherical function ℎ is invariant under permutations of the , so we may assume that 1 > · · · > . Then it may be shown [8], via a straightforward modification of the proof of the second part of Theorem 3.1 in [9], that almost surely, for any initial condition. In particular, (5.4) holds if, and only if, 1 < 0. This condition also ensures that for all ∈ P. Then, by the uniqueness property of the spherical functions on P with a given set of eigenvalues [ This implies that is bounded. Using the homogeneity property ℎ ( ) = ℎ ( ), where = , we see that, for any fixed ∈ P, Assuming that these limits extend to Proposition 3 would then imply that Again using the homogeneity property of ℎ , this is equivalent to the identity: where is distributed according to the probability measure To make this claim rigorous, one would need to establish the existence of the limits in (5.10). We will not pursue this here. We remark that, writing = − , we may compute, for = 1, 2, 3: is the beta function. The analogue of this formula in the complex case is given by (10.10) below, which is valid for all . It seems natural to expect (5.13) to be valid for all also. Returning to the general setting, let us define As before, with the change of variables = , we can also write Then (5.1) extends to: This intertwining relation has a probabilistic interpretation, as follows. Let be a probability measure on P and define a probability measure on P × P by Suppose that is such that the martingale problems associated with ( ( ) , ) and ( ( ) , ) are well-posed, and that ( , ) is a diffusion process with infinitesimal generator ( ) and initial law . Then we may apply Theorem 15 to conclude that, with respect to its own filtration, is a diffusion with generator ( ) and initial distribution ; moreover, the conditional law of , given , ≤ , only depends on and is given by ( ). These conditions certainly hold when ( ) = | | /2 , for any ∈ R, in which case we obtain the following generalisation of [42,Proposition 23]. Define ( ) = | | /2 ( ).
Theorem 5. Let , ≥ 0 be a Brownian motion in P with drift /2 started at , and let = ∫ 0 . Fix 0 ∈ P, choose˜ 0 at random, independent of , according to the distribution 0 ( ), and definẽ =˜ 0 started at 0 . In particular, as a degenerate case, the process −1 −1 , > 0 is a diffusion in P with infinitesimal generator .
Proof. The relevant martingale problems are well posed, as shown in Appendix B (Example 5), so the first claim follows from Theorem 15, as outlined above. For the second, we can let 0 = and consider the limit as → ∞. By Lemma 1, 1/2˜ 0 → in probability, as required.
The second statement was proved, under the condition 2| | > − 1, by Rider and Valkó [42]. Related results in the complex setting have been obtained by Bougerol [7]. In the case = 1, the above theorem is due to Matsumoto and Yor [30], see also Baudoin [2]. We note that, as observed in [42], the law of the process with generator is invariant under a change of sign of the underlying drift , since = − , cf. (2.18). More generally, if is such that, as −1 → 0, the measure ( ) is concentrated around = , and the relevant martingale problems are well-posed, then the corresponding statement should hold: if is a Brownian motion in P with drift and = ∫ 0 , then −1 −1 , > 0 is a diffusion in P with generator ( ) .

TWO PARTICLES WITH ONE-SIDED INTERACTION AND A 'REFLECTING WALL'
Let ∈ R, and define We first note that is self-adjoint with respect to the measure Then Then is a positive eigenfunction of with eigenvalue . Define Then the above intertwining relation extends to For example, if = 1, then ( ) = ( −1 ). We note that is related to the of the previous section, via = ( ) , where = −1 and ( ) = | | /2 .
These intertwining relations (and their probabilistic interpretations) may be viewed as non-commutative, 'positive temperature' versions of the following well known relation between reflecting Brownian motion and the three-dimensional Bessel process: for appropriate initial conditions, a Brownian motion, reflected off a [Brownian motion reflected at zero], is a three-dimensional Bessel process ( [13], [1,Prop. 3.5]). In the above, note that , so we may interpret the -process as a Brownian motion in P with 'soft reflection off the identity', and the -process as a second Brownian motion in P with 'soft reflection off '.
(i) The following integral converges: The proof is given in Appendix C.3. We note that, when = 2, . The following properties are straightforward to verify from the definition. We also anticipate that ( ) is symmetric in the parameters 1 , . . . , ; in the case = 2, this symmetry holds and follows from (2.18).

7.2.
Interpretation as eigenfunctions. Consider the differential operator This is a quantisation of the -particle non-Abelian Toda lattice on P.
For ∈ C and ( , ) ∈ P × P −1 define We identify ( ) with the integral operator defined, for appropriate , by Note that, for ∈ C , It is straightforward to show that with the convention (1) = Δ. It follows that, on a suitable domain, .
The intertwining relation (7.5) implies that, for any ∈ C , The integral representation of Proposition 6 is a generalisation of the Givental-type formula obtained in [21] for the eigenfunctions of the quantum Toda lattice, also known as ( )-Whittaker functions. We remark that a slightly richer family of eigenfunctions of ( ) can be obtained by taking (1) to be an arbitrary eigenfunction of Δ in the recursive definition (7.4), provided the corresponding integrals converge. 7.3. Feynman-Kac interpretation. Define Brownian motion in P with drift ∈ R to be the diffusion process in P with infinitesimal generator We begin with a lemma.

Lemma 8. let be a Brownian motion in
and is a Wishart random matrix with parameters and 2 .
This follows from the matrix Dufresne identity [42, Theorem 1] (Corollary 4 in the present paper), together with the fact that the eigenvalue process of 1 ( ) −1/2 2 ( ) 1 ( ) −1/2 has the same law as that of a Brownian motion in P with generator 2Δ − 2 tr , started at . A proof of the latter claim is given in Appendix C.4. Now let ( ), ≥ 0 be a Brownian motion in P with drift ∈ R started at . Suppose that − > ( − 1)/2 for all < , and define Proposition 9. Suppose that − > ( − 1)/2 for all < . Then Moreover, under these hypotheses, ( ) is the unique solution to (7.7), up to a constant factor, such that − ( ) ( ) is bounded.
Proof. Define It follows from Lemma 8 that lim By Feynman-Kac, (Δ − ) = 0, hence = satisfies the eigenvalue equation (7.7). A standard martingale argument (as in the proof of 3) then shows that is the unique solution to (7.7) such that − is bounded and It therefore suffices to show that − is bounded and We prove these statements by induction on , using the recursion (7.4). For = 1, the claim holds since ( ) = ( ) in this case. For ≥ 2, we have where ′ = 1/2 1/2 . By induction on , we see immediately that

Let us write
Here we are using Now observe that, for each ∈ P −1 , if ( ) → 0 then ( −1) ( ′ ) → 0. Thus the claim follows, again by induction, using the dominated convergence theorem.
Here we have made the change of variables ′ = ( −1 , . . . , −1 1 ), as in Proposition 7. We remark that, by Proposition 10, for all such that + ∈ P, we have ∫ This implies that the ℎ marginal of , is the inverse Wishart distribution with parameters Σ = /2 and = 2 , as defined in Section 2.7. 7.5. Triangular processes. Consider the differential operator on T defined for ∈ C by Then the following intertwining relation holds: This extends (7.5) and is readily verified by induction. Define As shown in Appendix B (Example 6), for any ∈ R , the martingale problems associated with and are well posed, for any initial conditions. We thus deduce from the intertwining relation (7.9), and Theorem 15, the following theorem.
Theorem 11. Let ∈ R and ∈ P , and suppose that ( ), ≥ 0 is a diffusion in T with generator and initial law Then ( ), ≥ 0 is a diffusion in P with generator . Moreover, for each ≥ 0, the conditional law of ( ), given ( ), ≤ , is ( ( ), ).

Remark 12. Theorem 11 is a matrix generalisation of Proposition 9.1 in [34]. It is not clear whether Theorem 3.1 in [34] admits a similar generalisation, other than in the case = 2, where it is given by the statement of Theorem 5.
We note that contains the autonomous These are special cases of the diffusions with one-sided interactions discussed in Section 3, see (3.5) and (3.6). It follows from Theorem 11 that, if ( ), ≥ 0 is a diffusion in P with generator (resp. ′ ), with appropriate initial conditions, then ( ), ≥ 0 is distributed as the first (resp. last) coordinate of a diffusion in P with generator . In the scalar case, this yields very precise information about the law of ( ), for special initial conditions, see for example [34,Corollary 4.1]. We note however that this application in the scalar case relies on the Plancherel theory for the quantum Toda lattice, currently unavailable for its non-Abelian generalisation (7.2). In the scalar case, for a particular (singular) initial condition, the random variable ( ) may be interpreted as the logarithmic partition function of the semi-discrete directed polymer in a Brownian environment introduced in [39]. Unfortunately this interpretation does not extend to the non-Abelian setting, other than in an intuitive sense, because the SDE's do not admit an explicit solution as an integral in this case. Nevertheless, to understand the law of ( ) when > 1, and also its asymptotic behaviour as , → ∞, seems to be an interesting topic for future research.

THE NON-ABELIAN TODA LATTICE
The non-Abelian Toda lattice is a Hamiltonian system which describes the evolution of a system of particles 1 , . . . , in the space of invertible × matrices. There is a standard version, introduced by A.M. Polyakov [10], which generalises the classical Toda lattice. There is also an indefinite version, in which the potential has the opposite sign, as considered by Popowicz [40,41]. As in the scalar case [35,37], it is the indefinite version which is relevant to our setting. Writing = +1 −1 and = −1 , the Hamiltonian is given by , and the equations of motion are  Here we are using the notation [ , ] = − , where multiplication of matrices with matrix-valued entries is ordered, that is, The Lax representation implies that, for each positive integer , = ( ) is a constant of motion for the system, that is = 0. Note that these constants of motion are matrixvalued. In the following we will use the notation = ( ) ( , ) for the above Lax matrix, and ( ) for the corresponding constants of motion.
The equations of motion (8.1) can be written, equivalently, as In the scalar case, writing = , these reduce to the (indefinite) Toda equations Observe that the space P × S is invariant under the evolution (8.3), where S denotes the set of real symmetric × matrices. The system therefore admits a natural quantisation in this space, with Hamiltonian given by (7.2). The diffusion with generator , defined by (7.8) with ∈ R , is in fact a stochastic version of a series of Bäcklund transformations between non-Abelian Toda systems with different numbers of particles, as we shall now explain.
Recall the kernel function ( ) ( , ) defined by (7.3), and consider the following evolution in P × P −1 : Here we are using the conventions +1 = = −1 0 = 0. If (8.4) and (8.5) hold, then it is straightforward to compute −1 , ′ = +1 −1 , with conventions 0 = = 0 and ′ 0 = ′ −1 = 0. As such, this defines a Bäcklund transformation between the -and ( − 1)-particle systems. In the case = 0, it can be seen as a degeneration of the auto-Bäcklund transformation for the semi-infinite non-Abelian Toda lattice described in the paper [41]. Moreover, the constants of motion are related by ( ) = ( −1) + . This follows from the relation, readily verified from (8.4) and (8.5): This Bäcklund transformation may be used to construct solutions for the -particle system, recursively, as follows. Let T , T ( ), F ( ) and ( ) be defined as in Section 7.1. Let ∈ R and set F ( ) = F ( ) − ln ( ). By Proposition 6 (ii), F ( ) → +∞ as → T ( ), hence there exists * ( , ) ∈ T ( ) at which F ( ) achieves its minimum value on T ( ). Moreover, this minimiser must satisfy Equivalently, . Denote by T the set of ∈ T which satisfy the critical point equations (8.8), and consider the evolution on T defined by 1 Note that this is equivalent to To conclude, Theorem 11 says that if we add noise to the evolution (8.9), and choose the initial law on T appropriately, then ( ), ≥ 0 evolves as a diffusion in P with generator given by a Doob transform of the quantised Hamiltonian (7.2).

A RELATED CLASS OF PROCESSES
Let be a right-invariant Brownian motion in ( ), satisfying = , where is a Brownian motion in ( , R) with each entry having infinitesimal variance 1/2. Then = is a Brownian motion in P. As the evolution of is governed by Δ, its law is invariant under the action of ( ) on P. Norris, Rogers and Williams [33] consider the closely related Markov process = . This process has the same eigenvalues as , but its eigenvectors behave quite differently. It satisfies = + , from which the Markov property is evident, and one may compute its infinitesimal generator: Here, ′ = ( ) . The differential operator Ω is easily seen to be ( )-invariant, but not ( )-invariant. Nevertheless, it bears many similarities to the Laplacian. For example, it agrees with the Laplacian when applied to radial functions on P, as can be seen by noting that ′ tr = tr , for all positive integers (cf. (2.9)). It is selfadjoint with respect to , and invariant under the change of variables = −1 . If ( , ) = etr (− −1 ), then Ω = Ω . An important difference is the associated product rule, cf. (2.7): Many of the diffusions we have considered have natural analogues in which the underlying motion of particles is governed by Ω rather than Δ. Let be a right-invariant Brownian motion in ( ) with drift /2, started at . Then = is a Brownian motion in P with drift /2, and = is a diffusion in P with generator We note that this is the Doob transform of Ω via the positive eigenfunction | | /2 . Consider the process in P defined by and 0 is independent of . This is a diffusion in P with infinitesimal generator This process (with a different normalisation) was studied in [42], where it was observed that R is self-adjoint with respect to the measure (6.1), as in the case of a Brownian particle.
For several particles with one-sided interactions, the interactions need to be modified on account of the product rule (9.1). For example, we may consider by (3.4). Assuming the associated martingale problem is well posed, this yields the following analogue of the 'Burke' Theorem 2: if ( , ) be a diffusion in P 2 with generator T and initial law 0 ( )Π ( , ) then, with respect to its own filtration, is a diffusion in P with generator Ω ( ) , started at 0 ; moreover, the conditional law of , given , ≤ , only depends on and is given by Π ( , ).

THE COMPLEX CASE
Most of the discussion in this paper carries over naturally to the complex setting. We remark in particular that the matrix Dufresne identity and related 2 − theorem are studied in some detail in the complex setting by Rider and Valkó [42] and Bougerol [7]. In this section, we briefly outline how the framework developed in this paper may be extended to the complex setting, with particular emphasis on a complex version of the intertwining relation (5.2) and some of its consequences. We will also briefly indicate how this relates to a remarkable identity of Fitzgerald and Warren [19] concerning Brownian motion in a Weyl chamber, and closely related work of Nguyen and Remenik [31] and Liechty, Nguyen and Remenik [28] on non-intersecting Brownian bridges.
Let us define a 'partial matrix derivative' on P, writing = ( ), by One can check that tr ( ) = and = . This implies that tr 2 satisfies (10.2) and hence Δ = tr 2 , as in the real case.
We define Brownian motion in P to be the diffusion process with generator Δ/2. If is a positive eigenfunction of the Laplacian with eigenvalue , we may consider the corresponding Doob transform Δ ( ) /2, and call the diffusion process with this generator a Brownian motion in P with drift . In particular, for ∈ R , we may consider the positive eigenfunction , where is such that + ∈¯ , where = ( − 2 + 1)/2 and = { ∈ R : 1 > · · · > }. Then, as is well known, the logarithmic eigenvalues of a Brownian motion on P with drift evolve as a standard Brownian motion in R with drift + conditioned never to exit . With these ingredients in place, all of the basic calculus and intertwining relations discussed previously carry over, with the obvious modifications. We will briefly illustrate this here in the context of the intertwining relation (5.2) and consider some of its consequences.
10.1. Matrix Dufresne identity. As in the real case, it is known [29,11] that for each , ∈ P, the equation = has a unique solution in P, namely Moreover, if is fixed, then Then, on a suitable domain, the following intertwining relation holds: The proof is identical to the real case. Note that the intertwining relation (10.4) implies where is the linear operator defined, for suitable : P → C by Now suppose is a positive eigenfunction of Δ with eigenvalue such that = < ∞. Then it follows from (10.5) that is a positive eigenfunction of with eigenvalue . Note that if we write ( ) = ( ) ( ), then this implies As the real case, for suitable , the function admits a natural probabilistic interpretation, via the Feynman-Kac formula.
Proposition 13. Let be a positive eigenfunction of Δ such that < ∞, and the martingale problem associated with Δ ( ) is well posed for any initial condition in P. Let be a Brownian motion in P with drift started at , and denote by E the corresponding expectation. Assume that, for any ∈ P, Again the proof is identical to the real case. We obtain the following corollary, in agreement with [42,Corollary 10]. Let ∈ R . It follows from the above that := , which is easily seen to be finite, is a eigenfunction of . Thus, setting ( ) = ( ) −1 ( ), we have (10.9) (Δ ( ) − tr ) ( ) = 0.
Let us now assume that ∈ , with 2 1 < 1 − . Then and, by uniqueness of the spherical functions, ∫ Recall the representation (10.1) and note that, writing = − − , where 1 , . . . , denote the eigenvalues of . Thus, using Schur's Pfaffian identity and de Bruijn's formula [15], we may compute Let be a Brownian motion in P with drift , started at , and denote by E the corresponding expectation. The condition 2 1 < 1 − ensures that < ∞, almost surely. As in the real case, using the fact that the law of is invariant under multiplication by scalars, it is easy to see that E − → 1 and ( ) → as → 0 along any ray in P. To prove that these limits exist as → 0 in P is somewhat technical, and we will not pursue it here, but rather state it as a hypothesis: Assuming Hypothesis 1, it follows from Proposition 13 that Again using the fact that the law of is invariant under multiplication by scalars, the identity (10.11) is equivalent to the statement that has the same law as tr ( ), where is distributed according to the probability measure As this is a statement about the eigenvalues of , we may rephrase it in terms of a Brownian motion in , as follows. Let − , ∈ and assume that 1 > 0. Let ( ) be a Brownian motion with drift − , started at and conditioned never to exit . Let Then, assuming Hypothesis 1, has the same law as tr ( ), where has eigenvalues 1 , . . . , and is distributed according to The measure (10.12) is a generalised Bures measure and defines a Pfaffian point process related to the BKP hierarchy [48,Proposition 4.4].
On the other hand, by Brownian scaling and the method of Laplace, as → 0 the random variable ln converges in law to sup 1 ( ), where ( ) is a Brownian motion in R with drift − /2, conditioned to stay in , and started at the origin. Putting these observations together we conclude (modulo technicalities) that sup 1 ( ) has the same law as the first coordinate of the ensemble in + with density given by (10.14), in agreement with [19,Theorem 1].
We note that one may compute the law of sup 1 ( ) directly, for a general initial condition, as follows. Consider the process ( ), a BM with drift ∈ + , started at ∈ + and conditioned never to exit . Let be the first exit time of from + . Then, using formulas from [4] for the exit probabilities of a Brownian motion with drift from and + , This implies that, if ( ) is a Brownian motion in R with drift − /2, conditioned to stay in , and started at ∈ , and = sup 1 ( ), then, for 1 > 0 and ≥ 1 , we have the formula ( ≤ ) = ( , ), where = − +1 /2 and = − − +1 .
We remark that ( , ) may also be interpreted as the probability that a collection of non-intersecting Brownian bridges of unit length, started at positions 1 , . . . , and ending at positions 1 , . . . , , do not exit the domain + . This follows from the reflection principles associated with and + , and gives a formula for the distribution function of the maximum of the 'top' bridge, namely ( ) = ( ′ , ′ ), where ′ = − and ′ = − . In this context, the connection to the LOE was first discovered by Nguyen and Remenik [31] for bridges starting and ending at the origin, and for general starting and ending positions by Liechty, Nguyen and Remenik [28], where further asymptotic results are obtained in terms of Painlevé II and the KdV equation.

APPENDIX A. MARKOV FUNCTIONS
The theory of Markov functions is concerned with the question: when does a function of a Markov process inherit the Markov property? The simplest case is when there is symmetry in the problem, for example, the norm of Brownian motion in R has the Markov property, for any initial condition, because the Laplacian on R is invariant under the action of ( ). A more general formulation of this idea is the well-known Dynkin criterion [17]. There is another, more subtle, criterion which has been proved at various levels of generality by, for example, Kemeny and Snell [25], Rogers and Pitman [43] and Kurtz [26]. It can be interpreted as a time-reversal of Dynkin's criterion [24] and provides sufficient conditions for a function of a Markov process to have the Markov property, but only for very particular initial conditions. For our purposes, the martingale problem formulation of Kurtz [26] is best suited, as it is quite flexible and formulated in terms of infinitesimal generators.
Let be a complete, separable metric space. Denote by ( ) the set of Borel measurable functions on , by ( ) the set of bounded continuous functions on and by P ( ) the set of Borel probability measures on . Let : D ( ) ⊂ ( ) → ( ) and ∈ P ( ). A progressively measurable -valued process = ( , ≥ 0) is a solution to the martingale problem for ( , ) if 0 is distributed according to and there exists a filtration F such that is a F -martingale, for all ∈ D ( ). The martingale problem for ( , ) is well-posed is there exists a solution which is unique in the sense that any two solutions have the same finite-dimensional distributions.
The following is a special case of Corollary 3.5 (see also Theorems 2.6, 2.9 and the remark at the top of page 5) in the paper [26].
Theorem 15 (Kurtz, 1998). Assume that is locally compact, that : where the are locally Lipschitz functions on P . The corresponding SDE therefore admits a unique strong solution, for any given initial condition, up to an explosion time . If = ∞ almost surely, then the corresponding martingale problem is well-posed [18,27].
To show that = ∞ almost surely, it suffices to find a positive function and constants , such that ( ) → ∞ as → P and ≤ + . Such a function is called a Lyapunov function. In the following, we exhibit Lyapunov functions for some of the generators considered in this paper. is a Lyapunov function for the Laplacian on P. Indeed, using (2.4) we have Another choice is This is positive, since > . Using (2.4), (2.14) and the inequality > 2 , we have as required.
The functions and are also Lyapunov functions for Δ ( ) = Δ + 2 tr , since More generally, suppose is a positive eigenfunction of Δ with eigenvalue , and Considering positive and negative powers separately, and repeatedly applying the Chebyshev sum inequality, yields (B.3) with = + and = − .

Example 6
Here we consider the generators and defined by (7.8) and (7.10), respectively, with ∈ R . We follow the same approach as in Examples 3 and 4 above, which treat the case = 2. If is a Lyapunov function for with Σ < ∞, then, by the intertwining relation (7.9),˜ ( ) = ( ) −1 Σ ( ) is a Lyapunov function for .
We also have, using (2.3) and (2.14), respectively, We thus obtain a Lyapunov function for with ≤ ′ + and ′ = + 2( + 2) ( − 1). It remains to show that Σ ′ < ∞. This is straightforward, using 2 tr ( −1 ) ≤ tr 2 + tr −2 , and proceeding as in the proof of Proposition 6. Continuing this procedure implies the statement for odd, in which case we can take = 2 ( −1)/2 , = 1/ and = 0. For even, we bound the last term in the sum by   These integrals are easily seen to be finite using (2.12) (and in fact can be expressed as Pfaffians using de Bruijn's formula).