Abstract
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 lattice.
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1 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
where \(\beta _i\) 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 \(\mathcal {P}\) of positive \(n\times n\) real symmetric matrices. One of the main examples we consider is a generalisation of the system (1.1), a diffusion process in \(\mathcal {P}^N\) with infinitesimal generator
where \(\Delta _X\) denotes the Laplacian, and \(\partial _X\) denotes the partial matrix derivative, on \(\mathcal {P}\). In the case \(n=1\) with \(x_i=\ln X_i\), 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 \(\mathcal {P}\).
In another direction, Matsumoto and Yor [30] obtained an analogue of Pitman’s \(2M-X\) 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 \(\mathcal {P}\) 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 Sects. 3–6, by a series of examples with small numbers of particles. In Sect. 7, we discuss the example (1.2) and its relation to a quantisation of the non-Abelian Toda lattice. In Sect. 8, we outline how this example is related to some Bäcklund transformations for the classical non-Abelian Toda lattice. In Sect. 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 Sect. 10, we briefly outline 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].
2 Preliminaries
We mostly follow the nomenclature of Terras [47], to which we refer the reader for more background. Let \(\mathcal {P}\) denote the space of positive \(n\times n\) real symmetric matrices. For \(a\in GL(n)\) and \(X\in \mathcal {P}\), write \(X[a]=a^t X a\). This defines an action of GL(n) on \(\mathcal {P}\).
For \(X\in \mathcal {P}\), we will use the notation
and denote by \(X^{1/2}\) the unique positive square root of X.
2.1 Differential operators
The partial derivative on \(\mathcal {P}\) is defined, writing \(X=(x_{ij})\), by
We define the Laplacian on \(\mathcal {P}\) by \(\Delta _X = \,\text{ tr }\,\vartheta _X^2\), where \(\vartheta _X = X \partial _X\). The Laplacian is a GL(n)-invariant differential operator on \(\mathcal {P}\), meaning \((\Delta f)^a=\Delta f^a\) for all \(a\in GL(n)\), where \(f^a(X)=f(X[a])\). In fact, the differential operators \(L_k=\,\text{ tr }\,(\vartheta _X)^k,\ k=1,2,\ldots \) are all GL(n)-invariant [47, Exercise 1.1.27]. This follows from the fact that, if \(Y=X[a]\) for some fixed \(a\in GL(n)\), then
If \(Y=X^{-1}\), then \(\vartheta _Y=-\vartheta _X'\), where \(\vartheta _X' f = (\partial _X f)X\). It follows that \(L_k^Y=(-1)^k L_k^X\). In particular, the Laplacian is invariant under this change of variable.
2.2 Chain rule for quadratic change of variables
It is known [11, 29] that for each \(X,Y\in \mathcal {P}\), the equation \(Y=AXA\) has a unique solution in \(\mathcal {P}\), namely
This fact may also be deduced from [49, Theorem 4.1]. If X is fixed, then
2.3 Calculus
For \(A\in \mathcal {P}\) fixed:
For positive integers k,
2.4 Integration
Denote the GL(n)-invariant volume element on \(\mathcal {P}\) by
where \(dx_{ij}\) is the Lebesgue measure on \(\mathbb {R}\). If we write \(X=a[k]\), where a is diagonal with entries \(a_1,\ldots ,a_n>0\) and \(k\in O(n)\), then
where dk is the normalised Haar measure on O(n) and \(c_n\) is a normalisation constant.
2.5 Power and Gamma functions
For \(s\in \mathbb {C}^n\), define the power function
where \(X^{(k)}\) denotes the \(k\times k\) upper left hand corner of X. For \(s\in \mathbb {C}^n\) satisfying
define
For \(s=(0,\ldots ,0,\nu )\), we will write \(p_s(X)=e_\nu (X)=|X|^\nu \) and \(\Gamma _n(\nu )=\Gamma _n(0,\ldots ,0,\nu )\). The spherical functions on \(\mathcal {P}\) are defined, for \(s\in \mathbb {C}^n\), by
where dk denotes the normalised Haar measure on O(n).
The power function \(p_s\) is an eigenfunction of the Laplacian on \(\mathcal {P}\), with eigenvalue
(See, for example, [47, Exercise 1.2.12 & Equation (1.93)].) The functions \(p_{s,k}(X)=p_s(X[k]),\ k\in O(n)\), and \(h_s(X)\), are also eigenfunctions of \(\Delta \) with eigenvalue \(\lambda _2(s)\).
For \(s=(0,\ldots ,0,\nu )\), so that \(p_s(X)=e_\nu (X)=|X|^\nu \), we note:
2.6 Bessel functions
Following Terras [47], for \(s\in \mathbb {C}^n\) and \(V,W\in \mathcal {P}\), define
As remarked in [47], following Exercise 1.2.16, we can always reduce one of the positive matrix arguments in \(K_n(s|V,W)\) to the identity: if \(W=I[g]\), where g is upper triangular with positive diagonal coefficients, then
In the following, if \(s=(0,\ldots ,0,\nu )\) we will write \(K_n(s|V,W)=K_n(\nu |V,W)\).
For \(\nu \in \mathbb {C}\) and \(X\in \mathcal {P}\), define
This function was introduced by Herz [22], and is related to \(K_n\) by
We note that \(B_{-\nu }(X)=e_\nu (X)B_\nu (X)\), which implies
The asymptotic behaviour of \(B_\nu (X)\) for large arguments has been studied via Laplace’s method in the paper [12], see also [20, Appendix B]. If we fix \(M\in \mathcal {P}\) and let \(X=z^2 M^2/2\), then it holds that, as \(z\rightarrow \infty \),
where
and \(m_i\) denote the eigenvalues of M. In particular, taking \(M=I\) and \(z^2/2=\alpha \), say, we deduce from the same application of Laplace’s method, the following lemma, which we record here for later reference.
Lemma 1
For \(\alpha >0\), let \(A(\alpha )\) be distributed according to the (matrix GIG) law
Then \(\alpha ^{1/2} A(\alpha )\rightarrow I\), in probability, as \(\alpha \rightarrow \infty \).
2.7 Standard probability distributions
The Wishart distribution on \(\mathcal {P}\) with parameters \(\Sigma \in \mathcal {P}\) and \(p>n-1\) has density
If \(p\ge n\) is an integer and A is a \(p\times n\) random matrix with independent standard normal entries, then \(\Sigma ^{1/2}A^tA\Sigma ^{1/2}\) is Wishart distributed with parameters \(\Sigma \) and p. The inverse Wishart distribution on \(\mathcal {P}\), with parameters \(\Sigma \in \mathcal {P}\) and \(p>n-1\), is the law of the inverse of a Wishart matrix with parameters \(\Sigma \) and p, and has density
The matrix GIG (generalised inverse Gaussian) distribution on \(\mathcal {P}\) with parameters \(\nu \in \mathbb {R}\) and \(A,B\in \mathcal {P}\) is defined by
2.8 Invariant kernels
A kernel k(X, Y) is invariant if, for all \(a\in GL(n)\),
For k sufficiently smooth, this implies that
Examples include
and, more generally, for \(\nu \in \mathbb {C}\),
We note that the kernel k defined by (2.19) satisfies
2.9 Brownian motion and diffusion
We define Brownian motion in \(\mathcal {P}\) with drift \(\nu \in \mathbb {R}\) to be the diffusion process in \(\mathcal {P}\) with generator
More generally, if \(\varphi \) is a positive eigenfunction of the Laplacian on \(\mathcal {P}\) with eigenvalue \(\lambda \), then we may consider the corresponding Doob transform
If \(\varphi (X)=|X|^\nu \) for some \(\nu \in \mathbb {R}\), then \(\lambda =\nu ^2 n\) and \(\Delta _X^{(\varphi )} \equiv \Delta _X^{(\nu )}\). We shall refer to the diffusion process with infinitesimal generator \(\Delta _X^{(\varphi )}\) as a Brownian motion in \(\mathcal {P}\) with drift \(\varphi \).
A Brownian motion in \(\mathcal {P}\), with drift \(\nu \), may be constructed as follows. Let \(b_t,\ t\ge 0\) be a standard Brownian motion in the Lie algebra \(\mathfrak {gl}(n,\mathbb {R})\) of real \(n\times n\) matrices, that is, each matrix entry evolves as a standard Brownian motion on the real line. Set \(\beta _t=b_t/\sqrt{2}+\nu t\). Define a Markov process \(G_t,\ t\ge 0\) in GL(n) via the Stratonovich SDE: \(\partial G_t = \partial \beta _t\, G_t\). When \(\nu =0\), this is called a right-invariant Brownian motion in GL(n); thus, we shall refer to G as a right-invariant Brownian motion in GL(n) with drift \(\nu \). Then \(Y=G^tG\) is a Brownian motion in \(\mathcal {P}\) with drift \(\nu \). Note that Y satisfies the Stratonovich SDE
By orthogonal invariance of the underlying Brownian motion in \(\mathfrak {gl}(n,\mathbb {R})\), one may replace the G and \(G^t\) factors in this equation by \(Y^{1/2}\) to obtain a closed SDE for the evolution of Y.
We will also consider more general diffusions on \(\mathcal {P}^r\), with generators of the form
where the \(a_i\) are locally Lipschitz functions on \(\mathcal {P}^r\). For such generators, we may take the domain to be \(C^2_c(\mathcal {P}^r)\), the set of continuously twice differentiable, compactly supported, functions on \(\mathcal {P}^r\). If \(\rho \) is a probability measure on \(\mathcal {P}^r\) and the martingale problem associated with \((L,\rho )\) is well posed, then we may construct a realisation of the corresponding Markov process by solving the (Stratonovich) SDE’s:
where \(b_1,\ldots ,b_r\) are independent standard Brownian motions in \(\mathfrak {gl}(n,\mathbb {R})\), \(S_i=(b_i+b_i^t)/\sqrt{2}\), and X(0) is chosen according to \(\rho \).
3 Brownian particles with one-sided interaction
Consider the differential operator on \(\mathcal {P}^2\) defined by
Let \(k(X,Y)=\, \text{ etr }\,(-YX^{-1})\) and consider the integral operator defined, for suitable test functions f on \(\mathcal {P}^2\), by
Then, on a suitable domain, the following intertwining relation holds:
Indeed, let us write \(k=k(X,Y)\), \(f=f(X,Y)\) and note the following identities:
It follows, using the fact that \(\Delta \) is self-adjoint with respect to \(\mu \), that
as required.
Now suppose \(\varphi \) is a positive eigenfunction of \(\Delta \) with eigenvalue \(\lambda \) such that
Then \(\tilde{\varphi }\) is also a positive eigenfunction of \(\Delta \) with eigenvalue \(\lambda \):
For example, if \(\varphi =p_s\), for some \(s\in \mathbb {R}^n\) satisfying (2.13), then \(\tilde{\varphi }=\Gamma _n(s)\varphi \) (see, for example, [47, Exercise 1.2.4]). Similarly, if \(s\in \mathbb {R}^n\) satisfies (2.13), and \(\varphi =h_s\) or \(\varphi =p_{s,k}\) for some \(k\in O(n)\), then it also holds that \(\tilde{\varphi }=\Gamma _n(s)\varphi \). More generally, \(\tilde{\varphi }\) is a constant multiple of \(\varphi \) whenever \(\varphi \) is a simultaneous eigenfunction of the Laplacian and the integral operator with kernel k(X, Y); note that these two operators commute, since \(\Delta _X k = \Delta _Y k\).
Define
and
Then (3.2) extends to:
The intertwining relation (3.3) has a probabilistic interpretation, as follows. Set
Let \(\rho \) be a probability measure on \(\mathcal {P}\) and define a probability measure on \(\mathcal {P}\times \mathcal {P}\) by
Suppose that \(\varphi \) is such that the martingale problems associated with \((\Delta ^{(\tilde{\varphi })},\rho )\) and \((T^{(\varphi )},\sigma )\) are well-posed, and that \((X_t,Y_t)\) is a diffusion process with infinitesimal generator \(T^{(\varphi )}\) and initial law \(\sigma \). Then it follows from the theory of Markov functions (see Appendix A) that, with respect to its own filtration, \(X_t\) is a Brownian motion with drift \(\tilde{\varphi }\) and initial distribution \(\rho \); moreover, the conditional law of \(Y_t\), given \(X_s,\ s\le t\), only depends on \(X_t\) and is given by \( \pi (X_t,Y) \mu (dY)\). This statement is analogous to the Burke output theorem for the M/M/1 queue, although in this context the ‘output’ (a Brownian motion with drift \(\tilde{\varphi }\)) need not have the same law as the ‘input’ (a Brownian motion with drift \(\varphi \)). Note however that these Brownian motions are equivalent whenever \(\tilde{\varphi }\) is a constant multiple of \(\varphi \), so whenever this holds the output does have the same law as the input. This is always the case when \(n=1\), as was observed in the paper [39].
We note that the intertwining relation (3.3) also implies that
where \((T^{(\varphi )})^*\) is the formal adjoint of \(T^{(\varphi )}\).
One can replace k by any invariant kernel \(k'\) and the above remains valid with
For example, if \(k'=k_{\nu }\), defined by (2.20), then
In this case, we require \(\tilde{\varphi }= k_\nu \varphi \) to be finite, where
For example, if \(\varphi (X)=|X|^\lambda \), then this holds provided \(2(\lambda -\nu )>n-1\), in which case \(k_\nu \varphi =\Gamma _n(\lambda -\nu )\varphi \). For this example, the associated martingale problems are well posed, as shown in Appendix B (Example 2), so we may state the following theorem.
For \(2a>n-1\), we define the Markov kernel
Theorem 2
Suppose \(2(\lambda -\nu )>n-1\), and let \((X_t,Y_t)\) be a diffusion process in \(\mathcal {P}^2\) with infinitesimal generator
and initial law \(\delta _{X_0}(dX) \Pi _{\lambda -\nu }(X,dY)\). Then, with respect to its own filtration, \(X_t\) is a Brownian motion with drift \(\lambda \) started at \(X_0\). Moreover, the conditional law of \(Y_t\), given \(X_s,\ s\le t\), only depends on \(X_t\) and is given by \( \Pi _{\lambda -\nu }(X_t,dY)\).
The above example extends naturally to a system of N particles with one-sided interactions, as follows. Let \(\nu _2,\ldots ,\nu _N\in \mathbb {R}\), and \(\varphi \) a positive eigenfunction of \(\Delta \) such that
For example, if \(\varphi (X)=|X|^{\nu _1}\) then this condition is satisfied provided \(\nu _i<\nu _1\) for all \(1<i\le N\), in which case we have
Define
Then
This implies that \(T^*\pi =0\) and, moreover, if \(\varphi \) is such that the relevant martingale problems are well posed and the system is started in equilibrium, then \(X_N\) is a Brownian motion, in its own filtration, with drift \(\tilde{\varphi }\). This certainly holds in the case \(\varphi (X)=|X|^{\nu _1}\), with \(\nu _i<\nu _1\) for all \(1<i\le N\). Note that this can also be seen as a direct consequence of Theorem 2.
Finally we remark that, by a simple change of variables, one may also consider
The operator \(T'\) is related to T as follows. Write \(T=T_X(\varphi ,\nu )\), \(T'=T'_X(\varphi ,\nu )\). Then, under the change of variables \(Y_i=X_i^{-1}\), \(T_X(\varphi ,\nu )=T'_Y(\bar{\varphi },\bar{\nu })\), where \(\bar{\varphi }(Y)=\varphi (X^{-1})\) and \(\bar{\nu }_i=-\nu _i\).
In this case, if we assume that \(\tilde{\varphi } = (k_{-\nu _N}\circ \cdots \circ k_{-\nu _2})\varphi \) is finite, and define
then it holds that \(\Delta _{X_N}^{(\tilde{\varphi })} \circ K' = K' \circ T'\), with the analogous conclusions.
4 Connection with Bessel functions
The previous example, with two particles, extends naturally to
Note that this is a combination of the T and \(T'\) of the previous section.
Writing \(X=(X_1,X_2)\), define
Consider the integral operator, defined for suitable f on \(\mathcal {P}^2\times \mathcal {P}\) by
Then the following intertwining relation holds:
Indeed, let us write \(q=q(X,Y)\), \(f=f(X,Y)\) and note that
The claim follows, using the fact that \(\Delta \) is self-adjoint with respect to \(\mu \).
Suppose that \(\varphi \) is a positive eigenfunction of \(\Delta \) with eigenvalue \(\lambda \) such that
Then \(\psi \) is a positive eigenfunction of H with eigenvalue \(\lambda \). We remark that, if \(\varphi =p_s\), then
Let us define
Then (4.1) extends to:
This intertwining relation has a probabilistic interpretation, as follows. Let \(\rho \) be a probability measure on \(\mathcal {P}^2\) and define a probability measure on \(\mathcal {P}^2\times \mathcal {P}\) by
Suppose that \(\varphi \) is such that the martingale problems associated with \((H^{(\psi )},\rho )\) and \((G^{(\varphi )},\sigma )\) are well-posed, and that \((X_t,Y_t)\) is a diffusion process with infinitesimal generator \(G^{(\varphi )}\) and initial law \(\sigma \). Then, with respect to its own filtration, \(X_t\) is a diffusion with generator \(H^{(\psi )}\) and initial distribution \(\rho \); moreover, the conditional law of \(Y_t\), given \(X_s,\ s\le t\), only depends on \(X_t\) and is given by
The above example is a special case of a more general construction which will be discussed in Sect. 7.
5 Matrix Dufresne identity and \(2M-X\) theorem
Let \(M=\Delta _Y +\,\text{ tr }\,(Y\partial _A)\). If \(Y=AXA\) then, in the variables (X, A), we can write
To see this, let \(f=f(X,A)=g(AXA,A)\) and first note that, by invariance,
Let us write \(g_1(Y,A)=\partial _Y g(Y,A)\), \(g_2(Y,A)=\partial _A g(Y,A)\). By (2.1) and (2.2),
It follows that
as required.
Let us define
and the corresponding integral operator
Then, on a suitable domain, the following intertwining relation holds:
Indeed, let us write \(p=p(X,A)\), \(f=f(X,A)\) and first note that
Now, using the fact that \(\Delta \) is self-adjoint with respect to \(\mu \), together with the identity
we have
It follows that \(J(Pf)(X)=P(Mf)(X)\), as required.
Note that the intertwining relation (5.1) implies
where D is the linear operator defined, for suitable \(f:\mathcal {P}\rightarrow \mathbb {C}\) by
The intertwining relation (5.2) is essentially equivalent to [42, Corollary 6].
Now suppose \(\varphi \) is a positive eigenfunction of \(\Delta \) with eigenvalue \(\lambda \) such that \(\beta =D\varphi <\infty \). Then it follows from (5.2) that \(\beta \) is a positive eigenfunction of J with eigenvalue \(\lambda \). Note that if we write \(\beta (X)=\varphi (X) B_\varphi (X)\), then this implies
This suggests that, for suitable \(\varphi \), the function \(B_\varphi \) admits a natural probabilistic interpretation, via the Feynman-Kac formula, and this is indeed the case.
Proposition 3
Let \(\varphi \) be a positive eigenfunction of \(\Delta \) such that \(D\varphi <\infty \), and the martingale problem associated with \(\Delta ^{(\varphi )}\) is well posed for any initial condition in \(\mathcal {P}\). Let Y be a Brownian motion in \(\mathcal {P}\) with drift \(\varphi \) started at X, and denote by \(\mathbb {E}_X\) the corresponding expectation. Assume that, for any \(X\in \mathcal {P}\),
almost surely, and define \(M_\varphi (X)=\mathbb {E}_X e^{-Z}\). Suppose that \(\lim _{X\rightarrow 0} M_\varphi (X) =1\) and
where \(C_\varphi >0\) is a constant. Then \(B_\varphi (X)=C_\varphi \ M_\varphi (X)\) and, moreover, \(B_\varphi \) is the unique bounded solution to (5.3) satisfying the boundary condition (5.5).
Proof
It follows from the Feynman-Kac formula that \(M_\varphi \) satisfies
To prove uniqueness, up to a constant factor, suppose U(X) is another bounded solution which vanishes as \(X\rightarrow 0\). Note that, by (5.4), it must hold that \(Y_t\rightarrow 0\) almost surely as \(t\rightarrow \infty \). Thus,
is a bounded martingale which converges to 0 almost surely, as \(t\rightarrow \infty \), hence must be identically zero almost surely, which implies \(U= 0\), as required. \(\square \)
If \(\varphi (X)=|X|^{-\nu /2}\), then \(B_\varphi =B_{-\nu }\) is the matrix K-Bessel function defined by (2.17). If we denote the eigenvalues of \(Y_t\) by \(\lambda _i(Y_t)\), arranged in decreasing order, then, as shown in [42], it holds almost surely that, for any initial condition \(X\in \mathcal {P}\),
In particular, if \(\nu >(n-1)/2\), then (5.4) holds. In this example, the process \(Y_t\) is \(GL(n,\mathbb {R})\)-invariant, so we may write
and it follows, using bounded convergence, that \(\lim _{X\rightarrow 0} M_\varphi (X) =1\). 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 \(n=1\), this is Dufresne’s identity [16].
Corollary 4
If Y is a Brownian motion in \(\mathcal {P}\) with drift \(-\nu /2\), started at the identity, then \(\int _0^\infty Y_s\ ds\) is inverse Wishart distributed with parameters I/2 and \(2\nu \).
More generally, suppose \(\varphi =h_s\), where \(s\in \mathbb {R}^n\). Define new variables \(r_i\) by
It is well known that the spherical function \(h_s\) is invariant under permutations of the \(r_i\), so we may assume that \(r_1>\cdots >r_n\). 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, \(r_1<0\). This condition also ensures that
for all \(X\in \mathcal {P}\). Then, by the uniqueness property of the spherical functions on \(\mathcal {P}\) with a given set of eigenvalues [47, Proposition 1.2.4],
where
This implies that
is bounded. Using the homogeneity property \(h_s(cX)=c^dh_s(X)\), where \(d=\sum _k ks_k\), we see that, for any fixed \(X\in \mathcal {P}\),
and
Assuming that these limits extend to
Proposition 3 would then imply that
Again using the homogeneity property of \(h_s\), this is equivalent to the identity:
where A 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 \(r=-\mu \), we may compute, for \(n=1,2,3\):
where \(B(x,y)=\Gamma (x)\Gamma (y)/\Gamma (x+y)\) is the beta function. The analogue of this formula in the complex case is given by (10.10) below, which is valid for all n. It seems natural to expect (5.13) to be valid for all n also.
Returning to the general setting, let us define
As before, with the change of variables \(Y=AXA\), we can also write
Then (5.1) extends to:
This intertwining relation has a probabilistic interpretation, as follows. Let \(\rho \) be a probability measure on \(\mathcal {P}\) and define a probability measure on \(\mathcal {P}\times \mathcal {P}\) by
where
Suppose that \(\varphi \) is such that the martingale problems associated with \((J^{(\beta )},\rho )\) and \((M^{(\varphi )},\sigma )\) are well-posed, and that \((X_t,A_t)\) is a diffusion process with infinitesimal generator \(M^{(\varphi )}\) and initial law \(\sigma \). Then we may apply Theorem 15 to conclude that, with respect to its own filtration, \(X_t\) is a diffusion with generator \(J^{(\beta )}\) and initial distribution \(\rho \); moreover, the conditional law of \(A_t\), given \(X_s,\ s\le t\), only depends on \(X_t\) and is given by \(\gamma _{X_t}(dA)\).
These conditions certainly hold when \(\varphi (X)=|X|^{\nu /2}\), for any \(\nu \in \mathbb {R}\), in which case we obtain the following generalisation of [42, Proposition 23]. Define \(\beta _\nu (X)=|X|^{\nu /2} B_\nu (X)\).
Theorem 5
Let \(Y_t,\ t\ge 0\) be a Brownian motion in \(\mathcal {P}\) with drift \(\nu /2\) started at I, and let \(A_t=\int _0^t Y_s ds\). Fix \(X_0\in \mathcal {P}\), choose \(\tilde{A}_0\) at random, independent of Y, according to the distribution \(\gamma _{X_0}(dA)\), and define
Then \(X_t=\tilde{A}_t^{-1} \tilde{Y}_t \tilde{A}_t^{-1},\ t\ge 0\) is a diffusion in \(\mathcal {P}\) with infinitesimal generator
started at \(X_0\). In particular, as a degenerate case, the process \(A_t^{-1} Y_t A_t^{-1},\ t>0\) is a diffusion in \(\mathcal {P}\) with infinitesimal generator \(L_\nu \).
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 \(X_0=mI\) and consider the limit as \(m\rightarrow \infty \). By Lemma 1, \(m^{1/2} \tilde{A}_0\rightarrow I\) in probability, as required. \(\square \)
The second statement was proved, under the condition \(2|\nu |>n-1\), by Rider and Valkó [42]. Related results in the complex setting have been obtained by Bougerol [7]. In the case \(n=1\), 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 \(L_\nu \) is invariant under a change of sign of the underlying drift \(\nu \), since \(\beta _\nu =\beta _{-\nu }\), cf. (2.18).
More generally, if \(\varphi \) is such that, as \(X^{-1}\rightarrow 0\), the measure \(\gamma _X(dA)\) is concentrated around \(AXA=I\), and the relevant martingale problems are well-posed, then the corresponding statement should hold: if \(Y_t\) is a Brownian motion in \(\mathcal {P}\) with drift \(\varphi \) and \(A_t=\int _0^t Y_s ds\), then \(A_t^{-1} Y_t A_t^{-1},\ t>0\) is a diffusion in \(\mathcal {P}\) with generator \(J^{(\beta )}\).
6 Two particles with one-sided interaction and a ‘reflecting wall’
Let \(\nu \in \mathbb {R}\), and define
We first note that R is self-adjoint with respect to the measure
Define
Then \(R_Q \, \text{ etr }\,(-QX^{-1}) = S_X \, \text{ etr }\,(-QX^{-1})\), which implies \(S \circ C = C \circ N\).
Now suppose \(\rho \) is a positive eigenfunction of R with eigenvalue \(\lambda \) such that
Then \(\gamma \) is a positive eigenfunction of S with eigenvalue \(\lambda \). Define
Then the above intertwining relation extends to
For example, if \(\rho =1\), then \(\gamma (X)=B_{\nu }(X^{-1})\). We note that S is related to the J of the previous section, via \(S_X=J_Y^{(\varphi )}\), where \(Y=X^{-1}\) and \(\varphi (Y)=|Y|^{\nu /2}\).
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 \(\partial _Q=Q^{-1}\vartheta _Q\) and \(Q\partial _X=QX^{-1}\vartheta _X\), so we may interpret the Q-process as a Brownian motion in \(\mathcal {P}\) with ‘soft reflection off the identity’, and the X-process as a second Brownian motion in \(\mathcal {P}\) with ‘soft reflection off Q’.
7 Whittaker functions and related processes
7.1 Whittaker functions of several matrix arguments
For \(X=(X_1,\ldots ,X_N)\in \mathcal {P}^N\) and \(\nu \in \mathbb {C}\), we define
and, for \(X=(X_1,\ldots ,X_N)\in \mathcal {P}^N\) and \(\lambda \in \mathbb {C}^N\),
For \(N\ge 1\), we define the product measure
Let \(\mathcal {T}=\mathcal {P}\times \mathcal {P}^2\times \cdots \times \mathcal {P}^N\) and, for \(X\in \mathcal {P}^N\), denote by \(\mathcal {T}(X)\) the set of \(Y=(Y^1,\ldots ,Y^N)\in \mathcal {T}\) such that \(Y^N=X\). For \(Y\in \mathcal {T}\), define
For \(Y\in \mathcal {T}\) and \(\lambda \in \mathbb {C}^N\), define
Proposition 6
Let \(X\in \mathcal {P}^N\) and \(\lambda \in \mathbb {C}^N\).
-
(i)
The following integral converges:
$$\begin{aligned} \psi _\lambda (X)=\int _{\mathcal {T}(X)} e_\lambda (Y) e^{-\mathcal {F}(Y)} \prod _{1\le m<N} \mu _m(dY^m). \end{aligned}$$ -
(ii)
The integrand \(e_\lambda (Y) e^{-\mathcal {F}(Y)}\) vanishes as \(Y\rightarrow \partial \mathcal {T}(X)\).
The proof is given in Appendix C.3. We note that, when \(N=2\),
The following properties are straightforward to verify from the definition.
Proposition 7
Let \(X\in \mathcal {P}^N\), \(a\in GL(n)\), \(\lambda \in \mathbb {C}^N\), \(\nu \in \mathbb {C}\) and write \(\lambda '_i=\lambda _i+\nu \) and \(X'=(X_N^{-1},\ldots ,X_1^{-1})\). Then
We also anticipate that \(\psi _\lambda (X)\) is symmetric in the parameters \(\lambda _1,\ldots ,\lambda _N\); in the case \(N=2\), this symmetry holds and follows from (2.18).
7.2 Interpretation as eigenfunctions
Consider the differential operator
This is a quantisation of the N-particle non-Abelian Toda lattice on \(\mathcal {P}\).
For \(\nu \in \mathbb {C}\) and \((X,Y)\in \mathcal {P}^N\times \mathcal {P}^{N-1}\) define
We identify \(Q^{(N)}_\nu \) with the integral operator defined, for appropriate f, by
Note that, for \(\lambda \in \mathbb {C}^N\),
It is straightforward to show that
with the convention \(H^{(1)}=\Delta \). It follows that, on a suitable domain,
For \(\lambda \in \mathbb {C}^N\), set
The intertwining relation (7.5) implies that, for any \(\lambda \in \mathbb {C}^N\),
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 GL(n)-Whittaker functions. We remark that a slightly richer family of eigenfunctions of \(H^{(N)}\) can be obtained by taking \(\psi _\lambda ^{(1)}\) to be an arbitrary eigenfunction of \(\Delta \) in the recursive definition (7.4), provided the corresponding integrals converge.
7.3 Feynman–Kac interpretation
Define Brownian motion in \(\mathcal {P}^N\) with drift \(\lambda \in \mathbb {R}^N\) to be the diffusion process in \(\mathcal {P}^N\) with infinitesimal generator
We begin with a lemma.
Lemma 8
let Y be a Brownian motion in \(\mathcal {P}^2\) with drift \(\lambda \) started at X, with \(\nu =\lambda _1-\lambda _2>(n-1)/2\). Then
where \(A=X_1^{-1/2} X_2 X_1^{-1/2}\) and W is a Wishart random matrix with parameters I and \(2\nu \).
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 \(Y_1(t)^{-1/2} Y_2(t) Y_1(t)^{-1/2}\) has the same law as that of a Brownian motion in \(\mathcal {P}\) with generator \(2\Delta -2\nu \, \,\text{ tr }\,\vartheta _{X}\), started at A. A proof of the latter claim is given in Appendix C.4.
Now let \(Y(t),\ t\ge 0\) be a Brownian motion in \(\mathcal {P}^N\) with drift \(\lambda \in \mathbb {R}^N\) started at X. Suppose that \(\lambda _i-\lambda _j>(n-1)/2\) for all \(i<j\), and define
Proposition 9
Suppose that \(\lambda _i-\lambda _j>(n-1)/2\) for all \(i<j\). Then
Moreover, under these hypotheses, \(\psi _\lambda (X)\) is the unique solution to (7.7), up to a constant factor, such that \(e_{-\lambda }(X) \psi _\lambda (X)\) is bounded.
Proof
Define
It follows from Lemma 8 that
By Feynman-Kac, \((\Delta _\lambda -V)\varphi _\lambda =0\), hence \(f_\lambda =e_\lambda \varphi _\lambda \) satisfies the eigenvalue equation (7.7). A standard martingale argument (as in the proof of 3) then shows that \(f_\lambda \) is the unique solution to (7.7) such that \(e_{-\lambda } f_\lambda \) is bounded and
It therefore suffices to show that \(e_{-\lambda } \psi _\lambda \) is bounded and
We prove these statements by induction on N, using the recursion (7.4).
For \(N=1\), the claim holds since \(\psi _\lambda (X)=e_\lambda (X)\) in this case. For \(N\ge 2\), we have
where
Let us write
and
where \(A_i=X_i^{-1/2} X_{i+1} X_i^{-1/2}\). Changing variables from \(Y_i\) to \(X_i^{-1/2} Y_i X_i^{-1/2} \), we can write
where \(Y'_i=X_i^{1/2} Y_i X_i^{1/2}\). By induction on N, we see immediately that
Here we are using
which implies
Now observe that, for each \(Y\in \mathcal {P}^{N-1}\), if \(V(X)\rightarrow 0\) then \(V^{(N-1)}(Y')\rightarrow 0\). Thus the claim follows, again by induction, using the dominated convergence theorem. \(\square \)
In the scalar case \(n=1\), the above proposition is a special case of [3, Proposition 5.1], see also [36, Corollary 3].
7.4 Whittaker measures on \(\mathcal {P}^N\)
The following generalises an integral identity due to Stade [46]. The proof is straightforward, by induction, using (7.4). Denote by \(e_n\) the unit vector \((0,0,\ldots ,0,1)\) in \(\mathbb {C}^n\).
Proposition 10
Let \(s\in \mathbb {C}^n\), \(A\in \mathcal {P}\) and \(\lambda ,\nu \in \mathbb {C}^N\). Set \(a=\sum _i(\lambda _i+\nu _i)\). Assume that \(\mathfrak {R}(\lambda _i+\nu _j)>(n-1)/2\) for all i and j, and \(\mathfrak {R}(s_k+\cdots +s_n+a)>(k-1)/2\) for all k. Then
In particular, if \(\lambda ,\nu \in \mathbb {R}^N\) satisfy \(\lambda _i+\nu _j>(n-1)/2\) for all i, j, then
is a probability measure on \(\mathcal {P}^N\) which generalises the Whittaker measures of [14, 38]. Here we have made the change of variables \(X'=(X_N^{-1},\ldots ,X_1^{-1})\), as in Proposition 7. We remark that, by Proposition 10, for all B such that \(I+B\in \mathcal {P}\), we have
This implies that the \(N^{th}\) marginal of \(W_{\lambda ,\nu }\) is the inverse Wishart distribution with parameters \(\Sigma =I/2\) and \(p=2a\), as defined in Sect. 2.7.
7.5 Triangular processes
Consider the differential operator on \(\mathcal {T}\) defined for \(\lambda \in \mathbb {C}^N\) by
where \(\epsilon _{ij}=1-\delta _{ij}\).
Define a kernel from \(\mathcal {P}^N\) to \(\mathcal {T}\) by
and note that
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 \(\lambda \in \mathbb {R}^N\), the martingale problems associated with \(G_\lambda \) and \(L_\lambda \) 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 \(\lambda \in \mathbb {R}^N\) and \(X\in \mathcal {P}^N\), and suppose that \(Y(t),t\ge 0\) is a diffusion in \(\mathcal {T}\) with generator \(G_\lambda \) and initial law
Then \(Y^N(t),\ t\ge 0\) is a diffusion in \(\mathcal {P}^N\) with generator \(L_\lambda \). Moreover, for each \(t\ge 0\), the conditional law of Y(t), given \(Y^N(s),\ s\le t\), is \(\sigma _\lambda (Y^N(t),dY)\).
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 \(N=2\), where it is given by the statement of Theorem 5.
We note that \(G_\lambda \) contains the autonomous
These are special cases of the diffusions with one-sided interactions discussed in Sect. 3, see (3.5) and (3.6). It follows from Theorem 11 that, if \(X(t),\ t\ge 0\) is a diffusion in \(\mathcal {P}^N\) with generator \(T_\lambda \) (resp. \(T'_\lambda \)), with appropriate initial conditions, then \(X_N(t),\ t\ge 0\) is distributed as the first (resp. last) coordinate of a diffusion in \(\mathcal {P}^N\) with generator \(L_\lambda \). In the scalar case, this yields very precise information about the law of \(X_N(t)\), 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 \(X_N(t)\) 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 \(X_N(t)\) when \(n>1\), and also its asymptotic behaviour as \(N,t\rightarrow \infty \), seems to be an interesting topic for future research.
8 The non-Abelian Toda lattice
The non-Abelian Toda lattice is a Hamiltonian system which describes the evolution of a system of particles \(X_1,\ldots ,X_N\) in the space of invertible \(n\times n\) 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 \(A_i=X_{i+1} X_i^{-1} \) and \(B_i=\dot{X}_i X_i^{-1}\), the Hamiltonian is given by
and the equations of motion are
This system admits the Lax representation \(\dot{L}=[L,M]\), where
Here we are using the notation \([L,M]=LM-ML\), where multiplication of matrices with matrix-valued entries is ordered, that is,
The Lax representation implies that, for each positive integer k, \(C_k=\sum _i (L^k)_{ii}\) is a constant of motion for the system, that is \(\dot{C}_k=0\). Note that these constants of motion are matrix-valued. In the following we will use the notation \(L=L^{(N)}(A,B)\) for the above Lax matrix, and \(C^{(N)}_k\) for the corresponding constants of motion.
The equations of motion (8.1) can be written, equivalently, as
In the scalar case, writing \(X_i=e^{x_i}\), these reduce to the (indefinite) Toda equations
Observe that the space \(\mathcal {P}^N\times \mathcal {S}^N\) is invariant under the evolution (8.3), where \(\mathcal {S}\) denotes the set of real symmetric \(n\times n\) matrices. The system therefore admits a natural quantisation in this space, with Hamiltonian given by (7.2).
The diffusion with generator \(G_\lambda \), defined by (7.8) with \(\lambda \in \mathbb {R}^N\), 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 \(Q^{(N)}_{\nu }(X,Y)\) defined by (7.3), and consider the following evolution in \(\mathcal {P}^N\times \mathcal {P}^{N-1}\):
Here we are using the conventions \(X_{N+1}=Y_N=Y_0^{-1}=0\). If (8.4) and (8.5) hold, then it is straightforward to compute
where \(A_i=X_{i+1} X_i^{-1} \), \(A_i'=Y_{i+1} Y_i^{-1}\), with conventions \(A_0=A_N=0\) and \(A'_0=A'_{N-1}=0\). As such, this defines a Bäcklund transformation between the N- and \((N-1)\)-particle systems. In the case \(\nu =0\), 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 \(C^{(N)}_k=C^{(N-1)}_k+\nu ^k I\). This follows from the relation, readily verified from (8.4) and (8.5):
where
and
This Bäcklund transformation may be used to construct solutions for the N-particle system, recursively, as follows. Let \(\mathcal {T}\), \(\mathcal {T}(X)\), \(\mathcal {F}(Y)\) and \(e_\lambda (Y)\) be defined as in Sect. 7.1. Let \(\lambda \in \mathbb {R}^n\) and set \(\mathcal {F}_\lambda (Y)=\mathcal {F}(Y)-\ln e_\lambda (Y)\). By Proposition 6 (ii), \(\mathcal {F}_\lambda (Y)\rightarrow +\infty \) as \(Y\rightarrow \partial \mathcal {T}(X)\), hence there exists \(Y^*(\lambda ,X)\in \mathcal {T}(X)\) at which \(\mathcal {F}_\lambda (Y)\) achieves its minimum value on \(\mathcal {T}(X)\). Moreover, this minimiser must satisfy
Equivalently,
with the conventions \(Y^{m-1}_m=0\) and \((Y^{m-1}_{0})^{-1} =0\). The equations with \(2\le m\le N-1\) are equivalent to
Denote by \(\mathcal {T}_\lambda \) the set of \(Y\in \mathcal {T}\) which satisfy the critical point Eq. (8.8), and consider the evolution on \(\mathcal {T}\) defined by \(B^1_1=\dot{Y}^1_1 (Y^1_1)^{-1}=\lambda _1\) and, for \(2\le m\le N\),
Note that this is equivalent to
with the conventions, as above, \(Y^{m-1}_m=0\) and \((Y^{m-1}_{0})^{-1} =0\). It corresponds precisely to the drift term of the diffusion in \(\mathcal {T}\) with infinitesimal generator \(G_\lambda /2\).
It follows from (8.6) that \(\mathcal {T}_\lambda \) is invariant under the evolution (8.9), as in the scalar (\(n=1\)) case [35, Proposition 8.1]. Thus, if \(Y(0)=Y^*(\lambda ,X)\) and we let Y(t) evolve according to (8.9), then \(Y^N(t),\ t\ge 0\) is a realisation of the N-particle non-Abelian Toda flow on \(\mathcal {P}^N\), with \(Y^N(0)=X\) and \(C^{(N)}_k=\sum _i \lambda _i^k I\). In the scalar case, this agrees with the statement of [35, Theorem 8.4].
To conclude, Theorem 11 says that if we add noise to the evolution (8.9), and choose the initial law on \(\mathcal {T}\) appropriately, then \(Y^N(t),\ t\ge 0\) evolves as a diffusion in \(\mathcal {P}^N\) with generator given by a Doob transform of the quantised Hamiltonian (7.2).
9 A related class of processes
Let G be a right-invariant Brownian motion in GL(n), satisfying \(\partial G = \partial \beta \, G\), where \(\beta \) is a Brownian motion in \(\mathfrak {gl}(n,\mathbb {R})\) with each entry having infinitesimal variance 1/2. Then \(Y=G^tG\) is a Brownian motion in \(\mathcal {P}\). As the evolution of Y is governed by \(\Delta \), its law is invariant under the action of GL(n) on \(\mathcal {P}\).
Norris, Rogers and Williams [33] consider the closely related Markov process \(X=GG^t\). This process has the same eigenvalues as Y, but its eigenvectors behave quite differently. It satisfies \(\partial X = \partial \beta \, X+X\, \partial \beta ^t\), from which the Markov property is evident, and one may compute its infinitesimal generator:
Here, \(\vartheta _X' f = (\partial _X f)X\). The differential operator \(\Omega \) is easily seen to be O(n)-invariant, but not GL(n)-invariant. Nevertheless, it bears many similarities to the Laplacian. For example, it agrees with the Laplacian when applied to radial functions on \(\mathcal {P}\), as can be seen by noting that \(\vartheta _X' \,\text{ tr }\,X^k = \vartheta _X \,\text{ tr }\,X^k\), for all positive integers k (cf. (2.9)). It is self-adjoint with respect to \(\mu \), and invariant under the change of variables \(Y=X^{-1}\). If \(k(X,Y)=\, \text{ etr }\,(-YX^{-1})\), then \(\Omega _Xk=\Omega _Y k\). An important difference is the associated product rule, cf. (2.72.8):
Many of the diffusions we have considered have natural analogues in which the underlying motion of particles is governed by \(\Omega \) rather than \(\Delta \). Let G be a right-invariant Brownian motion in GL(n) with drift \(\nu /2\), started at I. Then \(Y=G^t G\) is a Brownian motion in \(\mathcal {P}\) with drift \(\nu /2\), and \(X=GG^t\) is a diffusion in \(\mathcal {P}\) with generator
We note that this is the Doob transform of \(\Omega _X\) via the positive eigenfunction \(|X|^{\nu /2}\).
Consider the process in \(\mathcal {P}\) defined by \(Q = G (Q_0+A) G^{t}\), where \(A_t=\int _0^t Y_s^{-1} ds\) and \(Q_0\) is independent of G. This is a diffusion in \(\mathcal {P}\) with infinitesimal generator
This process (with a different normalisation) was studied in [42], where it was observed that \(\mathcal {R}\) 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
Assume \(2\lambda >n-1\). Then \(\Omega ^{(\lambda )} \circ K_\lambda = K_\lambda \circ \mathcal {T}\), where \((K_\lambda f)(X)= \int _\mathcal {P}f(X,Y) \Pi _\lambda (X,dY)\) and \(\Pi _\lambda \) is defined by (3.4). Assuming the associated martingale problem is well posed, this yields the following analogue of the ‘Burke’ Theorem 2: if \((X_t,Y_t)\) be a diffusion in \(\mathcal {P}^2\) with generator \(\mathcal {T}\) and initial law \(\delta _{X_0}(dX) \Pi _{\lambda }(X,dY)\) then, with respect to its own filtration, \(X_t\) is a diffusion in \(\mathcal {P}\) with generator \(\Omega ^{(\lambda )}_X\), started at \(X_0\); moreover, the conditional law of \(Y_t\), given \(X_s,\ s\le t\), only depends on \(X_t\) and is given by \( \Pi _{\lambda }(X_t,dY)\).
10 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 \(2M-X\) 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.
In this section, \(\mathcal {P}\) will denote the space of positive \(n\times n\) Hermitian matrices. For \(a\in GL(n,\mathbb {C})\) and \(X\in \mathcal {P}\), write \(X[a]=a^\dagger X a\). This defines an action of \(GL(n,\mathbb {C})\) on \(\mathcal {P}\). The \(GL(n,\mathbb {C})\)-invariant volume element on \(\mathcal {P}\) is given by \(\mu (dX)=|X|^{-n} dX\), where dX denotes the Lebesgue measure on \(\mathcal {P}\). We note that, writing \(A=a[k]\), where \(k\in U(n)\) and a is the diagonal matrix with entries given by the eigenvalues \(z_1,\ldots ,z_n\) of A, with \(z\in C_+=\{z_1>\cdots>z_n>0\}\), we have the decomposition, on \(U(n)\times \mathbb {R}_+^n\),
The Laplacian on \(\mathcal {P}\) may be characterised (see, for example, [32]) as the unique invariant differential operator on \(\mathcal {P}\) which satisfies
Let us define a ‘partial matrix derivative’ \(\partial _X\) on \(\mathcal {P}\), writing \(X=(x_{ij})\), by
where
If \(Y=X[a]\) for some fixed \(a\in GL(n)\), then \( \partial _X=a\, \partial _Y \, a^\dagger \). This implies that \(\,\text{ tr }\,\vartheta _X^2\) is \(GL(n,\mathbb {C})\)-invariant, where \(\vartheta _X = X \partial _X\). One can check that \( \partial _X \,\text{ tr }\,(X)=I\) and \(\partial _X X=n I\). This implies that \(\,\text{ tr }\,\vartheta _X^2\) satisfies (10.2) and hence \(\Delta _X = \,\text{ tr }\,\vartheta _X^2\), as in the real case.
In this setting, the spherical functions are defined as follows. For \(\lambda ,x\in \mathbb {C}^n\), define \(a_\lambda (x)=\det (x_i^{\lambda _j})\) and \(s_\lambda (x)=a_{\delta +\lambda }(x)/a_\delta (x)\), where \(\delta _i=n-i\). For \(X\in \mathcal {P}\), define \(s_\lambda (X)\) to be the function \(s_\lambda \) evaluated at the eigenvalues of X.
We define Brownian motion in \(\mathcal {P}\) to be the diffusion process with generator \(\Delta /2\). If \(\varphi \) is a positive eigenfunction of the Laplacian with eigenvalue \(\gamma \), we may consider the corresponding Doob transform \(\Delta _X^{(\varphi )}/2\), and call the diffusion process with this generator a Brownian motion in \(\mathcal {P}\) with drift \(\varphi \). In particular, for \(\lambda \in \mathbb {R}^n\), we may consider the positive eigenfunction \(s_\lambda \), where \(\lambda \) is such that \(\rho +\lambda \in \bar{C}\), where \(\rho _i=(n-2i+1)/2\) and \(C=\{x\in \mathbb {R}^n:\ x_1>\cdots >x_n\}\). Then, as is well known, the logarithmic eigenvalues of a Brownian motion on \(\mathcal {P}\) with drift \(s_\lambda \) evolve as a standard Brownian motion in \(\mathbb {R}^n\) with drift \(\rho +\lambda \) conditioned never to exit C.
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 [11, 29] that for each \(X,Y\in \mathcal {P}\), the equation \(Y=AXA\) has a unique solution in \(\mathcal {P}\), namely
Moreover, if X is fixed, then
Let \(M=\Delta _Y +\,\text{ tr }\,(Y\partial _A)\). If \(Y=AXA\) then, in the variables (X, A), we can write
This follows from (10.3), as in the real case.
Let us define
and the corresponding integral operator
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 D is the linear operator defined, for suitable \(f:\mathcal {P}\rightarrow \mathbb {C}\) by
Now suppose \(\varphi \) is a positive eigenfunction of \(\Delta \) with eigenvalue \(\gamma \) such that \(\beta =D\varphi <\infty \). Then it follows from (10.5) that \(\beta \) is a positive eigenfunction of J with eigenvalue \(\gamma \). Note that if we write \(\beta (X)=\varphi (X) B_\varphi (X)\), then this implies
As the real case, for suitable \(\varphi \), the function \(B_\varphi \) admits a natural probabilistic interpretation, via the Feynman-Kac formula.
Proposition 13
Let \(\varphi \) be a positive eigenfunction of \(\Delta \) such that \(D\varphi <\infty \), and the martingale problem associated with \(\Delta ^{(\varphi )}\) is well posed for any initial condition in \(\mathcal {P}\). Let Y be a Brownian motion in \(\mathcal {P}\) with drift \(\varphi \) started at X, and denote by \(\mathbb {E}_X\) the corresponding expectation. Assume that, for any \(X\in \mathcal {P}\),
almost surely, and define \(M_\varphi (X)=\mathbb {E}_X e^{-Z}\). Suppose also that \(\lim _{X\rightarrow 0} M_\varphi (X) =1\) and
where \(C_\varphi >0\) is a constant. Then \(B_\varphi (X)=C_\varphi \ M_\varphi (X)\) and, moreover, \(B_\varphi \) is the unique bounded solution to (10.6) satisfying the boundary condition (10.8).
Again the proof is identical to the real case. We obtain the following corollary, in agreement with [42, Corollary 10].
Corollary 14
If \(\varphi (X)=|X|^{-\nu /2}\), where \(\nu >n-1\), then Z is inverse complex Wishart distributed with density proportional to
Let \(\lambda \in \mathbb {R}^n\). It follows from the above that \(\beta _\lambda :=Ds_\lambda \), which is easily seen to be finite, is a eigenfunction of J. Thus, setting \(B_\lambda (X)=s_\lambda (X)^{-1}\beta _\lambda (X)\), we have
Let us now assume that \(\lambda \in C\), with \(2\lambda _1<1-n\). Then
and, by uniqueness of the spherical functions,
Recall the representation (10.1) and note that, writing \(\mu =-\lambda -\rho \),
where \(z_1,\ldots ,z_n\) denote the eigenvalues of A. Thus, using Schur’s Pfaffian identity and de Bruijn’s formula [15], we may compute
Let Y be a Brownian motion in \(\mathcal {P}\) with drift \(s_\lambda \), started at X, and denote by \(\mathbb {E}_X\) the corresponding expectation. The condition \(2\lambda _1<1-n\) ensures that \(Z<\infty \), almost surely. As in the real case, using the fact that the law of Y is invariant under multiplication by scalars, it is easy to see that \(\mathbb {E}_X e^{-Z} \rightarrow 1\) and \(B_\lambda (X)\rightarrow c_\lambda \) as \(X\rightarrow 0\) along any ray in \(\mathcal {P}\). To prove that these limits exist as \(X\rightarrow 0\) in \(\mathcal {P}\) is somewhat technical, and we will not pursue it here, but rather state it as a hypothesis:
Hypothesis 1
As \(X\rightarrow 0\) in \(\mathcal {P}\), \(\mathbb {E}_X e^{-Z} \rightarrow 1\) and \(B_\lambda (X)\rightarrow c_\lambda \).
Assuming Hypothesis 1, it follows from Proposition 13 that
Again using the fact that the law of Y is invariant under multiplication by scalars, the identity (10.11) is equivalent to the statement that Z has the same law as \(\,\text{ tr }\,(AX)\), where A is distributed according to the probability measure
As this is a statement about the eigenvalues of Y, we may rephrase it in terms of a Brownian motion in C, as follows. Let \(-\mu ,x\in C\) and assume that \(\mu _1>0\). Let \(\xi (t)\) be a Brownian motion with drift \(-\mu \), started at x and conditioned never to exit C. Let
Then, assuming Hypothesis 1, Z has the same law as \(\,\text{ tr }\,(AX)\), where X has eigenvalues \(e^{x_1},\ldots ,e^{x_n}\) and A is distributed according to
If \(\xi (t)\) is started at the origin, then Z has the same law as \(\sum _i z_i\), where z has density
on \(C_+\). The normalisation constant \(b_\mu \) is given by
The measure (10.12) is a generalised Bures measure and defines a Pfaffian point process related to the BKP hierarchy [48, Proposition 4.4].
To conclude, consider the function
where \(K_\nu \) is the Macdonald function, and \(e^{x_1},\ldots ,e^{x_n}\) are the eigenvalues of X. It can be shown that \(\tilde{B}_\lambda (X)\) satisfies (10.9) and \(\lim _{X\rightarrow 0}B_\lambda (X)=c_\lambda \), so if Hypothesis 1 holds then \(B_\lambda (X) =\tilde{B}_\lambda (X)\).
10.2 Maximum of Dyson Brownian motion with negative drift
Let us write \(2\mu =\epsilon \alpha \), where \(-\alpha \in C\), \(\alpha _1>0\) and \(\epsilon >0\). Changing variables to \(y_i=\epsilon \ln z_i\), the density (10.12) becomes, up to a constant factor,
In the limit as \(\epsilon \rightarrow 0\), this reduces to
on the set \(C_+=\{y_1>\cdots>y_n>0\}\). Note that, in this scaling limit,
If \(\alpha _i=\alpha \), for all i, the measure (10.14) reduces to a special case of the Laguerre Orthogonal Ensemble (LOE) of random matrix theory.
On the other hand, by Brownian scaling and the method of Laplace, as \(\epsilon \rightarrow 0\) the random variable \(\epsilon \ln Z\) converges in law to \(\sup _t \eta _1(t)\), where \(\eta (t)\) is a Brownian motion in \(\mathbb {R}^n\) with drift \(-\alpha /2\), conditioned to stay in C, and started at the origin. Putting these observations together we conclude (modulo technicalities) that \(\sup _t \eta _1(t)\) has the same law as the first coordinate of the ensemble in \(C_+\) with density given by (10.14), in agreement with [19, Theorem 1].
We note that one may compute the law of \(\sup _t \eta _1(t)\) directly, for a general initial condition, as follows. Consider the process \(\xi (t)\), a BM with drift \(\nu \in C_+\), started at \(y\in C_+\) and conditioned never to exit C. Let T be the first exit time of \(\xi \) from \(C_+\). Then, using formulas from [4] for the exit probabilities of a Brownian motion with drift from C and \(C_+\),
This implies that, if \(\eta (t)\) is a Brownian motion in \(\mathbb {R}^n\) with drift \(-\alpha /2\), conditioned to stay in C, and started at \(x\in C\), and \(M=\sup _t \eta _1(t)\), then, for \(\alpha _1>0\) and \(z\ge x_1\), we have the formula \(P(M\le z)=f(\nu ,y)\), where \(\nu _i=\alpha _{n-i+1}/2\) and \(y_i=z-x_{n-i+1}\).
We remark that f(v, y) may also be interpreted as the probability that a collection of non-intersecting Brownian bridges of unit length, started at positions \(\nu _1,\ldots ,\nu _n\) and ending at positions \(y_1,\ldots ,y_n\), do not exit the domain \(C_+\). This follows from the reflection principles associated with C and \(C_+\), and gives a formula for the distribution function of the maximum of the ‘top’ bridge, namely \(F(r)=f(\nu ',y')\), where \(\nu _i'=r-\nu _i\) and \(y_i'=r-y_i\). 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.
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Acknowledgements
Thanks to the anonymous referees for many helpful comments on an earlier version, and to Philippe Bougerol for helpful correspondence.
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Appendices
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 \(\mathbb {R}^n\) has the Markov property, for any initial condition, because the Laplacian on \(\mathbb {R}^n\) is invariant under the action of O(n). 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 E be a complete, separable metric space. Denote by B(E) the set of Borel measurable functions on E, by \(C_b(E)\) the set of bounded continuous functions on E and by \(\mathcal {P}(E)\) the set of Borel probability measures on E. Let \(A:\mathcal {D}(A)\subset B(E)\rightarrow B(E)\) and \(\nu \in \mathcal {P}(E)\). A progressively measurable E-valued process \(X=(X_t,\ t\ge 0)\) is a solution to the martingale problem for \((A,\nu )\) if \(X_0\) is distributed according to \(\nu \) and there exists a filtration \(\mathcal {F}_t\) such that
is a \(\mathcal {F}_t\)-martingale, for all \(f\in \mathcal {D}(A)\). The martingale problem for \((A,\nu )\) is well-posed is there exists a solution X 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
[26] Assume that E is locally compact, that \(A:\mathcal {D}(A)\subset C_b(E) \rightarrow C_b(E)\), and that \(\mathcal {D}(A)\) is closed under multiplication, separates points and is convergence determining. Let F be another complete, separable metric space, \(\gamma :E\rightarrow F\) continuous and \(\Lambda (x,dz)\) a Markov transition kernel from F to E such that \(\Lambda (g\circ \gamma )=g\) for all \(g\in B(F)\), where \(\Lambda f(x)=\int _E f(z) \Lambda (x,dz)\) for \(f\in B(E)\). Let \(B:\mathcal {D}(B)\subset B(F)\rightarrow B(F)\), where \(\Lambda (\mathcal {D}(A))\subset \mathcal {D}(B)\), and suppose
Let \(\mu \in \mathcal {P}(F)\) and set \(\nu =\int _F \mu (dx) \Lambda (x,dz)\in \mathcal {P}(E)\). Suppose that the martingale problems for \((A,\nu )\) and \((B,\mu )\) are well-posed, and that Z is a solution to the martingale problem for \((A,\nu )\). Then \(X=\gamma \circ Z\) is a Markov process and a solution to the martingale problem for \((B,\mu )\). Furthermore, for each \(t\ge 0\) and \(f\in B(E)\) we have, almost surely,
To apply this theorem to the examples considered in Sects. 1–9, we take \(E=\mathcal {P}^a\), \(F=\mathcal {P}^b\), where \(a>b\), and \(\gamma (X,Y)=X\). In all of our examples, we have
where the \(a_i\) and \(b_i\) are locally Lipschitz functions on \(\mathcal {P}^a\) and \(\mathcal {P}^b\), respectively. For such generators, we may take \(\mathcal {D}(A)=C^2_c(\mathcal {P}^a)\) and \(\mathcal {D}(B)=C^2_c(\mathcal {P}^b)\). In our examples, the intertwining operator always has the form
where \(k\in C^2(\mathcal {P}^a)\). This ensures that \(\Lambda (g\circ \gamma )=g\) for all \(g\in B(\mathcal {P}^b)\), and \(\Lambda (\mathcal {D}(A))\subset \mathcal {D}(B)\). Thus, in all of our examples, as long as the martingale problems associated with A and B are well posed, for arbitrary initial conditions, and the intertwining relation \(B\Lambda f=\Lambda Af\) holds for \(f\in C^2_c(\mathcal {P}^a)\), then the conclusions of Theorem 15 are valid.
B: Well-posedness of martingale problems
In the following, \(\mathcal {P}\) is the space of positive \(n\times n\) real symmetric matrices. In all of the examples we consider (in Sections 1–9) we have generators of the form
where the \(b_i\) are locally Lipschitz functions on \(\mathcal {P}^r\). The corresponding SDE therefore admits a unique strong solution, for any given initial condition, up to an explosion time \(\tau \). If \(\tau =\infty \) almost surely, then the corresponding martingale problem is well-posed [18, 27].
To show that \(\tau =\infty \) almost surely, it suffices to find a positive function U and constants c, d such that \(U(X)\rightarrow \infty \) as \(X\rightarrow \partial \mathcal {P}^r\) and \(LU\le cU+d\). Such a function U is called a Lyapunov function. In the following, we exhibit Lyapunov functions for some of the generators considered in this paper.
Example 1
The function
is a Lyapunov function for the Laplacian on \(\mathcal {P}\). Indeed, using (2.4) we have
Another choice is
This is positive, since \(e^x>x\). Using (2.4), (2.14) and the inequality \(e^x>2x\), we have
as required.
The functions C and D are also Lyapunov functions for \(\Delta ^{(\nu )}_X=\Delta _X+2\nu \,\,\text{ tr }\,\vartheta _X\), since
More generally, suppose \(\varphi \) is a positive eigenfunction of \(\Delta \) with eigenvalue \(\lambda \), and
For positive integers k, l, define
By Lemma 18, there exist constants \(c_{k,l}\) such that \(\Delta U_{k,l} \le c_{k,l} U_{k,l}\), and hence
Thus, if there exist positive integers k, l such that \(U^{(\varphi )}_{k,l} (X) \rightarrow +\infty \) as \(X\rightarrow \partial \mathcal {P}\), then \(U^{(\varphi )}_{k,l}\) is a Lyapunov function for \(\Delta _X^{(\varphi )}\). For example, if \(\varphi =h_s\), where \(s\in \mathbb {R}^n\), then we may find \(p,q>0\) such that
and hence \(U^{(\varphi )}_{k,l}\) is a Lyapunov function for \(\Delta _X^{(\varphi )}\) provided \(k>p\) and \(l>q\). Indeed, recall that
where \(X^{(i)}\) denotes the \(i\times i\) upper left hand corner of X. Now, for each i,
hence
Considering positive and negative powers separately, and repeatedly applying the Chebyshev sum inequality, yields (B.3) with \(p=\sum _i i s_i^+\) and \(q=\sum _i is_i^-\).
Example 2
In Sect. 3, we encounter
In the following we continue to make use of the functions C and D defined in the previous example. Let
We compute
and
For the first identity we have used (2.4) and (2.15), and for the second we have used (2.3) and (2.14). It follows that
Now, using \(X^{-1/2}YX^{-1/2}\in \mathcal {P}\) and \(e^x>2x\), it holds that
and so \(TV\le 2(n+3)V\). For the Lyapunov function we can now take
Recalling that \( \Delta _Y C(Y) =(n+1)C(Y)/2\), we obtain \(TU\le 2(n+3)U\), as required.
One may also consider the same process with drifts:
In this case, we have \(T' U\le cU+d\), where \(d=2(\nu -\lambda )^+ n\) and \(c=2(n+3)+4(|\lambda |+|\nu |)\).
Example 3
For
we let \(U=C(Y)+ V'\), where C is defined by (B.1) and
with V defined by (B.4). Then it holds that
and hence
Example 4
Consider
where
Let G and U be as in the previous example. From (7.5), we have the intertwining
Together with (B.6), this implies that
where
By Lemma 16,
Together with Lemma 19, this implies that \(\tilde{U}(X)<\infty \) for all \(X\in \mathcal {P}^2\). Finally, by Fatou’s lemma, \(\tilde{U}(X)\rightarrow \infty \) as \(X\rightarrow \partial \mathcal {P}^2\), as required.
Example 5
In Sect. 5, especially Theorem 5, we encounter
Recall that \(L_\nu = J^{(\beta _\nu )}=\beta _\nu (X)^{-1}\circ (J-\lambda ) \circ \beta _\nu (X)\), where \(J=\Delta _X-\,\text{ tr }\,X\), \(\beta _\nu (X)=|X|^{\nu /2} B_\nu (X)\) and \(\lambda =n\nu ^2/4\).
For the first process, we can take
Then it holds that \(M_\nu U \le c U\), where \(c=(n+|\nu |+3)/2\). It follows, using the intertwining relation (5.14) together with Fatou’s lemma (as in the previous example), that \(\tilde{U}= P_{\nu } U\) is a Lyapunov function for the generator \(L_\nu \), where
Note that, by Lemma 16,
which together with Lemma 19 implies
When applying Fatou’s lemma, we note that, for fixed \(A\in \mathcal {P}\), \(X\rightarrow \partial \mathcal {P}\iff AXA\rightarrow \partial \mathcal {P}\).
Example 6
Here we consider the generators \(G_\lambda \) and \(L_\lambda \) defined by (7.8) and (7.10), respectively, with \(\lambda \in \mathbb {R}^N\). We follow the same approach as in Examples 3 and 4 above, which treat the case \(N=2\). If U is a Lyapunov function for \(G_\lambda \) with \(\Sigma _\lambda U<\infty \), then, by the intertwining relation (7.9), \(\tilde{U}(X)=\psi _\lambda (X)^{-1} \Sigma _\lambda U (X)\) is a Lyapunov function for \(L_\lambda \).
Consider the directed graph with vertices \(\{(i,m):\ 1\le i\le m\le N\}\), and edges
Let us simplify notation and write, for \(a=(i,m)\), \(Y_a=Y^m_i\), \(\partial _a=\partial _{Y_a}\), \(\vartheta _a = \vartheta _{Y_a}\), \(\Delta _a=\Delta _{Y_a}\), etc. We will also write \(\nu _a=\lambda _m\), for \(a=(i,m)\). Finally, for \(a=(i,m)\) denote \(a'=(i,m-1)\) and \(a''=(i-1,m-1)\), whenever these neighbouring vertices exist. Then we can write
Here we adopt the convention that the term involving \(a'\) is defined to be zero when \(a=(m,m)\) and similarly the term involving \(a''\) is defined to be zero when \(a=(1,m)\).
Let
where
and the sum is over all pairs of distinct vertices a, b such that there is a directed path starting at a and ending at b. We first note that
and for each pair \(a<b\), using (2.4) and (B.5),
Hence
We also have, using (2.3) and (2.14), respectively,
and
Combining these gives
where \(c=2(n+1)+2|\lambda _1|+4\max _{a<b}(\nu _a-\nu _b)^+\) and \(d=2n\sum _{a<b}(\nu _b-\nu _a)^+\).
Finally, we note that
and
hence
We thus obtain a Lyapunov function U for \(G_\lambda \) with \(G_\lambda U \le c' U+d\) and \(c'=c+2(N+2)(N-1)\). It remains to show that \(\Sigma _\lambda U' <\infty \). This is straightforward, using \(2\,\text{ tr }\,(Y_{a} Y_b^{-1} ) \le \,\text{ tr }\,Y_a^2+\,\text{ tr }\,Y_b^{-2}\), and proceeding as in the proof of Proposition 6.
C Some additional lemmas and proofs
1.1 C.1 Matrix inequalities
Lemma 16
For \(A,B,C\in \mathcal {P}\), we have
Proof
Let \(X=B^{1/2}A-B^{-1/2} C\). Then
and \(\,\text{ tr }\,X^tX \ge 0\) implies the result. \(\square \)
Note that this implies, for \(A,B,C\in \mathcal {P}\),
Also, taking \(B=C\), this becomes
and we note that iterating this gives
Lemma 17
For \(A_1,\ldots ,A_m\in \mathcal {P}\), there exist constants \(\alpha ,c>0\) and \(d\ge 0\) (depending only on m) such that
Proof
From the above lemma,
and
Summing these and applying the lemma again gives
Continuing this procedure implies the statement for m odd, in which case we can take \(c=2^{(m-1)/2}\), \(\alpha =1/c\) and \(d=0\). For m even, we bound the last term in the sum by
with \(k=(m-2)/2\), and then applying the lemma again implies the statement with \(c=2^{m/2}\), \(\alpha =1/c\) and \(d=n(2^k-1)\). \(\square \)
Lemma 18
For any integer k,
Proof
From (2.10) we have, for k positive,
By Chebyshev’s sum inequality, \(\,\text{ tr }\,X^j \,\text{ tr }\,X^{k-j}\le n\,\text{ tr }\,X^k\) for each j. Hence
Similarly, using (2.11), we have for l positive,
\(\square \)
1.2 C.2 Convergence lemma
Lemma 19
For any \(V,W\in \mathcal {P}\), \(\nu ,p\in \mathbb {C}\) and \(\alpha >0\), the integrals
and
are convergent.
Proof
Without loss of generality we can assume that \(\nu ,p\in \mathbb {R}\). If \(\kappa \) is the smallest among the set of eigenvalues of V and W, then
and
These integrals are easily seen to be finite using (2.12) (and in fact can be expressed as Pfaffians using de Bruijn’s formula). \(\square \)
1.3 C.3 Proof of Proposition 6
First note that, without loss of generality, we can assume \(\lambda \in \mathbb {R}^N\). For each \(1\le i<j\le N\), define
and observe that
By Lemma 17, there exist constants \(\alpha ,c,d>0\) such that
It follows that
where
These integrals converge by Lemma 19. For the second claim, set \(\mathcal {F}_\lambda (Y)=\mathcal {F}(Y)-\ln e_\lambda (Y)\). It follows from the above lower bound that there exist positive constants \(C,D,\alpha \) such that
Now, for any \(A,B\in \mathcal {P}\), \(a>0\) and \(b\in \mathbb {R}\), \(\,\text{ tr }\,(AZ^a+BZ^{-a})+b\ln |Z| \rightarrow +\infty \) as \(Z\rightarrow \partial \mathcal {P}\). Hence, \(\mathcal {F}_\lambda (Y)\rightarrow +\infty \) as \(Y\rightarrow \partial \mathcal {T}(X)\), as required.
1.4 C.4 Proof of Lemma 8
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 \(Y_1(t)^{-1/2} Y_2(t) Y_1(t)^{-1/2}\) has the same law as that of a Brownian motion in \(\mathcal {P}\) with generator \(2\Delta -2\nu \, \,\text{ tr }\,\vartheta _{X}\) started at A. The latter can be seen as follows. Consider a realisation of the process Y defined by \(Y_i=G_i^t G_i\), where \(G_i\) are independent, right-invariant Brownian motions on \(GL(n,\mathbb {R})\), with respective drifts \(\lambda _i\). Then the eigenvalues of \(Y_1(t)^{-1/2} Y_2(t) Y_1(t)^{-1/2}\) are the same as those of \(X=(G_2 G_1^{-1})(G_2 G_1^{-1})^t\). The process X is Markov with infinitesimal generator \(\Sigma =\Delta ^{(-\lambda _1)}+\Omega ^{(\lambda _2)}\), where \(\Omega ^{(\lambda _2)}\) is the generator of \(G_2 G_2^t\), as discussed in Sect. 9. The claim follows from the fact that \(\Sigma \) is O(n)-invariant and has the same radial part as \(\Delta ^{(-\lambda _1)}+\Delta ^{(\lambda _2)}=2\Delta -2\nu \, \,\text{ tr }\,\vartheta _{X}\).
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O’Connell, N. Interacting diffusions on positive definite matrices. Probab. Theory Relat. Fields 180, 679–726 (2021). https://doi.org/10.1007/s00440-021-01039-3
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DOI: https://doi.org/10.1007/s00440-021-01039-3
Keywords
- Whittaker functions of matrix argument
- Diffusion processes with one-sided interactions
- Intertwining relations
- Non-Abelian Toda lattice