Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions
aa r X i v : . [ m a t h . P R ] J un Brownian motions and eigenvalues on complex Grassmannianand Stiefel manifolds
Fabrice Baudoin ∗ , Jing Wang † June 11, 2020
Abstract
We study Brownian motions and related random matrices diffusions on the complexGrassmannian and Stiefel manifolds. In particular, the distribution of eigenvalues processesrelated to those Brownian motions is proved to be the law of a conditioned Karlin-McGregordiffusion associated to a Jacobi process and is shown to converge in large time to the dis-tribution of a Coulomb gas corresponding to a complex Jacobi ensemble. We then use theStiefel fibration to lift the Brownian motion of the complex Grassmannian to the complexStiefel manifold and deduce a skew-product decomposition from which we prove asymptoticwinding laws for the block entries of the unitary Brownian motion.
Contents G n,k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Invariant probability and convergence to equilibrium . . . . . . . . . . . . . . . . 9 J process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Eigenvalues process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Distribution and limit law of the eigenvalues . . . . . . . . . . . . . . . . . . . . . 14 ∗ F.B. was partly supported by the NSF grant DMS 1901315. † J.W. was partly supported by the NSF grant DMS 1855523. Proofs 26 G n,k . . . . . . . . . . . . . . . . . . . . . 265.2 Computation of the volume measure on G n,k . . . . . . . . . . . . . . . . . . . . 295.3 Stochastic differential equation for J . . . . . . . . . . . . . . . . . . . . . . . . . 305.4 Determinant of the J process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5 Stochastic differential equations for the eigenvalues . . . . . . . . . . . . . . . . . 325.6 Non-collision property for the eigenvalues . . . . . . . . . . . . . . . . . . . . . . 33 Motivation
The complex Grassmannian G n,k is the set of k -dimensional complex subspaces in C n . Grass-mannians have a very rich geometrical, combinatorial and topological structure and they natu-rally appear in algebraic topology, differential geometry, analysis, combinatorics, statistics andmathematical physics; see for instance [36]. The Riemannian metric of interest on G n,k is thecanonical one inherited from the Stiefel fibration U ( k ) → V n,k → G n,k , where V n,k denotes the complex Stiefel manifold that is given by the set of all unitary k -framesin C n , and where U ( k ) is the unitary group acting on V n,k . A goal of the paper is to studyBrownian motions and related processes on V n,k and G n,k with a view toward random matricestheory. Random matrix theory is a very rich and vibrant research topic with connections tomany areas of pure or applied mathematics and mathematical physics, see for instance [1] andthe references therein. The fruitful idea to introduce a time-dynamic on random matrices goesback at least to Freeman Dyson [23] who quantitatively described the eigenvalues dynamics of theHermitian Brownian motion and put forward the fundamental non-collision property exhibitedby this eigenvalues process. We also refer to the early work by Eugene Dynkin [22]. Since then,non colliding processes associated with random matrices models have extensively been studied,we refer for instance to [15, 31, 25, 32, 33]. The connection between the Brownian motionson V n,k or G n,k and random matrices theory is made through the unitary Brownian motionwhich is an extensively studied object in random matrix theory (see for instance [17, 35] andreferences therein). Indeed, we prove in Theorem 2.1 and Corollary 2.2 that if U t = (cid:18) X t Y t Z t W t (cid:19) is a Brownian motion on the unitary group U ( n ), then the C n × k matrix valued process (cid:18) X t Z t (cid:19) isa Brownian motion on V n,k and that the C ( n − k ) × k matrix valued process given by W t = X t Z − t is a Brownian motion on G n,k . As a consequence, one can study the block decomposition ofthe unitary Brownian motion using the Brownian motions on V n,k and G n,k . Both the complexmanifolds V n,k and G n,k have a rich differential geometric structure: G n,k is an irreducible rank k symmetric K¨ahler manifold and V n,k is a Riemannian homogeneous space that can be seenas a U ( k )-principal bundle over G n,k . Those rich structures provide many tools and techniquesto study Brownian motions and a novelty of the paper is to bring those tools in the field ofrandom matrices. In particular, we will see for instance how the connection form of the Stiefel2bration controls the windings of the process Z t and how the K¨ahler form of G n,k plays a rolein asymptotic winding laws for the process det Z t . Main results
We first study the Brownian motion W t on G n,k . We prove that it is a symmetric diffusionprocess on C ( n − k ) × k with generator12 ∆ G n,k = 2 X ≤ i,i ′ ≤ n − k, ≤ j,j ′ ≤ k ( I n − k + WW ∗ ) ii ′ ( I k + W ∗ W ) j ′ j ∂ ∂ W ij ∂ W i ′ j ′ , and invariant probability measure given by dµ = c n,k det( I k + W ∗ W ) − n d W . Since the complexGrassmannian manifold G n,k is an irreducible rank k symmetric K¨ahler manifold, its Riccicurvature can be computed explicitly following Calabi-Vesentini [14]. As a consequence, weobtain several quantitative functional inequalities satisfied by this invariant measure like thefamily of Beckner-Sobolev inequalities which include the Poincar´e and log-Sobolev inequalities.In particular, one obtains an explicit rate of convergence to equilibrium for W t . We stressthat since G n,k is a K¨ahler manifold, the constants we can get from [7] for the Poincar´e orlog-Sobolev inequality are sharper than the ones obtained by just applying classical curvaturedimension criteria. We then turn to the study of the eigenvalue process of W ∗ t W t and provethe following: Theorem 1.1.
Let ( W t ) t ≥ be a Brownian motion on the complex Grassmannian G n,k . Theordered eigenvalues process ( ρ ( t )) t ≥ of the diffusion (cid:0) ( I k − W ∗ t W t )( I k + W ∗ t W t ) − (cid:1) t ≥ hasgenerator k X i =1 (1 − ρ i ) ∂ i − k X i =1 (cid:18) n − k + ( n − k + 2) ρ i + 2 X ℓ = i − ρ i ρ ℓ − ρ i (cid:19) ∂ i and a density with respect to the Lebesgue measure dx given by e k ( k − n − k +2) t Q i>j ( x i − x j ) Q i>j ( ρ i (0) − ρ j (0)) det (cid:16) p n − k, t ( ρ i (0) , x j ) (cid:17) ≤ i,j ≤ k ∆ k ( x ) , where p n − k, t is the heat kernel of a one-dimensional Jacobi diffusion and ∆ k = { x ∈ [ − , k , − ≤ x < · · · < x k ≤ } . Moreover, when t → + ∞ , ρ ( t ) converges in distribution to the invariant probability measure dν = c n,k Y ≤ i Let ( W t ) t ≥ be a Brownian motion on G n,k started at W ∈ G n,k and let (Ω t ) t ≥ be a Brownian motion on the unitary group U ( k ) independent from ( W t ) t ≥ . Let (Θ t ) t ≥ be the U ( k ) valued solution of the Stratonovitch stochastic differential equation ( d Θ t = ◦ d a t Θ t Θ = I k , where a t is the Stratonovitch line integral R W [0 ,t ] η and η is a push-forward to G n,k of the con-nection form of the Stiefel fibration. Then, the process (cid:18) X t Z t (cid:19) := (cid:18) W t I k (cid:19) ( I k + W ∗ t W t ) − / Θ t Ω t , t ≥ is a Brownian motion on V n,k . Finally, in the last part of the paper we study limit laws for the windings of the blockdeterminants of the unitary Brownian motion and prove the following. Theorem 1.3. Let U t = (cid:18) X t Y t Z t W t (cid:19) be a Brownian motion on the unitary group U ( n ) with det( Z ) = 0 . One has the polar decom-position det( Z t ) = ̺ t e iθ t where < ̺ t = det( I k + W ∗ t W t ) − / ≤ and θ t is a real-valued continuous martingale suchthat: • If k = 1 , the following convergence holds in distribution when t → + ∞ θ t t → C n − , where C n − is a Cauchy distribution of parameter n − . • If k > , the following convergence holds in distribution when t → + ∞ θ t √ t → N (cid:18) , k ( n − k ) k − n (cid:19) , where N (cid:16) , k ( n − k ) k − + 2 n (cid:17) is a Gaussian random variable with mean 0 and variance k ( n − k ) k − +2 n . 4e note that for k = 1, the theorem therefore yields a Spitzer’s type theorem (see [37]) forthe windings of each of the entries of the unitary Brownian motion. The proof of Theorem 1.3relies on Theorem 1.2 and is rather long and difficult. The main step, interesting on its own, isto obtain the angular motion representation θ t = i tr( ω t ) + Z W [0 ,t ] α where ω t is a Brownian motion on u ( k ) independent from W and α is an almost everywheredefined one-form on G n,k whose exterior derivative yields the K¨ahler form. To study the func-tional R W [0 ,t ] α , which might be interpreted as a generalized stochastic area process in thesense of [8], we prove that R W [0 ,t ] α is a time-changed Brownian motion with independent clock R t tr( W ∗ s W s ) ds and finally study the distribution of this latter additive functional using a Gir-sanov transform in the spirit of [8, 21, 40] . Structure of the paper The paper is organized as follows. In Section 2, we first describe the geometry of the Stiefelfibration U ( k ) → V n,k → G n,k , and construct the Brownian motion on G n,k from the unitary Brownian motion. We thencompute its invariant probability measure and study convergence to equilibrium. In Section 3,we study the process J t := W ∗ t W t , t ≥ 0. We first show that it is a matrix diffusion processsolving the stochastic differential equation dJ = p I k + J d B ∗ p I k + J √ J + √ J p I k + J d B p I k + J + 2( n − k + t r ( J ))( I k + J ) dt where ( B t ) t ≥ is a k × k -complex-matrix-valued Brownian motion. The study of the properties ofthe process J is then inspired by the existing body of results concerning the Wishart processes,see for instance [13] and [19] and some of the techniques we use are similar to the techniquespresented by Yan Doumerc in his Phd thesis [21]. Actually the process J or rather ( I k − J )( I k + J ) − might be thought of as a complex projective analogue of the real Doumerc-Jacobi processesintroduced in [21], see also [18], [26] and [27]. We then turn to the study of the eigenvalues processof J . We show that it is a diffusion process with the non-colliding property and then give theproof of Theorem 1.1. As an interesting byproduct of Theorem 1.1 we can give an algebraicformula for the zonal spherical eigenfunctions of G n,k , see Remark 3.12. In Section 4, we studythe horizontal lift to V n,k of the Brownian motion on G n,k and then prove Theorem 1.2 andTheorem 1.3 . In Section 5, we collect some of the most computational or routine proofsAs a conclusion, let us point out that it has been a central problem in random matrix theoryto study the limiting behavior of eigenvalues in high dimensions. Free stochastic calculus thatwas introduced by Kummerer-Speicher [34] and then further developed by Biane-Speicher [12]studies the limiting process of certain type of n × n random matrices diffusions when n → ∞ .From [16], we expect that in our situation the study of the complex Grassmannian G n,αn andcomplex Stiefel spaces V n,αn when n → + ∞ and α is a fixed parameter will yield interesting5imit results for the objects we have been studying in this paper. This will possibly be addressedin a later research project. Notations: • If M ∈ C n × n is a n × n matrix with complex entries, we will sometimes denote M ∗ = M T its adjoint. • If z i = x i + iy i is a complex coordinate system ∂∂z i = 12 (cid:18) ∂∂x i − i ∂∂y i (cid:19) , ∂∂ ¯ z i = 12 (cid:18) ∂∂x i + i ∂∂y i (cid:19) . • For k ≥ S k denotes the permutation group of the set { , · · · , k } and for σ ∈ S k , wedenote sgn( σ ) its signature. • Throughout the paper we work on a filtered probability space (Ω , ( F t ) t ≥ , P ) that satisfiesthe usual conditions. • If X and Y are semimartingales, we denote R XdY the Itˆo integral, R X ◦ dY the Stratonovichintegral and R dXdY or h X, Y i the quadratic covariation. • For matrix-valued semimartingales M and N , the quadratic variation R dM dN is a matrixsuch that (cid:0)R dM dN (cid:1) ij = P ℓ R dM iℓ dN ℓj . • If X and Y are semimartingales, we write X ∼ Y to indicate that X − Y is a boundedvariation process. Let n ∈ N , n ≥ 2, and k ∈ { , . . . , n } . The complex Stiefel manifold V n,k is the set of unitary k -frames in C n . In matrix notation we have V n,k = { M ∈ C n × k | M ∗ M = I k } . As such V n,k is therefore an algebraic compact embedded submanifold of C n × k and inheritsfrom C n × k a Riemannian structure. We note that V n, is isometric to the unit sphere S n − .There is a right isometric action of the unitary group U ( k ) on V n,k , which is simply given bythe right matrix multiplication: M g , M ∈ V n,k , g ∈ U ( k ). The quotient space by this action G n,k := V n,k / U ( k ) is the complex Grassmannian manifold. It is a compact manifold of complexdimension k ( n − k ). We note that G n,k can be identified with the set of k -dimensional subspacesof C n . In particular G n, is the complex projective space C P n − . Since G n,k and G n,n − k can beidentified with each other via orthogonal complement, without loss of generality we can thereforeassume throughout the paper that k ≤ n − k .6et us quickly comment on the Riemannian structure of G n,k we will be using and whichis induced from the one of V n,k . From Example 2.3 in [5] , there exists a unique Riemannianmetric on G n,k such that the projection map π : V n,k → G n,k is a Riemannian submersion. FromExample 2.5 in [5] and Theorem 9.80 in [11] the fibers of this submersion are totally geodesicsubmanifolds of V n,k which are isometric to U ( k ). This therefore yields a fibration: U ( k ) → V n,k → G n,k which is often referred to as the Stiefel fibration, see also [3, 30]. We note that for k = 1 it isnothing else but the classical Hopf fibration considered from the probabilistic viewpoint in [8]: U (1) → S n − → C P n − . For further details on the Riemannian geometry of the complex Grassmannian manifolds we alsorefer to [38, 39], see in particular Theorem 4 in [38].More concretely, the computation of the Riemannian metric (or equivalently of the Laplace-Beltrami operator) on G n,k will be carried out explicitly in the next section in a convenient setof local coordinates that we now describe.In the following, we will use the block notations as below: For any U ∈ U ( n ) and A ∈ u ( n )we will write U = (cid:18) X YZ W (cid:19) , A = (cid:18) α βγ ǫ (cid:19) where X ∈ C ( n − k ) × k , Y ∈ C ( n − k ) × ( n − k ) , Z ∈ C k × k , W ∈ C k × ( n − k ) and α ∈ C k × k , β ∈ C k × ( n − k ) , γ ∈ C ( n − k ) × k , ǫ ∈ C ( n − k ) × ( n − k ) . We note that since (cid:18) X ∗ Z ∗ Y ∗ W ∗ (cid:19) (cid:18) X YZ W (cid:19) = (cid:18) X YZ W (cid:19) (cid:18) X ∗ Z ∗ Y ∗ W ∗ (cid:19) = I n , we have that: X ∗ X + Z ∗ Z = I k , X ∗ Y + Z ∗ W = 0 , Y ∗ Y + W ∗ W = I n − k and XX ∗ + Y Y ∗ = I n − k , ZX ∗ + W Y ∗ = 0 , ZZ ∗ + W W ∗ = I k . (2.1)We consider then the open set O ⊂ V n,k given by O = (cid:26)(cid:18) XZ (cid:19) ∈ V n,k , det Z = 0 (cid:27) and the smooth map p : O → C ( n − k ) × k given by p (cid:18) XZ (cid:19) = XZ − . It is clear that for every g ∈ U ( k ) and M ∈ V n,k , p ( M g ) = p ( M ). Since p is a submersion from O onto its image p ( O ) ⊂ C ( n − k ) × k we deduce that there exists a unique diffeomorphism Ψ from an open set of G n,k onto C ( n − k ) × k such that Ψ ◦ π = p. (2.2)The map Ψ induces a (local) coordinate chart on G n,k that we call inhomogeneous by analogywith the case k = 1 which corresponds to the complex projective space.7 .2 Brownian motion on G n,k In this section, we study the Brownian motion on G n,k and show how it can be constructedfrom a Brownian motion on the unitary group U ( n ). First, we recall that the Lie algebra u ( n )consists of all skew-Hermitian matrices u ( n ) = { X ∈ C n × n | X = − X ∗ } , which we equip with the inner product h X, Y i u ( n ) = − tr( XY ). This induces a Riemannianmetric on U ( n ). With respect to this inner product, an orthonormal basis of u ( n ) can be givenby { E ℓj − E jℓ , i ( E ℓj + E jℓ ) , T ℓ , ≤ ℓ < j ≤ n } where E ij = ( δ ij ( k, ℓ )) ≤ k,ℓ ≤ n , T ℓ = √ iE ℓℓ . A Brownian motion on u ( n ) is then of the form A t = X ≤ ℓ Let U t = (cid:18) X t Y t Z t W t (cid:19) be the solution of (2.3) . Then, P (inf { t > , det Z t = 0 } < + ∞ ) = 0 and the process ( W t ) t ≥ := (cid:0) X t Z − t (cid:1) t ≥ is a diffusion process with generator given by the diffu-sion operator ∆ G n,k , where ∆ G n,k = 4 X ≤ i,i ′ ≤ n − k, ≤ j,j ′ ≤ k ( I n − k + WW ∗ ) ii ′ ( I k + W ∗ W ) j ′ j ∂ ∂ W ij ∂ W i ′ j ′ . Proof. See Appendix 5.1.The following corollary shows that ( W t ) t ≥ is a Brownian motion on G n,k which is read ininhomogeneous coordinates. Corollary 2.2. Let ( W t ) t ≥ = (cid:0) X t Z − t (cid:1) t ≥ be the C ( n − k ) × k -valued process defined in Theorem2.1 and Ψ the map defined by (2.2) then the process (Ψ − ( W t )) t ≥ is a Brownian motion on G n,k and therefore ∆ G n,k is the Laplace-Beltrami operator of G n,k in inhomogeneous coordinates. roof. The smooth map p : O ⊂ V n,k → C ( n − k ) × k given by p (cid:18) XZ (cid:19) = XZ − is a submersionand the process (cid:18) X t Z t (cid:19) is a Brownian motion on V n,k . Let us now observe that∆ G n,k = 4 X ≤ i,i ′ ≤ n − k, ≤ j,j ′ ≤ k ( I n − k + WW ∗ ) ii ′ ( I k + W ∗ W ) j ′ j ∂ ∂ W ij ∂ W i ′ j ′ is the Laplace-Beltrami operator of a Riemannian metric on C ( n − k ) × k which is easy to compute.From Theorem 2.1, W t = p (cid:18) X t Z t (cid:19) is a Brownian motion for this Riemannian metric. Thisimplies that p is a Riemannian submersion and thus, since Ψ ◦ π = p , that Ψ is an isometry. Weconclude that (Ψ − ( W t )) t ≥ is indeed a Brownian motion on G n,k Thanks to this corollary, we can refer to W as a Brownian motion on G n,k . Since Ψ is anisometry, if needed, we can also identify C ( n − k ) × k with an open subset of G n,k . Note that inthis description of G n,k we are “missing” the boundary set det Z = 0, but that this set is polarfor the Brownian motion (according to Lemma 5.1). When k = 1 we recover the expression forthe Laplacian in inhomogeneous coordinates:∆ C P n − = 4(1 + | w | ) n − X k =1 ∂ ∂w k ∂w k + 4(1 + | w | ) RR where R = n − X j =1 w j ∂∂w j . We refer to [8] and [9] for a review of the Brownian motion on C P n − . We now study the invariant probability measure on G n,k and the convergence to equilibrium ofthe Brownian motion to this measure. Let us consider on G n,k the probability measure given ininhomogeneous coordinates by dµ := c n,k det( I k + W ∗ W ) − n d W where c n,k is the normalization constant and d W the Lebesgue measure on C ( n − k ) × k . Proposition 2.3. The probability measure µ is invariant and symmetric for the operator ∆ G n,k .More precisely, for every smooth and compactly supported functions f, g on C ( n − k ) × k the follow-ing integration by parts formula holds Z (∆ G n,k f ) g dµ = Z f (∆ G n,k g ) dµ = − Z Γ( f, g ) dµ, where the carr´e du champ operator Γ( f, g ) := 12 (cid:0) ∆ G n,k ( f g ) − (∆ G n,k f ) g − (∆ G n,k g ) f (cid:1) s given by Γ( f, g ) = 2 X ≤ i,i ′ ≤ n − k, ≤ j,j ′ ≤ k ( I n − k + WW ∗ ) ii ′ ( I k + W ∗ W ) j ′ j (cid:18) ∂f∂ W ij ∂g∂ W i ′ j ′ + ∂g∂ W ij ∂f∂ W i ′ j ′ (cid:19) . Proof. See Section 5.2.We obtain an interesting corollary about the distribution of some random matrices. Corollary 2.4. Let U = (cid:18) X YZ W (cid:19) be a random variable on U ( n ) be distributed according tothe (normalized) Haar measure on U ( n ) . Then, the random variable W = XZ − ∈ C ( n − k ) × k has density c n,k det( I k + W ∗ W ) − n with respect to the Lebesgue measure.Proof. If U = (cid:18) X YZ W (cid:19) is distributed according to the (normalized) Haar measure on U ( n ),then (cid:18) XZ (cid:19) is distributed according to the (normalized) Riemannian volume measure on theStiefel manifold V n,k . Thus, since p (cid:18) XZ (cid:19) = XZ − is a totally geodesic Riemannian submer-sion, one deduces that XZ − is distributed according to the Riemannian volume of G n,k ininhomogeneous coordinates, which is µ thanks to Proposition 2.3.We now discuss the convergence of the Brownian motion on G n,k to the invariant probability µ and related functional inequalities. The basic lemma is the following. Lemma 2.5. The Riemannian manifold G n,k is an Einstein manifold with constant Ricci cur-vature n .Proof. The complex Grassmannian manifold G n,k is an irreducible rank k symmetric K¨ahlermanifolds and thus is an Einstein manifold, see Calabi-Vesentini [14]. The value of the Einsteinconstant can be seen from the expansion of the Calabi diastasis D (0 , W ) in a neighborhood ofthe origin (see [14] page 502 for further details): D (0 , W ) = log det( I k + W ∗ W ) − = ∞ X ℓ =1 ℓ t r (( W ∗ W ) ℓ )= X i,j | W ij | − X i,j,p,q W ij W iq W pq W pj + o ( | W | ) . From the expansion we know that W ij are canonical coordinates at the origin, hence we cancompute the (complex) curvature tensor R ij pq st uv at the origin by differentiating four times D (0 , W ): R ij pq st uv (0) = − ( δ ip δ su δ jv δ qt + δ iu δ ps δ jq δ tv ) . R ij uv = − X p,q,s,t R ij pq st uv (0) g (0) pq st = kδ iu δ jv + ( n − k ) δ iu δ jv = ng (0) ij uv . The Einstein constant of G n,k is thus 2 n .An explicit formula for the Ricci curvature provides information about various functionalinequalities satisfied by the invariant probability measure µ . In particular, since G n,k is ad-ditionally a K¨ahler manifold, one deduces (see [7]) that µ satisfies the following log-Sobolevinequality Z C ( n − k ) × k f ln f dµ − (cid:18)Z C ( n − k ) × k f dµ (cid:19) ln (cid:18)Z C ( n − k ) × k f dµ (cid:19) (2.4) ≤ k ( n − k )( k ( n − k ) + 1) n Z C ( n − k ) × k Γ( f, f ) dµ, and the following Poincar´e inequality Z C ( n − k ) × k f dµ − Z C ( n − k ) × k f dµ ≤ n Z C ( n − k ) × k Γ( f, f ) dµ. (2.5)We note that the constant n is sharp for the Poincar´e inequality (2.5) because the firsteigenvalue of G n,k is indeed equal to 4 n , see Theorem 3.9. We do not know if the constant k ( n − k )( k ( n − k )+1) n is sharp or not for the log-Sobolev inequality (2.4). From this, one can easily deducesthat ( W t ) t ≥ converges exponentially fast to equilibrium with an explicit rate that can beestimated. In particular, we obtain for instance: Corollary 2.6. When t → + ∞ , W t → µ in distribution. Moreover, we have the followingquantitative estimate: There exists a constant C > such that for any bounded Borel function f on C ( n − k ) × k and t ≥ , (cid:12)(cid:12)(cid:12)(cid:12) E ( f ( W t )) − Z C ( n − k ) × k f dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − nt k f k ∞ . Proof. The estimate classically follows from the Poincar´e inequality (2.5) by heat semigrouptheory. We now turn to the study of the eigenvalues process of the random matrices ( W ∗ t W t ) t ≥ J process In this section we study the C k × k valued stochastic process J t := W ∗ t W t where, as before, W t is a Brownian motion on G n,k i.e. a diffusion with generator ∆ G n,k . Let H k denote the set11f k × k Hermitian matrices and let ˆ H k be the definite positive cone in H k . We assume that J ∈ ˆ H k , and consider the stopping time T = inf { t > , J t ˆ H k } = inf { t > , det( J t ) = 0 } . Theorem 3.1. Let ( J t ) t ≥ be given as above, then up to time T , it satisfies the followingstochastic differential equation dJ = p I k + J d B ∗ p I k + J √ J + √ J p I k + J d B p I k + J + 2( n − k + t r ( J ))( I k + J ) dt (3.6) where ( B t ) t ≥ is a Brownian motion in C k × k .Proof. See Section 5.3.Our next goal is to prove that P ( T < + ∞ ) = 0 so that the stochastic differential equation(3.6) is actually defined for all t ≥ 0. First we compute determinants related to J in the lemmabelow. Lemma 3.2. Let ( J t ) t ≥ be as previously defined. We have for any t ≥ that d (det( J )) = det( J )t r (cid:16) J − / ( I k + J )( d B + d B ∗ ) (cid:17) + 2 det( J ) (cid:18) k − k + nk + t r ( J ) + ( n + 1 − k )t r ( J − ) (cid:19) dt, (3.7) where B is a k × k -matrix-valued Brownian motion. As a consequence we have d (log det( J )) = t r (cid:16) J − / ( I k + J )( d B + d B ∗ ) (cid:17) + 2 (cid:18) k ( n − k ) + ( n − k )t r ( J − ) (cid:19) dt. (3.8) Proof. See Section 5.4. Proposition 3.3. Assume J ∈ ˆ H k . Then, we have almost surely that T = + ∞ .Proof. Consider the process (Γ t := log(det( J t ))) t ≥ . From the above lemma we have d Γ = t r ( H ( d B + d B ∗ )) + V dt where H = J − / ( I k + J ) and V = 2 (cid:18) k ( n − k ) + ( n − k )t r ( J − ) (cid:19) . The local martingale partis a time-changed Brownian motion β C t and V ≥ k ( n − k ). Hence we haveΓ t − Γ − k ( n − k ) t ≥ β C t . On { T < + ∞} , we then have lim t → T β C t = −∞ . This implies that P ( T < + ∞ ) = 0.12 .2 Eigenvalues process In this section we study the eigenvalues of the process ( J t ) t ≥ . We denote by λ ( t ) = ( λ i ( t )) ≤ i ≤ k the eigenvalues of J t , t ≥ 0. Let N = { λ ∈ (0 , ∞ ) k , λ i = λ j , ∀ i = j } , and consider the stoppingtime τ N = inf { t > , λ ( t ) 6∈ N } . (3.9) Theorem 3.4. Assume λ (0) ∈ N . Then up to time τ N , the eigenvalues λ ( t ) = ( λ , . . . , λ k )( t ) , t ≥ satisfy the following stochastic differential equation dλ i = 2(1 + λ i ) p λ i dB i + 2(1 + λ i ) (cid:18) n − k + 1 − (2 k − λ i + 2 λ i (1 + λ i ) X ℓ = i λ i − λ ℓ (cid:19) dt. (3.10) where ( B t ) t ≥ is a Brownian motion in R k .Proof. See Section 5.5.The next theorem establishes the non-collision property for the eigenvalues process. Theorem 3.5. Let λ be the eigenvalues process of J and τ N the stopping time as given in (3.9) .Assume that at time t = 0 , λ (0) ∈ N . Then for all t ≥ , λ ( t ) ∈ N a.s. and therefore P ( τ N < + ∞ ) = 0 . Proof. See Section 5.6.We can simplify the stochastic differential equation (3.10) with a simple algebraic transfor-mation. Corollary 3.6. Let ρ i = − λ i λ i , i = 1 , . . . , k . Then, we have dρ i = − q − ρ i dB i − (cid:18) ( n − k + ( n − k + 2) ρ i ) + 2 X ℓ = i − ρ i ρ ℓ − ρ i (cid:19) dt, where ( B t ) t ≥ is the same Brownian motion as in (3.10) .Proof. From (3.10) we have dρ i = − λ i ) dλ i + 2(1 + λ i ) d h λ i i = − q − ρ i dB i − (cid:18) ( n − k + ( n − k + 2) ρ i ) + 2 X ℓ = i − ρ i ρ ℓ − ρ i (cid:19) dt. .3 Distribution and limit law of the eigenvalues In this section, we denote as before by λ the eigenvalues process of J and ρ i = − λ i λ i . Corollary3.6 and Theorem 3.5 show that ρ is a diffusion process with generator given by L n,k = 2 k X i =1 (1 − ρ i ) ∂ i − k X i =1 (cid:18) n − k + ( n − k + 2) ρ i + 2 X ℓ = i − ρ i ρ ℓ − ρ i (cid:19) ∂ i . Note that we can also write L n,k = 2 G n − k, − k X i =1 X ℓ = i − ρ i ρ ℓ − ρ i ∂ i where G α,β = P ki =1 (1 − ρ i ) ∂ i − ( α − β + ( α + β + 2) ρ i ) ∂ i is a sum of Jacobi diffusion operators on[ − , L n,k turns out to be the groundstate conditioned Karlin-McGregor semigroup associated with 2 G n − k, . We refer to [2] for ageneral overview of Karlin-McGregor semigroups. Lemma 3.7. Consider the Vandermonde function h ( ρ ) = Y i>j ( ρ i − ρ j ) . We have for every smooth function f on [ − , k : L n,k f = 2 (cid:18) h G n − k, ( hf ) + 16 k ( k − 1) (3 n − k + 2) f (cid:19) . Proof. Let Γ( f, g ) := 12 ( G α,β ( f g ) − f G α,β g − g G α,β ( f ))be the carr´e du champ operator associated to G α,β . We haveΓ( h, f ) = k X i =1 (1 − ρ i )( ∂ i h )( ∂ i f ) . From the definition of h it is clear that ∂ i h = h X ℓ = i ρ i − ρ ℓ , thus, we obtain Γ(log h, f ) = k X i =1 X j = i − ρ i ρ i − ρ ℓ ∂ i f. 14n the other hand, thanks to a direct computation (or Proposition 12.1.1 in [21]) G α,β h = k X i =1 (1 − ρ i ) ∂ i h − k X i =1 ( α − β + ( α + β + 2) ρ i ) ∂ i h = − k ( k − (cid:18) k − 23 + α + β + 22 (cid:19) h. In particular, one has G n − k, ( h ) = − k ( k − 1) (3 n − k + 2) h. We conclude1 h G n − k, ( hf ) = 1 h ( G n − k, ( h ) f + G n − k, ( f ) h + 2Γ( f, h ))= − k ( k − 1) (3 n − k + 2) f + G n − k, ( f ) + 2Γ(log h, f )= − k ( k − 1) (3 n − k + 2) f + 12 L n,k f. Thanks to this lemma, we can compute the density at time t > ρ ( t ).To fix notations we first give some reminders about one-dimensional Jacobi diffusion operators.Let J α,β = (1 − x ) ∂ ∂x − (( α + β + 2) x + α − β ) ∂∂x be the one-dimensional Jacobi operator. The spectrum and eigenfunctions of J α,β are knownand can be described in terms of the Jacobi polynomials. Let us denote by P α,βm ( x ), m ∈ Z ≥ the Jacobi polynomials given by P α,βm ( x ) = ( − m m m !(1 − x ) α (1 + x ) β d m dx m ((1 − x ) α + m (1 + x ) β + m ) . The family { P α,βm ( x ) } m ≥ is orthonormal in L ([ − , , − α − β − (1 + x ) β (1 − x ) α dx ) and satisfies J α,β P α,βm ( x ) = − m ( m + α + β + 1) P α,βm ( x ) . If we denote by p α,βt ( x, y ) the transition density, with respect to the Lebesgue measure, of thediffusion with generator 2 J α,β and initiated from x ∈ ( − , p α,βt ( x, y ) = (1 + y ) β (1 − y ) α α + β +1 + ∞ X m =0 c m,α,β e − m ( m + α + β +1) t P α,βm ( x ) P α,βm ( y ) , (3.11)where c m,α,β = (2 m + α + β + 1) Γ( m + α + β +1)Γ( m +1)Γ( m + α +1)Γ( m + β +1) .We can now state the main theorem of the section:15 heorem 3.8. Let λ be the eigenvalues process of J and ρ i = − λ i λ i . Let us assume that ρ (0) < · · · < ρ k (0) . The density at time t > of ρ ( t ) with respect to the Lebesgue measure dx on [ − , k is given by e k ( k − n − k +2) t h ( x ) h ( ρ (0)) det (cid:16) p n − k, t ( ρ i (0) , x j ) (cid:17) ≤ i,j ≤ k ∆ k ( x ) , where ∆ k := {− ≤ x < · · · < x k ≤ } . Proof. Let f be a smooth function defined on the simplex ∆ k . We almost everywhere extend f to [ − , k by symmetrization, i.e. for every permutation σ ∈ S k , f ( x σ (1) , · · · , x σ ( k ) ) = f ( x , · · · , x k ) . It follows from the intertwining of generators L n,k = 2 (cid:18) h G n − k, k − ( h · ) + 16 k ( k − 1) (3 n − k + 2) (cid:19) that for the corresponding semigroups e t L n,k f ( ρ (0))= e k ( k − n − k +2) t h ( ρ (0)) e t G n − k, ( hf )( ρ (0))= e k ( k − n − k +2) t h ( ρ (0)) Z [ − , k h ( x ) p n − k, t ( ρ (0) , x ) · · · p n − k, t ( ρ k (0) , x k ) f ( x ) dx = e k ( k − n − k +2) t h ( ρ (0)) X σ ∈ S k Z − Let λ be the eigenvalues process of J and ρ i = − λ i λ i . Assume that ρ (0) < · · · < ρ k (0) . Then, when t → + ∞ , ρ ( t ) converges in distribution to the probability measure on [ − , k givenby dν = c n,k Y ≤ i 1) and, up to a constant, is given by e − k ( k − n − k +2) t det (cid:16) P n − k, i − ( x j ) (cid:17) ≤ i,j ≤ k det (cid:16) P n − k, i − ( ρ j (0)) (cid:17) ≤ i,j ≤ k which up to a constant is e − k ( k − n − k +2) t h ( x ) h ( ρ (0))where, for the computation of the Vandermonde determinant, we used the fact that the Jacobipolynomial P m is a polynomial of degree m . The next order in t corresponds to ( m , · · · , m k ) =(0 , , · · · , k − , k ) which yields e − nt in (3.12).17 emark 3.10. The limit law ν is therefore the distribution of a Coulomb gas at inverse tem-perature 2 with a logarithmic confinement potential V ( x ) = − ( n − k ) ln(1 − x ) . It corresponds to a complex Jacobi ensemble in random matrix theory, see for instance [29]. Remark 3.11. Since ρ i = − λ i λ i , we easily deduce the distribution and the limit law for theeigenvalues process ( λ ( t )) t ≥ . Remark 3.12. As a byproduct, the previous proof yields a spectral expansion for the heat kernelof L n,k with respect to the Lebesgue measure of the form: e k ( k − n − k +2) t h ( x ) h ( ρ (0)) X m < ··· 1) (3 n − k + 2) + 2 k X i =1 m i ( m i + n − k + 1) . This recovers the Berezin-Karpeleviˇc formula [10] for the zonal spherical eigenfunctions on G n,k , see also [28]. In fact, our approach yields an algebraic representation of such eigenfunc-tions. Indeed Φ m , ··· ,m k is a symmetric polynomial (a multivariate Jacobi polynomial) and if weconsider the unique polynomial function Φ ∗ m , ··· ,m k defined on the set of k × k positive definiteHermitian matrices such that for every unitary M ∈ U ( k ) , every positive definite Hermitian X ∈ C k × k and every diagonal matrix D = diag( ρ , · · · , ρ k ) : ( Φ ∗ m , ··· ,m k ( M ∗ XM ) = Φ ∗ m , ··· ,m k ( X )Φ ∗ m , ··· ,m k ( D ) = Φ m , ··· ,m k ( ρ , · · · , ρ k ) , then the function Φ ∗ m , ··· ,m k (cid:0) ( I k − W ∗ W )( I k + W ∗ W ) − (cid:1) is an eigenfunction of ∆ G n,k . Remark 3.13. Let us observe that many functional inequalities for the invariant measure ν andthe law of ρ ( t ) can be obtained as a result of Bakry- ´Emery theory. Indeed, as before, considerthe generator of the diffusion ( ρ ( t )) t ≥ : L n,k = 2 k X i =1 (1 − ρ i ) ∂ i − k X i =1 (cid:18) n − k + ( n − k + 2) ρ i + 2 X ℓ = i − ρ i ρ ℓ − ρ i (cid:19) ∂ i . Let now Γ ( f, f ) := 12 ( L n,k Γ( f, f ) − Γ( L n,k f, f ) − Γ( f, L n,k f ))18 e the Bakry’s Γ operator where Γ denotes the carr´e du champ operator of L n,k . By Lemma2.5 we deduce that L n,k satisfies the curvature dimension inequality CD(2 n, k ( n − k )) , i.e. Γ ( f, f ) ≥ k ( n − k ) ( L n,k f ) + 2 n Γ( f, f ) . We refer to the book [4] for the numerous consequences of CD(2 n, k ( n − k )) . Those applicationsinclude log-Sobolev inequality or Sobolev inequalities for ν , Gaussian concentration properties,etc... To conclude, let us remark that by combining Theorem 3.9 with Corollary 2.6 we immediatelyobtain the following result. Corollary 3.14. Let W be a random variable on C ( n − k ) × k distributed according to the probabil-ity law µ . Then the ordered eigenvalues of ( I k − W ∗ W )( I k + W ∗ W ) − are distributed accordingto the probability measure ν . Note that this last result could be proved by more direct random matrices computations.Indeed, let g be a bounded Borel function on ˆ H k the set of positive definite Hermitian matricesand let W be a random variable on C ( n − k ) × k distributed according to c n,k det( I k + W ∗ W ) − n d W .Then, from Proposition 1 in [24] one has for some normalization constant c ′ n,k E ( g ( W ∗ W )) = c n,k Z C ( n − k ) × k g ( W ∗ W ) det( I k + W ∗ W ) − n d W = c ′ n,k Z ˆ H k g ( S ) det( I k + S ) − n det( S ) n − k dS. Thus, S = W ∗ W is distributed as c ′ n,k det( I k + S ) − n det( S ) n − k dS . The ordered eigenvalues λ i ’s of S are thus distributed as c ′′ n,k Y i>j ( λ i − λ j ) k Y i =1 (1 + λ i ) ! − n k Y i =1 λ i ! n − k λ > ··· >λ n > dλ · · · dλ n , from which we deduce Corollary 3.14 after the change of variables ρ i = − λ i λ i . Let us consider the Stiefel fibration U ( k ) → V n,k → G n,k that was described in Section 2.1. According to it, one can see the complex Stiefel manifold V n,k as a U ( k ) principal bundle over G n,k . The next lemma gives a formula for the connection formof this bundle. 19 emma 4.1. Consider on V n,k = (cid:26)(cid:18) XZ (cid:19) ∈ C n × k , X ∗ X + Z ∗ Z = I k (cid:27) the u ( k ) -valued one form ω := 12 (cid:18) ( X ∗ Z ∗ ) d (cid:18) XZ (cid:19) − d ( X ∗ Z ∗ ) (cid:18) XZ (cid:19)(cid:19) = 12 ( X ∗ dX − dX ∗ X + Z ∗ dZ − dZ ∗ Z ) . Then, ω is the connection form of the fibration U ( k ) → V n,k → G n,k .Proof. We first observe that if p = (cid:18) XZ (cid:19) ∈ V n,k , then the tangent space to V n,k at p is given by T p V n,k = (cid:26)(cid:18) AB (cid:19) ∈ C n × k , A ∗ X + X ∗ A + B ∗ Z + Z ∗ B = 0 (cid:27) . Then, if θ ∈ u k , one easily computes that the generator of the one-parameter group { p → pe tθ } t ∈ R is given by the vector field on V n,k whose value at p is (cid:18) XθZθ (cid:19) . Applying ω to this vector fieldyields θ . To show that ω is the connection form it remains therefore to prove that the kernel of ω is the horizontal space of the Riemannian submersion (cid:18) XZ (cid:19) → XZ − . This horizontal spaceat p , say H p , is the orthogonal complement of the vertical space at p , which is the subspace V p of T p V n,k tangent to the fiber of the submersion. The previous argument shows that V p = (cid:26)(cid:18) XθZθ (cid:19) , θ ∈ u ( k ) (cid:27) . Therefore we have H p = (cid:26)(cid:18) AB (cid:19) ∈ T p V n,k , ∀ θ ∈ u ( k ) , tr ( A ∗ Xθ + B ∗ Zθ ) = 0 (cid:27) . We deduce H p = (cid:26)(cid:18) AB (cid:19) ∈ T p V n,k , A ∗ X + B ∗ Z = X ∗ A + Z ∗ B (cid:27) , from which it is clear that ω |H = 0.Next, our goal is to describe the horizontal lift to V n,k of a Brownian motion on G n,k . Westill denote by p : V n,k → G n,k the Riemannian submersion. A continuous semimartingale( M t ) t ≥ on V n,k is called horizontal if for every t ≥ R M [0 ,t ] ω = 0, where R M [0 ,t ] ω denotes theStratonovich line integral of ω along the paths of M . If ( N t ) t ≥ is a continuous semimartingaleon G n,k with N ∈ G n,k , then if ˜ N ∈ V n,k is such that p ( ˜ N ) = N , there exists a uniquehorizontal continuous semimartingale ( ˜ N t ) t ≥ on V n,k such that p ( ˜ N t ) = N t for every t ≥ N t ) t ≥ is then called the horizontal lift at ˜ N of ( N t ) t ≥ to V n,k . We referto [6] for a more general description of the horizontal lift of a semimartingale in the context offoliations. We consider on G n,k the u ( k ) valued one-form η given in inhomogeneous coordinatesby η := 12 (cid:16) ( I k + W ∗ W ) − / ( d W ∗ W − W ∗ d W )( I k + W ∗ W ) − / − ( I k + W ∗ W ) − / d ( I k + W ∗ W ) / + d ( I k + W ∗ W ) / ( I k + W ∗ W ) − / (cid:17) heorem 4.2. Let ( W t ) t ≥ be a Brownian motion on G n,k started at W ∈ G n,k . Let (cid:18) X Z (cid:19) ∈ V n,k such that X Z − = W . The process ˜ W t := (cid:18) W t I k (cid:19) ( I k + W ∗ t W t ) − / Θ t is the horizontal lift at (cid:18) X Z (cid:19) of ( W t ) t ≥ to V n,k where a t = R W [0 ,t ] η and where (Θ t ) t ≥ is the U ( k ) valued solution of the Stratonovich stochastic differential equation ( d Θ t = ◦ d a t Θ t Θ = ( Z Z ∗ ) − / Z . Proof. As before we denote by p the submersion (cid:18) XZ (cid:19) → XZ − . It is easy to check that forevery t ≥ p ( ˜ W t ) = W t and that ˜ W = (cid:18) X Z (cid:19) . It is therefore enough to prove that ˜ W is ahorizontal semimartingale, i.e. that R ˜ W [0 ,t ] ω = 0. Denote X t = W t ( I k + W ∗ t W t ) − / Θ t , Z t = ( I k + W ∗ t W t ) − / Θ t A long, but routine, computation shows that12 ( X ∗ ◦ dX − ◦ dX ∗ X + Z ∗ ◦ dZ − ◦ dZ ∗ Z )= − (cid:18) ◦ d Θ ∗ Θ − Θ ∗ ◦ d Θ + Θ ∗ (cid:18) ◦ d ( I k + J ) − / ( I k + J ) / − ( I k + J ) / ◦ d ( I k + J ) − / (cid:19) Θ+ Θ ∗ ( I k + J ) − / ( ◦ d W ∗ W − W ∗ ◦ d W )( I k + J ) − / Θ (cid:17) . where J = W ∗ W . Since ◦ d Θ ∗ = ◦ d Θ − = − Θ − ◦ d Θ Θ − and ◦ d Θ = ◦ d a Θ with ◦ d a = 12 ( I k + J ) − / ( ◦ d W ∗ W − W ∗ ◦ d W )( I k + J ) − / − (cid:16) ( I k + J ) − / ◦ d ( I k + J ) / − ◦ d ( I k + J ) / ( I k + J ) − / (cid:17) we conclude that 12 ( X ∗ ◦ dX − ◦ dX ∗ X + Z ∗ ◦ dZ − ◦ dZ ∗ Z ) = 0and thus R ˜ W [0 ,t ] ω = 0. We now turn to the description of the Brownian motion on V n,k as a skew-product.21 heorem 4.3. Let ( W t ) t ≥ be a Brownian motion on G n,k started at W ∈ G n,k and let (Ω t ) t ≥ be a Brownian motion on the unitary group U ( k ) independent from ( W t ) t ≥ . Let (Θ t ) t ≥ be the U ( k ) valued solution of the Stratonovitch stochastic differential equation ( d Θ t = ◦ d a t Θ t Θ = I k , where a t = R W [0 ,t ] η . The process (cid:18) X t Z t (cid:19) := (cid:18) W t I k (cid:19) ( I k + W ∗ t W t ) − / Θ t Ω t is a Brownian motion on V n,k .Proof. We denote by ∆ H the horizontal Laplacian and by ∆ V the vertical Laplacian of the Stiefelfibration, see [5]. Since the submersion V n,k → G n,k is totally geodesic, the operators ∆ H and∆ V commute. We note that the Laplace-Beltrami operator of V n,k is given by ∆ V n,k = ∆ H + ∆ V and that the horizontal lift of the Brownian motion on G n,k is a diffusion with generator ∆ H ,see [6]. The fibers of the submersion V n,k → G n,k are isometric to U ( k ), thus if f is a smoothfunction on V n,k , one has e t ∆ V f (cid:18) XZ (cid:19) = E (cid:18) f (cid:18) X Ω t Z Ω t (cid:19)(cid:19) . Since e t ∆ V e t ∆ H = e t ∆ Vn,k , we conclude from Theorem 4.2. We now give a limit theorem for the process (cid:16)R t t r ( W ∗ s W s ) ds ) (cid:17) t ≥ that shall be used in thenext subsection. The method we use, a Girsanov transform, takes its root in the paper by M.Yor [40] and was further developed in the situation of matrix Wishart diffusions in [20] and inthe situation of the Doumerc-Jacobi matrix processes in Section 9.4.2 of the thesis [21]. Ourresult is the following: Theorem 4.4. As before, let J = W ∗ W . • If k = 1 , the following convergence holds in distribution when t → + ∞ t Z t t r ( J ) ds → X, where X is a random variable on [0 , + ∞ ) with density ( n − √ πx / e − ( n − x (i.e. X is theinverse of a gamma distributed random variable). • If k > , the following convergence holds in probability when t → + ∞ t Z t t r ( J ) ds → k ( n − k ) k − . Lemma 4.5. For every α ≥ the process M αt = e kα ( n − k ) t (cid:18) det( I k + J )det( I k + J t ) (cid:19) α exp (cid:18) − Z t ( α ( k − 1) + α )t r ( J ) ds (cid:19) is a martingale.Proof. Consider the exponential local martingale M αt := exp (cid:18) − α Z t t r ( √ J ( d B + d B ∗ )) − α Z t t r ( J ) ds (cid:19) , where B is the Brownian motion as given in Theorem 3.1. If we denote V = 2 k ( n − k ) − k − r ( J ), then similar computations as in Section 5.4 yield d (log det( I k + J )) = t r (cid:16) √ J ( d B + d B ∗ ) (cid:17) + V dt. Therefore (cid:18) det( I k + J t )det( I k + J ) (cid:19) α = exp (cid:18) α (cid:18)Z t t r (cid:16) √ J ( d B + d B ∗ ) (cid:17) + V ds (cid:19) (cid:19) , and thus M αt = e kα ( n − k ) t (cid:18) det( I k + J )det( I k + J t ) (cid:19) α exp (cid:18) − Z t ( α ( k − 1) + α )t r ( J s ) ds (cid:19) . From this expression, it is clear that there exists a constant C > | M αt | ≤ Ce kα ( n − k ) t and thus M αt is a martingale.We are now ready for the proof of Theorem 4.4. Proof of Theorem 4.4. Consider the probability measure P α defined by P α | F t = M αt · P | F t . From Girsanov theorem, the process β t = B t + 2 α Z t √ J ds is under P α a k × k -matrix-valued Brownian motion. Therefore, under P α , J solves the stochasticdifferential equation dJ = p I k + J dβ ∗ p I k + J √ J + √ J p I k + J dβ p I k + J + 2 ( n − kαJ + t r ( J )) ( I k + J ) dt. As a consequence the distribution of the eigenvalues of J under P α can therefore be computedsimilarly as in Theorem 3.8. We now note that E (cid:16) e − α ( k − α ) R t t r ( J ) ds (cid:17) = e − k ( n − k ) αt E α (cid:18) det( I k + J t )det( I k + J ) (cid:19) α µ = (cid:16) α ( k − α ) k +3 (cid:17) , we obtain E (cid:18) e − ( k +3) µ R t t r ( J ) ds (cid:19) = e − k ( n − k ) αt E α (cid:18) det( I k + J t )det( I k + J ) (cid:19) α where α = q ( k +3) µ + ( k − ) − k − . Hence when k > t →∞ E (cid:18) e − ( k +3) µ t R t t r ( J ) ds (cid:19) = e − k ( n − k )( k +3)2( k − µ lim t →∞ E q ( k +3) µ t +( k − ) − k − (cid:18) det( I k + J t )det( I k + J ) (cid:19) q ( k +3) µ t +( k − ) − k − = e − k ( n − k )( k +3)2( k − µ where the last limit can be justified using the formula for the density of the eigenvalues of J under a probability P α . When k = 1 we have µ = α and a similar proof yieldslim t →∞ E (cid:18) e − µ t R t t r ( J ) ds (cid:19) = e − ( n − µ . By definition of η , we note that Z W [0 ,t ] tr( η ) = 12 tr (cid:20)Z t ( I k + J ) − / ( ◦ d W ∗ W − W ∗ ◦ d W )( I k + J ) − / (cid:21) = 12 tr (cid:20)Z t ( I k + J ) − / ( d W ∗ W − W ∗ d W )( I k + J ) − / (cid:21) (4.13)where as before J = W ∗ W .From simple computations one can verify thatt r ( dη ) = ∂∂ log det( I k + W ∗ W ) , which implies that i t r ( dη ) is the K¨ahler form on G n,k . Therefore i R W [0 ,t ] tr( η ) can be consideredas a generalized stochastic area process on G n,k . In the theorem below we deduce large timelimit distributions of such processes. Theorem 4.6. Let ( W t ) t ≥ be a Brownian motion on G n,k started at W ∈ G n,k . • If k = 1 , the following convergence holds in distribution when t → + ∞ it Z W [0 ,t ] tr( η ) → C n − , where C n − is a Cauchy distribution of parameter n − . If k > , the following convergence holds in distribution when t → + ∞ i √ t Z W [0 ,t ] tr( η ) → N (cid:18) , k ( n − k ) k − (cid:19) . Proof. From (5.18) we know that d W ∗ W − W ∗ d W = p I k + J d B ∗ p I k + J √ J − √ J p I k + J d B p I k + J where ( B t ) t ≥ is a k × k -matrix-valued Brownian motion. Therefore,( I k + J ) − / ( d W ∗ W − W ∗ d W )( I k + J ) − / = d B ∗ √ J − √ J d B Consider the diagonalization of J = V Λ V ∗ , where V ∈ U ( k ) and Λ = diag { λ , . . . , λ k } , then d B ∗ √ J − √ J d B = V ( V − d B ∗ V √ Λ − √ Λ V − d B V ) V − . Therefore from (4.13), we have in distribution that Z W [0 ,t ] tr( η ) = i B R t tr( J ) ds where B is a one-dimensional Brownian motion independent from the process tr( J ). We concludefrom Theorem 4.4Let (cid:18) X t Z t (cid:19) be a Brownian motion on V n,k . We are interested in the windings of the complexvalued process det( Z t ). From Theorem 4.3, we have the polar decompositiondet( Z t ) = det( I k + W ∗ t W t ) − / det Θ t det Ω t . Lemma 4.7. For every t ≥ , det Θ t = exp (cid:16)R W [0 ,t ] tr( η ) (cid:17) . Proof. We have ( d Θ t = ◦ d (cid:16)R W [0 ,t ] η (cid:17) Θ t Θ = I k . Thus from the Chen-Strichartz expansion formula det Θ t = exp (cid:16) tr (cid:16)R W [0 ,t ] η (cid:17)(cid:17) = exp (cid:16)R W [0 ,t ] tr( η ) (cid:17) .We immediately deduce the following corollary. Corollary 4.8. Let (cid:18) X t Z t (cid:19) be a Brownian motion on V n,k with det( Z ) = 0 . One has the polardecomposition det( Z t ) = ̺ t e iθ t where < ̺ t ≤ and θ t is a continuous semimartingale such that: If k = 1 , the following convergence holds in distribution when t → + ∞ θ t t → C n − , where C n − is a Cauchy distribution of parameter n − . • If k > , the following convergence holds in distribution when t → + ∞ θ t √ t → N (cid:18) , k ( n − k ) k − n (cid:19) Proof. From the decomposition det( Z t ) = det( I k + W ∗ t W t ) − / det Θ t det Ω t one deduces ̺ t = det( I k + J t ) − / , iθ t = tr( ω t ) + Z W [0 ,t ] tr( η )where ω t is a Brownian motion on u ( k ) independent from W . The conclusion follows. G n,k We divide the proof of of Theorem 2.1 in two parts, the first one proves the a.s. invertibility of Z t and the second one proves that W t = X t Z − t is a diffusion with generator ∆ G n,k . Let usconsider the block decomposition A t = (cid:18) α t β t γ t ǫ t (cid:19) , with α t ∈ C k × k . Note that α t , β t = − γ ∗ t and ǫ t are independent. From (2.3) we obtain thefollowing system of stochastic differential equations: dX = X ◦ dα + Y ◦ dγ = Xdα + Y dγ + 12 ( dXdα + dY dγ ) dY = X ◦ dβ + Y ◦ dǫ = Xdβ + Y dǫ + 12 ( dXdβ + dY dǫ ) dZ = Z ◦ dα + W ◦ dγ = Zdα + W dγ + 12 ( dZdα + dW dγ ) (5.14) dW = Z ◦ dβ + W ◦ dǫ = Zdβ + W dǫ + 12 ( dZdβ + dW dǫ ) . Lemma 5.1. Let Z t be the bottom left corner of the U ( n ) valued Brownian motion U t as definedabove and let τ Z := inf { t > , det Z t = 0 } . Then τ Z = + ∞ a.s.Proof. Let ( J t ) := ( Z t Z ∗ t ) t ≥ . From (5.14) and (5.15) we have dZ = Zdα + W dγ − nZdt , hence dZZ ∗ = ZdαZ ∗ + W dγZ ∗ − n J dt. Note that I k − ZZ ∗ = W W ∗ , we have dZdZ ∗ = ( Zdα + W dγ )( dα ∗ Z ∗ + dγ ∗ W ∗ ) = Zdαdα ∗ Z ∗ + W dγdγ ∗ W ∗ = 2 kdt ZZ ∗ + 2 kdt ( I k − ZZ ∗ ) = 2 kI k dt. d J = dZZ ∗ + ZdZ ∗ + dZdZ ∗ = Zdγ ∗ W ∗ + W dγZ ∗ + (cid:18) kI k − n J (cid:19) dt. Let B t = R t ( J ) − / Zdγ ∗ W ∗ ( I k − J ) − / . We can easily check that B t , t ≥ H k –the collection of k × k Hermitian matrices. The martingale part of d J is thengiven by Zdγ ∗ W ∗ + W dγZ ∗ = √J d B p I k − J + p I k − J d B ∗ √J . Next we prove τ Z = ∞ a.s. Apply similar calculation as for (3.8) (see Section 5.4) to J wehave that d (log det( J )) = t r (cid:16) J − / p I k − J ( d B + d B ∗ ) (cid:17) − k ( n − k ) dt. Since the local martingale part of the above SDE is a time-changed Brownian motion β C t , wehave log det( J t ) − log det( J ) − k ( n − k ) t = β C t . On { τ Z < ∞} , we have lim t → τ Z log det( J t ) = −∞ . This implies that lim t → τ Z β C t = ∞ , SinceBrownian motion never goes to infinity without oscillating, we conclude that P ( τ Z < ∞ ) = 0. Proof of Theorem 2.1. Since dα = X ≤ i ZdZ − = − dZZ − − dZdZ − , hence we have d W = dXZ − − W dZZ − − W dZdZ − + dXdZ − = ( Xdα + Y dγ − nXdt ) Z − − W ( Zdα + W dγ − nZdt ) Z − − W dZdZ − + dXdZ − = Y dγZ − − W W dγZ − − W dZdZ − + dXdZ − . Note that for the finite variation part of d W we have − W dZdZ − + dXdZ − = W dZZ − dZZ − − dX Z − dZ Z − = W ( Zdα + W dγ ) Z − ( Zdα + W dγ ) Z − − ( Xdα + Y dγ ) Z − ( Zdα + W dγ ) Z − = W ZdαdαZ − − X ( dαdα ) Z − = 0 . Hence we have d W = Y dγZ − − W W dγZ − . We are now in position to prove that W is a matrix diffusion process using the above formula.Since for 1 ≤ i ≤ n − k , 1 ≤ j ≤ k , d W ij = k X ℓ =1 ( Y − W W ) iℓ ( dγZ − ) ℓj , we have d W ij d W i ′ j ′ = k X ℓ,m =1 ( Y − W W ) iℓ ( Y − W W ) i ′ m ( dγZ − ) ℓj ( dγZ − ) mj ′ . Moreover, since( dγZ − ) ℓj ( dγZ − ) mj ′ = k X p,q =1 ( dγ ) ℓp ( Z − ) pj ( dγ ) mq ( Z − ) qj ′ = 2 δ mℓ dt k X p =1 ( Z − ) pj ( Z − ) pj ′ we have d W ij d W i ′ j ′ = 2(( Y − W W )( Y − W W ) T ) ii ′ (( ZZ ∗ ) − ) j ′ j dt. (5.16)From (2.1) we know that − XZ − W Y ∗ = XX ∗ , plug into (5.16) we then obtain d W ij d W i ′ j ′ = 2( I n − k + WW ∗ ) ii ′ (( ZZ ∗ ) − ) j ′ j dt = 2( I n − k + WW ∗ ) ii ′ ( I k + W ∗ W ) j ′ j dt. (5.17)Therefore, we conclude that ( W t ) t ≥ is a diffusion whose generator is given by ∆ G n,k .28 .2 Computation of the volume measure on G n,k Proof. We denote ∂ ij = ∂∂ W ij , ∂ ij = ∂∂ W ij , A ii ′ jj ′ = ( δ ii ′ + ( WW ∗ ) ii ′ )( δ j ′ j + ( W ∗ W ) j ′ j ), and ρ = c n,k det( I k + W ∗ W ) − n . Let us denote T ( f, g ) = 2 X ≤ i,i ′ ≤ n − k, ≤ j,j ′ ≤ k ( I n − k + WW ∗ ) ii ′ ( I k + W ∗ W ) j ′ j (cid:18) ∂f∂ W ij ∂g∂ W i ′ j ′ + ∂g∂ W ij ∂f∂ W i ′ j ′ (cid:19) By integration by parts we have − Z (∆ G n,k f ) g dµ = X ≤ i,i ′ ≤ n − k, ≤ j,j ′ ≤ k Z ( ∂ ij f ) ∂ i ′ j ′ ( A ii ′ jj ′ gρ ) dm + Z ( ∂ i ′ j ′ f ) ∂ ij ( A ii ′ jj ′ gρ ) dm = 12 T ( f, g ) + X ≤ i,i ′ ≤ n − k, ≤ j,j ′ ≤ k (cid:18) Z [( ∂ ij f ) ( ∂ i ′ j ′ A ii ′ jj ′ ) + ( ∂ i ′ j ′ f ) ( ∂ ij A ii ′ jj ′ )] gρdm + Z [( ∂ ij f ) ( ∂ i ′ j ′ ρ ) + ( ∂ i ′ j ′ f ) ( ∂ ij ρ )] gA ii ′ jj ′ dm (cid:19) = 12 T ( f, g ) + R. Since ∂ i ′ j ′ A ii ′ jj ′ = W ij ′ ( δ j ′ j + ( W ∗ W ) j ′ j ) + ( δ ii ′ + ( WW ∗ ) ii ′ ) W i ′ j , and ∂ ij A ii ′ jj ′ = W i ′ j ( δ j ′ j + ( W ∗ W ) j ′ j ) + ( δ ii ′ + ( WW ∗ ) ii ′ ) W ij ′ we have X ≤ i ′ ≤ n − k, ≤ j ′ ≤ k ∂ i ′ j ′ A ii ′ jj ′ = n ( W ( I k + J )) ij and X ≤ i ≤ n − k, ≤ j ≤ k ∂ ij A ii ′ jj ′ = n ( W ( I k + ¯ J )) i ′ j ′ . Moreover, since ∂ i ′ j ′ det( I k + J ) = det( I k + J ) X ≤ p,q ≤ k (cid:0) ( I k + J ) − (cid:1) qp ∂ i ′ j ′ ( I k + J ) pq = det( I k + J ) (cid:18) W ( I k + J ) − (cid:19) i ′ j ′ and ∂ ij det( I k + J ) = det( I k + J ) (cid:18) W ( I k + ¯ J ) − (cid:19) ij , 29e have ∂ i ′ j ′ ρ = − nρ (cid:18) W ( I k + J ) − (cid:19) i ′ j ′ , ∂ ij ρ = − nρ (cid:18) W ( I k + ¯ J ) − (cid:19) ij . We then have X i ′ ,j ′ ( ∂ i ′ j ′ A ii ′ jj ′ ) gρ + ( ∂ i ′ j ′ ρ ) gA ii ′ jj ′ = 0and X i,j ( ∂ ij A ii ′ jj ′ ) gρ + ( ∂ ij ρ ) gA ii ′ jj ′ = 0 . This gives R = 0. J Proof of Theorem 3.1. We use the notations of the proof of Theorem 2.1. Recall that d W = ( Y − W W ) dγ Z − . We first compute the martingale part of dJ : dJ ∼ d W ∗ W + W ∗ d W ∼ ( Z − ) ∗ dγ ∗ ( Y − W W ) ∗ W + W ∗ ( Y − W W ) dγ Z − ∼ ( p I k + J ) ∗ ( d B ) ∗ ( p I k + J ) ∗ ( √ J ) ∗ + √ J p I k + J d B p I k + J (5.18)where B t is a k × k -matrix-valued stochastic process that satisfies d B = p I k + J − √ J − W ∗ ( Y − W W ) dγ Z − p I k + J − . Since for any 1 ≤ i, j ≤ m , d B ij = X k,ℓ,s,t,p,q ( p I k + J − ) ik ( √ J − ) kℓ ( W ∗ ) ℓs ( Y − W W ) st ( dγ ) tp ( Z − ) pq ( p I k + J − ) qj , we obtain that d B ij d B i ′ j ′ dt = X k,k ′ ℓ,ℓ ′ ,s,s ′ ,q,q ′ ( p I k + J − ) ik ( p I k + J − ) i ′ k ′ ( √ J − ) kℓ ( √ J − ) k ′ ℓ ′ ( W ∗ ) ℓs ( W ) s ′ ℓ ′ (( Y − W W )( Y − W W ) ∗ ) ss ′ (( Z − ) ∗ Z − ) q ′ q ( p I k + J − ) qj ( p I k + J − ) q ′ j ′ . Here we use the fact that ( dγ ) tp ( dγ ) t ′ p ′ = 2 dt δ tt ′ δ pp ′ . Moreover, since( Y − W W )( Y − W W ) ∗ = I n − k + WW ∗ , we have X s,s ′ ( W ) ℓs (( Y − W W )( Y − W W ) ∗ ) ss ′ W s ′ ℓ ′ = ( J + J ) ℓℓ ′ . Z − ) ∗ Z − = I k + J , we obtain d B ij d B i ′ j ′ dt = δ ii ′ δ jj ′ . Hence B is a k × k -matrix-valued Brownian motion. It remains to compute the bounded varia-tion part of J . We see that the bounded variation part in dJ is given by d W ∗ d W . From (5.17)we have for any 1 ≤ i, j ≤ k ,( d W ∗ d W ) ij = n − k X ℓ =1 ( d W ) ℓi ( d W ) ℓj = 2 n − k X ℓ =1 ( I n − k + WW ∗ ) ℓℓ ( I k + W ∗ W ) ij dt Therefore we have d W ∗ d W = 2( n − k + tr( WW ∗ ))( I k + J ) dt and the proof is complete. Proof of lemma 3.2. By Itˆo’s formula we know that d (det( J )) = k X i,j =1 ∂ det( J ) ∂J ij dJ ij + 12 k X i,j,i ′ ,j ′ =1 ∂ det( J ) ∂J ij ∂J i ′ j ′ dJ ij dJ i ′ j ′ . First we know that ∂ det( J ) ∂J ij = ∂ P kℓ =1 J iℓ ˜ J ℓi ∂J ij = ˜ J ji where ˜ J = det( J ) J − is the cofactor of J . Hence the first order term in the above SDE isdet( J )t r ( J − dJ ). Next, for any 1 ≤ i, j, i, j ′ ≤ k we have ∂ det( J ) ∂J ij ∂J i ′ j ′ = ∂ ˜ J ji ∂J i ′ j ′ = ∂ det( J ) ∂J i ′ j ′ ( J − ) ji + det( J ) ∂ ( J − ) ji ∂J i ′ j ′ . The first term on the right hand side is obviously det( J )( J − ) j ′ i ′ ( J − ) ji . To compute the secondterm, note that ∂J∂J i ′ j ′ J − + J ∂J − ∂J i ′ j ′ = 0 , which gives that ∂ ( J − ) ji ∂J i ′ j ′ = − (cid:18) J − ∂J − ∂J i ′ j ′ J − (cid:19) ji = − ( J − ) ji ′ ( J − ) j ′ i . Hence ∂ det( J ) ∂J ij ∂J i ′ j ′ = (det( J )) (cid:18) ( J − ) ji ( J − ) j ′ i ′ − ( J − ) j ′ i ( J − ) ji ′ (cid:19) . dJ ij dJ i ′ j ′ = 2 dt (cid:18) ( J + J ) i ′ j ( I k + J ) ij ′ + ( J + J ) ij ′ ( I k + J ) i ′ j (cid:19) , thus d (det( J )) = det( J )t r ( J − dJ )+ k X i,j,i ′ ,j ′ =1 det( J ) (cid:18) ( J − ) ji ( J − ) j ′ i ′ − ( J − ) j ′ i ( J − ) ji ′ (cid:19)(cid:18) ( J + J ) i ′ j ( I k + J ) ij ′ + ( J + J ) ij ′ ( I k + J ) i ′ j (cid:19) dt = det( J )t r ( J − dJ ) + 2 det( J ) (cid:18) t r (2 I k + J + J − ) − t r ( I k + J )t r ( I k + J − ) (cid:19) dt = det( J )t r ( J − dJ ) + 2 det( J ) (cid:18) k − k − ( k − r ( J ) + t r ( J − )) − tr ( J )t r ( J − ) (cid:19) dt. From (3.6) we knowt r ( J − dJ ) = t r (cid:16) J − / ( I k + J )( d B + d B ∗ ) (cid:17) + 2( n − k + t r ( J )) t r ( I k + J − ) dt. Hence we obtain (3.7). As a direct consequence of d h det( J ) , det( J ) i t = 4 t r (2 I k + J + J − ) dt wethen have (3.8). Proof of Theorem 3.4. We label the eigenvalues by λ ≥ · · · ≥ λ k . Note J is Hermitian, hence itcan be diagonalized by J = V Λ V ∗ where V ∈ U ( k ) and Λ = diag { λ , . . . , λ k } . Let dU = dV ∗ ◦ V and dN = V ∗ ◦ dJ V . Then d Λ = dU ◦ Λ − Λ ◦ dU + dN hence dλ i = dN ii , dU ij = 1 λ i − λ j ◦ dN ij for i = j. From (3.6) we know that( dJ ) ij ( dJ ) i ′ j ′ dt = ( J + J ) i ′ j ( I k + J ) ij ′ + ( J + J ) ij ′ ( I k + J ) i ′ j . We can then compute that( dN ) ij ( dN ) i ′ j ′ dt = X p,p ′ ,ℓ,ℓ ′ V ∗ ip V ∗ i ′ p ′ V ℓj V ℓ ′ j ′ (cid:0) ( J + J ) p ′ ℓ ( I k + J ) pℓ ′ + ( J + J ) pℓ ′ ( I k + J ) p ′ ℓ (cid:1) = ( V ∗ ( J + J ) V ) i ′ j ( V ∗ ( I k + J ) V ) ij ′ + ( V ∗ ( J + J ) V ) ij ′ ( V ∗ ( I k + J ) V ) i ′ j = (Λ + Λ ) i ′ j ( I k + Λ) ij ′ + (Λ + Λ ) ij ′ ( I k + Λ) i ′ j . 32f we denote by dM the local martingale part of dN and dF the finite variation part, then from(3.6) we know that dF dt = V ∗ ( n − k + tr( J ))( I k + J ) V + 12 ( dV ∗ dJ V + V ∗ dJ dV )2 dt = ( n − k + tr( J ))( I k + Λ) + 12 dU dN + dN ∗ dU ∗ dt . Since ( dU dN ) ij dt = X ℓ = i λ i − λ ℓ dN iℓ dN ℓj dt = δ ij X ℓ = i (1 + λ i )(1 + λ ℓ )( λ i + λ ℓ ) λ i − λ ℓ we obtain that ( dU dN ) ∗ = dU dN . Hence dF ij = 2 dtδ ij (cid:18) ( n − k + k X ℓ =1 λ ℓ )(1 + λ i ) + X ℓ = i (1 + λ i )(1 + λ ℓ )( λ i + λ ℓ ) λ i − λ ℓ (cid:19) = 2 dtδ ij (1 + λ i ) (cid:18) n − k + 1 − (2 k − λ i + 2 λ i (1 + λ i ) X ℓ = i λ i − λ ℓ (cid:19) . At last, we have that dM ii dM jj = dN ii dN jj = 2 dt ((Λ + Λ ) ji ( I k + Λ) ij + (Λ + Λ ) ij ( I k + Λ) ji )= 4 dtδ ij λ i (1 + λ i ) . Hence dM ii = 2(1 + λ i ) p λ i dB i where the B i ’s are independent standard real Brownian motions. We conclude dλ i = dM ii + dF ii = 2(1+ λ i ) p λ i dB i +2(1+ λ i ) (cid:18) n − k +1 − (2 k − λ i +2 λ i (1+ λ i ) X ℓ = i λ i − λ ℓ (cid:19) dt. Proof of Theorem 3.5. Let ρ i = − λ i λ i , i = 1 , . . . , k , and let τ = inf { t > | ∃ i < j, ρ i ( t ) = ρ j ( t ) } be the first colliding time. We then want to show that P ( τ < + ∞ ) = 0, namely for every t ≥ ρ ( t ) < · · · < ρ k ( t ) . Let h = Π i>j ( ρ i − ρ j ). Using similar idea as previously, let us consider the processΩ t := V ( ρ ( t ) , . . . , ρ k ( t )) , V ( ρ , . . . , ρ k ) = log h = P i>j log( ρ i − ρ j ). We can compute that d Ω t = k X i =1 (cid:18) ∂ i V dρ i + 12 ∂ i V d h ρ i i (cid:19) = L n,k V dt + dM t (5.19)where M t is a local martingale satisfying dM t = − P ki =1 q − ρ i ( ∂ i V ) dB it and L n,k = 2 k X i =1 (1 − ρ i ) ∂ i − k X i =1 (cid:18) n − k + ( n − k + 2) ρ i + 2 X ℓ = i − ρ i ρ ℓ − ρ i (cid:19) ∂ i . For any 1 ≤ i ≤ k , ∂ i V = ∂ i h h , ∂ i V = ∂ i h h − ( ∂ i h ) h . Since P ki =1 ∂ i h = 0 and P ki =1 ρ i ∂ i h = k ( k − h , we that k X i =1 ∂ i V = 0 , k X i =1 ρ i ∂ i V = k ( k − . Hence L n,k V = k X i =1 (1 − ρ i ) (cid:18) ∂ i hh + ( ∂ i h ) h (cid:19) − ( n − k + 2) k ( k − . 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