Distributions of truncations of the heat kernel on the complex projective space

Let (Ut)t≥0 be a Brownian motion valued in the complex projective space CPN−1. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of |U1 t |2 and of (|U1 t |2, |U2 t |2), and express them through Jacobi polynomials in the simplices of R and R2 respectively. More generally, the distribution of (|U1 t |2, . . . , |U t |2), 2 ≤ k ≤ N − 1 may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group U(N −k+ 1) yet computations become tedious. We also revisit the approach initiated in [13] and based on a partial differential equation (hereafter pde) satisfied by the Laplace transform of the density. When k = 1, we invert the Laplace transform and retrieve the expression already derived using spherical harmonics. For general 1 ≤ k ≤ N −2, integrations by parts performed on the pde lead to a heat equation in the simplex of R. 1. Motivation The complex unit sphere S2N−1 = {(z1, . . . , zN ), |z1| + · · ·+ |zN |2 = 1}, N ≥ 1, is a compact manifold without boundary and therefore carries a Brownian motion (Ut)t≥0 defined by means of its Laplace-Beltrami operator. This process is stationary and the random variable Ut converges weakly as t→∞ to a uniformly-distributed random vector U∞. For the latter, it is already known that (|U1 ∞|, . . . , |U ∞|), 1 ≤ k ≤ N − 1, follows the Dirichlet distribution ([9]) sk(u)du = (1− u1 − u2 − · · · − uk)1Σk(u) k ∏ i=1 dui, (1.1)


Motivation
The complex unit sphere S 2N −1 = {(z 1 , . . . , z N ), |z 1 | 2 + · · · + |z N | 2 = 1}, N ≥ 1, is a compact manifold without boundary and therefore carries a Brownian motion (U t ) t≥0 defined by means of its Laplace-Beltrami operator. This process is stationary and the random variable U t converges weakly as t → ∞ to a uniformly-distributed random vector U ∞ . For the latter, it is already known that

N. Demni
where Σ k = {u i > 0, 1 ≤ i ≤ k, u 1 + · · · + u k < 1} is the standard simplex. Motivated by quantum information theory, the investigations of the distribution of U (k) t = (|U 1 t | 2 , . . . , |U k t | 2 ), 1 ≤ k ≤ N, started in [13] yet have not been completed. There, a linear pde for the Laplace transform of this distribution was obtained and partially solved only when k = 1. Recall that for the Brownian motion on the Euclidian sphere S N −1 , the density of a single coordinate is given by a series involving products of ultraspherical polynomials of index (N − 2)/2 ( [10]). The main ingredients leading to this series are the expansion of the heat kernel on S N −1 in the basis of O(N )-spherical harmonics and on Gegenbauer addition Theorem ( [15], p.369). In the complex setting, it is very likely known that (|U 1 t | 2 ) t≥0 is a real Jacobi process (see [3] and references therein). Nonetheless, one wonders how does the proof written in [10] carry to the Brownian motion on S 2N −1 and how does it extend in order to derive the density of U (k) t . In the first part of this paper, we answer these questions by considering the heat kernel on the complex projective space CP N −1 = S 2N −1 /S 1 rather than S 2N −1 . This is by no means a loss of generality since we are interested in the joint distribution of the moduli of k coordinates of U t . Besides, the space of continuous functions on CP N −1 decomposes as the direct sum of subspaces of U(N )-spherical harmonics that are homogeneous of degree zero, while the decomposition of continuous functions on S 2N −1 involves all spherical harmonics ( [8]). Accordingly, the heat kernel on CP N −1 is expressed as a series of normalized Jacobi polynomials (P N −2,0 n /P N −2,0 n (1)) n≥0 which for each n, gives the n-th reproducing kernel on CP N −1 ([11]). Hence, the integration over the sphere S 2N −3 together with an application of Koornwinder's addition Theorem ( [11]) lead to first result proved here: Up to an additional ingredient, the derivation of the density of U (2) t is quite similar. Loosely speaking, we would like to integrate the heat kernel over the sphere S 2N −5 (we assume N large enough) and as such, we need to decompose degree zero homogeneous spherical harmonics in S 2N −1 under the action of the unitary group U(N −1). This decomposition is stated in [12], Theorem 5.1, and the n-th reproducing kernel in turn decomposes as a weighted sum of reproducing kernels on S 2N −3 . Consequently, Koornwinder's addition Theorem again leads to the sought density which may be expressed through Jacobi polynomials in the simplex Σ 2 ([6], Proposition 2.3.8 p.47). More precisely Proposition 1.2. Let (Q (N ) j,n−j ) n≥0,0≤j≤n denote the family of Jacobi polynomials in the simplex Σ 2 . Then the density of U More generally, the derivation of the distribution of U (k) t , 2 ≤ k ≤ N − 1 relies on the decomposition of the spherical harmonics under the action of U(N − k + 1). The resulting density with respect to Lebesgue measure du is expressed through orthonormal Jacobi polynomials in Σ k as where (c 1 , . . . , c k ) = (|U 1 0 | 2 , . . . , |U k 0 | 2 ). Yet computations become tedious and we are not willing to exhibit them here. Rather, we shall revisit and complete the investigations started in [13]. Actually, an expression for the Laplace transform of the density of |U 1 t | 2 was obtained there and involves the following sequence (a n = a n (c, N )) n≥0 of real numbers determined recursively by ( [13], eq. 4.23) p n=0 a n p n where a 0 = 1 and (x) p = Γ(x + p)/Γ(p) is the Pochhammer symbol. In particular, the following was proved ([13] eq. 4.24. and eq. 4.25): a n (0, N ) = (−1) n (N + n − 1) n , a n (1, N ) = (N − 1) n n!(N + n − 1) n , 3 N. Demni which we can rewrite as 1 respectively. Using a Neumann series for Bessel functions ( [15]), we shall prove that Proposition 1.3. For any c ∈ [0, 1], a n = a n (c, N ) = 1 Having these coefficients in hands, we can then invert the Laplace transform and retrieve (1.2). At this level, we point out that the pde satisfied by the Laplace transform of the density of |U 1 t | 2 leads after integrations by parts to the heat equation associated with the Jacobi operator More generally, the pde satisfied by the Laplace transform of the joint distribution of U gives rise to the heat equation on the standard simplex associated with the generalized Jacobi operator ( [1], see also [6] p.46 but consult the list of errata available on the webpage of Y. Xu): This is an elliptic operator admitting different orthogonal basis of eigenpolynomials corresponding to the sequence of eigenvalues {−n(N +n−1), n ≥ 0}. Among them figure the Jacobi polynomials in the simplex, which agrees with our previous computations. The paper is organized as follows. The two following sections are devoted to the derivations of the densities displayed in (1.2) and (1.3). In section 4, we prove proposition 1.3 and invert the Laplace transform of the density of |U 1 t | 2 . In section 5, we perform integrations by parts on the pde satisfied by the Laplace transform of the density of U (k) t , omitting for a while the boundary terms. In the last section, we write down the latters and show that all of them vanish unless k = N − 1.
coincides with the random variable studied in [13] is justified as follows. The sphere S 2N −1 ≈ U N /U N −1 is a homogeneous space and if ρ : is the Hopf projection then we also have from Proposition G. III. 15 in [5]: Acknowledgements. We would like to thank D. Bakry, C.F. Dunkl, T. Hmidi and Y. Xu for stimulating discussions and for their help with appropriate references.

The heat kernel on CP N −1 and the distribution of
Let m, n be non negative integers and recall from [11] that (m, n)-complex spherical harmonics are the restriction to S 2N −1 of harmonic polynomials in the variables which are m-homogenous in the variables (z i ) N i=1 and n-homogeneous in the variables (z i ) N i=1 . Taking m = n, we obtain the (n, n)-complex spherical harmonics that are homogenous of degree zero with respect to the action of S 1 . Their restrictions to CP N −1 form a dense algebra in the space of continuous functions on CP N −1 endowed with the uniform norm (see [8], p.189). Moreover, the spectrum of the Laplace-Beltrami operator on CP N −1 is given by the sequence {−n(n + N − 1), n ≥ 0} 1 . Hence, the corresponding heat kernel is expanded in any orthonormal (with respect to the volume measure vol CP N −1 ) basis of homogeneous spherical harmonics of degree zero (Y j ) j≥1 as: 1 We normalize the Laplacian on CP N −1 by a factor 1/4.

N. Demni
Here d(n, N ) is the dimension of the eigenspace of (n, n)-complex spherical harmonics given by (Theorem 3.6 in [11]) Besides, the reproducing kernel formula (Theorem 3.8 in [11] 2 ) shows that the kernel R t is real and does not depend on the choice of the basis (this is the analogue of (22) in [10]): , p.99). Thus Gasper's Theorem entails the positivity of R t ( [7]). Now, we proceed to the derivation of the density of |U 1 t | 2 and start with the decompositions where e 1 is the first vector of the canonical basis of C N , θ 1 , θ 2 ∈ (0, π/2), ξ 1 , ξ 2 ∈ S 2N −3 . The volume measure of CP N −1 in turn splits as (see [11], and the next step is to integrate (2.1) over ξ 2 . But U (N − 1) acts transitively on S 2N −3 therefore we can take ξ 1 = e 2 to be the second vector of the canonical basis. As such, we are left with the volume of S 2N −5 (if N is large enough) and with the integration over the distribution of the first coordinate of ξ 2 . If this coordinate is parametrized by (r, ψ) then its distribution reads Consequently, the density of |U 1 t | 2 displayed in (1.2) follows from the product formula (4.12) in [11] together with the variables change u = cos 2 θ 2 (c = cos 2 θ 1 ).
Remark 2.1. The eigenvalue of a (n, n)-spherical harmonic equals the eigenvalue of a O(2N )-spherical harmonic of degree 2n in S 2N −1 viewed as a real Euclidian sphere. This coincidence is due to the fact that both polynomials are homogenous with the same total degree 2n and since the corresponding eigenvalue comes from the action of the Euler operator Up to an additional ingredient, the lines of the previous proof enable to derive the density of U t . More precisely, we start with the decompositions w = cos θ 1 e 1 + sin θ 1 ξ 1 = cos θ 1 e 1 + sin θ 1 cos β 1 e iφ 1 e 2 + sin θ 1 sin β 1 η 1 , z = cos θ 2 e 1 + sin θ 2 ξ 2 = cos θ 2 e 1 + sin θ 2 cos β 2 e iφ 2 e 2 + sin θ 2 sin β 2 η 2 , where β 1 , β 2 ∈ (0, π/2), φ 1 , φ 2 ∈ (0, 2π), η 1 , η 2 ∈ S 2N −5 and e 2 is the second vector of the canonical basis of C N . We also split the volume measure Now comes the needed additional ingredient, which is the special instance m = n, φ 1 = φ 2 = 0 in the formula stated in the bottom of p.5 in [12]. In order to recall it, let be the j-th normalized Jacobi polynomial and define the complex-valued polynomial ( [11], eq.3.15) as well as ( [12], p.6) Then the n-th reproducing kernel on CP N −1 admits the following expansion Substituting in (2.1), we see that the next step towards the joint distribution of (U 1 t , U 2 t ) consists in integrating To this end, we can assume without loss of generality that η 1 = e 3 (the third vector of the canonical basis) and use formula 8

The distribution of |U 1 t | 2 : another proof
In this section, we shall prove proposition 1.3. To this end, we rewrite (1.4) as p n=0 a n 1 n!(p − n)!
multiply both sides of (4.1) by (−1) p (x/2) 2p+N −1 for x lying in some neighborhood of zero then sum over p ≥ 0. Interchanging the order of summation, the system (1.4) is equivalent to n≥0 a n n!

N. Demni
Specializing it to ν = N − 2, we get Substituting in (4.1), then the uniqueness of the solution of (1.4) yields a n n!

From the Laplace transform to the generalized Jacobi operator
Another way to come from ϕ t (c, λ) to f t (c, λ) is as follows. For sake of simplicity, we shall drop the dependence on the parameter c. So, recall from [13] Proposition 4.2 that ϕ satisfies with the initial conditions ϕ 0 (c, λ) = e λc , ϕ t (c, 0) = 1. Assume the density f is unknown and is smooth in both variables (t, u), then integration by parts yield Otherwise, write f t (u) = g t (u)s 1 (u) for a smooth function g and note that L (s 1 ) = 0. As a result, where the RHS is the operator displayed in (1.6). Since f t (1) = 0 when N ≥ 3 then we always have But the set of monomials (u n ) n≥0 is total in L 2 ([0, 1], (1−u) N −2 du) ( [6], Theorem 3.17) then g solves the heat equation More generally, the Laplace transform of the density of U Hoping there will be no confusion, set again where du is the Lebesgue measure in the simplex Σ. Then, integration by parts where this time L denotes the operator If k = N − 1 then L reduces to the operator displayed in (1.7). Otherwise, set where this time g is a smooth function in both variables (t, u), u ∈ Σ k and note that the relations acting on g t . Consequently, if the boundary terms vanish then Theorem 3.17 in [6] implies that g t solves the heat equation We shall see below that this is the case provided that 1 ≤ k ≤ N − 2.

Analysis of the boundary terms
Recall from the previous section that the integration by parts performed in the one-variable setting gave rise to the boundary term which vanishes at u = 1 since f t (1) = 0 when N ≥ 3 (note that there is no such condition when N = 2). For higher values k ≥ 2, the situation is similar provided that 1 ≤ k ≤ N − 2 and is different when k = N − 1 due to the interactions between u i and u j for i = j. Indeed, the boundary terms are given by If N ≥ k + 2 and f t = g t s k vanishes on the hyperplane {u 1 + · · · + u k = 1} and the boundary terms reduce to But since ∂ i s k = ∂ j s k and since s k vanishes on {u 1 + · · · + u k = 1}, then for any 1 ≤ i = j ≤ k ∂ i f t (u) = ∂ j f t (u), u 1 + · · · + u k = 1 so that all boundary terms vanish. When k = N − 1 the boundary terms read