Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces

We find necessary and sufficient conditions for a finite K–bi–invariant measure on a compact Gelfand pair (G,K) to have a square–integrable density. For convolution semigroups, this is equivalent to having a continuous density in positive time. When (G,K) is a compact Riemannian symmetric pair, we study the induced transition density for G–invariant Feller processes on the symmetric space X = G/K. These are obtained as projections of K–bi–invariant Lévy processes on G, whose laws form a convolution semigroup. We obtain a Fourier series expansion for the density, in terms of spherical functions, where the spectrum is described by Gangolli’s Lévy–Khintchine formula. The density of returns to any given point on X is given by the trace of the transition semigroup, and for subordinated Brownian motion, we can calculate the short time asymptotics of this quantity using recent work of Bañuelos and Baudoin. In the case of the sphere, there is an interesting connection with the Funk–Hecke theorem.


Introduction
Let X = (X(t), t ≥ 0) be a (time-homogeneous) Feller-Markov process, which takes values in a locally compact space M that is equipped with a positive regular Borel measure μ on its Borel σ -algebra. Key quantities of interest are the transition kernel K t (x, A) which is the probability that X(t) ∈ A given that X(0) = x, where x ∈ M and A is a Borel set, and the Feller semigroup (P t , t ≥ 0) defined on the space of real-valued continuous functions on M that vanish at infinity by There are a number of interesting and fundamental (linked) questions that we can ask: (I) Does a transition density k t (·, ·) exist for t > 0, so that and does the function k : (0, ∞)×M ×M → [0, ∞) have good regularity properties, such as continuity, differentiability or finite L p -norm? (II) Do the operators P t extend to form a semigroup on L 2 (M, μ). If so, when is it selfadjoint or trace-class? (III) Does there exists a complete set of (normalised) eigenfunctions {φ n , n ∈ N} for P t (with t > 0) so that for all x, y ∈ M we can write k t (x, y) = n∈N λ n (t)φ n (x)φ n (y), (1.1) where P t φ n = λ n (t)φ n ? (IV) When do we have a trace formula: M k t (x, x)μ(dx) = trace(P t )?
Assuming (I) and some imposed regularity, a quite general approach was taken to (II) and (III) in [20] with M assumed to be a compact separable metric space. A key role here is played by the requirement that k t exists and is square-integrable. The case where X is a compact Riemannian manifold, μ is the Riemannian volume measure μ and X is Brownian motion has been extensively studied. In this case k t is the "ubiquitous" heat kernel [23], and (I) to (IV) all have positive answers (see e.g. Chapter 3 of [31] or Chapter VI of [12]).
In [5] (and references therein) positive answers to (1) to (IV) were obtained for a class of central symmetric Lévy processes in compact Lie groups (where μ is normalised Haar measure), which are obtained by subordinating Brownian motion. Here the key techniques used were harmonic analytic, arising from the representation theory of G. The restriction to central (conjugate-invariant) processes was key as it ensured that the Fourier transform of the measure was a scalar, and these scalars give us the required eigenvalues, through the Lévy-Khintchine formula.
In this paper, we present another class of processes for which (I) to (IV) are valid. The object of interest is a Feller-Markov process defined on a compact symmetric space M, which has G-invariant transition probabilities, where G is the identity component of the isometry group of M, and is conditioned to start at the point fixed by a closed subgroup K of G (so M = G/K). The reference measure μ is the unique (normalised) G-invariant measure on M. It is well-known that all such processes arise as the projection to M of a Lévy process in G whose laws form a K-bi-invariant convolution semigroup of probability measures (see [9,24,25]). We cannot assert that this is a special case of the theory developed in [5], as the measures we consider are not, in general, central (see Proposition 4.3); however many of the techniques developed in [5], may be applied here. In particular, we find that the eigenvalues we need are given by Gangolli's Lévy-Khintchine formula [19,27], and the eigenvectors are the spherical functions (so we deal with a complex form of Eq. 1.1).
The plan of the paper is as follows. In Section 2, we develop some general considerations concerning measures on homogeneous spaces. Some, but not all, of the results presented there are known. The purpose of Section 3, is to extend the work of [3] to find necessary and sufficient conditions, in terms of the Fourier transform, for a finite measure associated to a compact Gelfand pair to have a square-integrable density. In Section 4, we consider convolution semigroups of probability measures, where we present a recent result of Liao [25] which tells us that the measures have a continuous density if and only if they have a square-integrable one. So from the work of Sections 3 and 4 together, we have necessary and sufficient conditions for a measure within a convolution semigroup, as above, to have a continuous density (in positive time). In Section 5 we specialise to compact symmetric spaces, where we develop the Fourier expansion of the transition density, and obtain the required trace formula. It is worth pointing out that we don't require our measures to be symmetric (or equivalently P t to be self-adjoint), in contrast to [20] and [5]. We also apply the theory of [7] to study short-time asymptotics of the transition density corresponding to subordinated Brownian motion on M. Finally in Section 6, we present the example of the n-sphere in a little more detail, and consider the implications of the Funk-Hecke theorem within our context.
Notation If X is a locally compact Hausdorff space, then B(X) is its Borel σ -algebra, C c (X) will denote the linear space of continuous functions from X to R having compact support, M(X) is the linear space of positive regular Borel measures on (X, B(X)), and M F (X) is the subspace comprising finite measures. If G is a locally compact Hausdorff group, we will denote its neutral element by e. We equip the space M(G) with the binary operation of convolution * , so that if μ 1 , μ 2 ∈ M(G), their convolution μ 1 * μ 2 is the unique element of M(G) such that for all f ∈ C c (G), Then M(G) is a monoid, with neutral element given by the measure δ e , where for all A ∈ for all A ∈ B(G), and A −1 := {g −1 ; g ∈ G}.
All L p spaces appearing in this paper comprise complex-valued functions. The space of all d × d complex-valued matrices is denoted by M d (C), and the trace of A ∈ M d (C) is written tr(A).

Absolute Continuity of Measures on Homogeneous Spaces
Let G be a locally compact Hausdorff group, which we equip with a a left Haar measure m G . We will tend to write m G (dg) = dg within integrals. The modular homomorphism from G to the multiplicative group (0, ∞) will be denoted G . It is uniquely defined by the fact that Let K be a closed subgroup of G, with fixed Haar measure m K , and X denote the homogeneous space G/K of left cosets of G, i.e. X = {gK, g ∈ G}. We equip X with the usual (Hausdorff) topology which is such that the canonical surjection ξ : G → X is both continuous and open. We will write o := ξ(K). The group G acts on X by homeomorphisms via the action τ (g)g K = gg K, for all g, g ∈ G. We will make frequent use of the fact that for all g ∈ G, where l g (h) = gh for all g, h ∈ G. If K is compact, we will always normalise so that m K (K) = 1. Define . It is shown that P is surjective in [17], pp.61-2. Let C c,K (G) denote the linear subspace of C c (G) comprising functions that are Kright-invariant. If K is compact, then any F ∈ C c (X) gives rise to F ξ ∈ C c,K (G) by the assignment F ξ = F • ξ . It is not difficult to see that this induces a linear isomorphism between the two spaces. If K is not compact, then ξ −1 (C c (X)) may contain no functions of compact support (e.g. consider the case G = R and K = Z).
If μ ∈ M(G) then μ ξ := μ • ξ −1 ∈ M(X). Moreover for all F ∈ C c (X) we have (see e.g. [11], Proposition 3.6.1, pp. 190-1) If we begin with functions defined on G, rather than on X, then given μ ∈ M(G), there exists a unique μ ∈ M(X) such that for all f ∈ C c (G), if and only if for all f ∈ C c (G), k ∈ K, If we take μ = m G , then it follows from Eqs. 2.5 and 2.1 that m G exists and is unique if and only if (2.6) for all k ∈ K. In this case we will write σ := m G . It is easily seen from Eq. 2.4 that σ is G-invariant, in that σ (τ (g)A) = σ (A), for all g ∈ G, A ∈ B(X), and in fact, if it exists, σ is (up to multiplication by a positive constant) the unique such measure on (X, B(X)). If K is compact, then Eq. 2.6 holds with both sides of the equation being equal to one.
We could repeat the above discussion, with X being replaced by X = K\G, the space of right cosets of G, which is again a locally compact Hausdorff space, with topology such that the natural surjection ξ : G → X is open and continuous. The natural action of G on X is τ (h)(Kg) = Kgh, for g, h ∈ G, and if Eq. 2.6 holds, there is a unique (up to positive scalar multiplication) G-invariant measure σ on X such that σ (τ (g )A) = σ (A),for all g ∈ G, A ∈ B(X). This measure is related to right Haar measure on G in the same way that σ is related to left Haar measure. The mapping C which takes gX to Xg for all g ∈ G is easily seen to be a homeomorphism between X and X , and we then have σ = σ • C −1 .
From now on we will always assume that K is compact. (2.7) If Eq. 2.5 is satisfied, and μ ∈ M K (G), then it follows from Eqs. 2.3, 2.4 and 2.7 that μ = μ ξ . In particular, m G ∈ M K (G), as can be seen from Eq. 2.1 and the fact that G (k) = 1 for all k ∈ K (see also Proposition 1.10 (a) in [25]), so σ = (m G ) ξ . The mapping μ → μ ξ is an isometric isomorphism between M K (G) and M(X), and It is well-known (and easy to deduce) that L g is an isometric isomorphism of L p (G), and if 1 ≤ p < ∞, then the mapping g → L g f is continuous from G to L p (G) (see e.g. Proposition 1.2.1 in [6]). For each g ∈ G, F ∈ L p (X), define T g F := F • τ (g −1 ). Then T g is an isometric isomorphism. Moreover we have Proof Since for all g, h ∈ G, T gh = T g • T h and T h is an isometry, its sufficient to prove continuity at e. First observe that by Eq. 2.2 and the result follows by continuity of the map g → L g F ξ . Now suppose that μ ∈ M K (G) is absolutely continuous with respect to m G and write the Radon-Nikodym derivative h := dμ/dm G . In the sequel we will frequently identify h with a particular member of the equivalence class that it defines in L 1 (G), and in common with standard probabilistic usage, we may refer to any version of h as the density of μ (with respect to m G ).
and the result follows.

Proposition 2.3 The measure μ ∈ M K (G) is absolutely continuous with respect to Haar measure on G, having Radon Nikodym H ξ if and only if μ ξ is absolutely continuous with respect to the
Proof Firstly let μ ∈ M K (G) be absolutely continuous as stated. Then its Radon-Nikodym derivative h is K-right-invariant by Proposition 2.2, and so h = H ξ for some H ∈ L 1 (X). Then for all F ∈ C c (X), and the result follows.
Conversely, if μ ξ is absolutely continuous with Radon-Nikodym derivative H , then and the result again follows since μ is determined by its action on C c,K (G). The result on continuity follows from the fact that the mapping H → H ξ is a bijection between C(X) and C K (G). The integrability statement follows similarly.
We have the following partial generalisations of a known result on locally compact groups due to Raikov-Williamson (see [33] and [6], Theorem 4.4.1 p.98).

Proposition 2.4 If μ ∈ M(X) is absolutely continuous with respect to σ then for all
Proof Writing h := dμ/dσ , we have by Proposition 2.1.
We conjecture that the converse to Proposition 2.4 also holds, but the proof of the corresponding result on a group G requires both the left and right action of G on itself, and we do not have analogues of both tools available to us. The The set E is a locally compact Hausdorff space when equipped with the topology which makes the canonical surjection ζ from G to E continuous and open. If N is the normaliser

Square-Integrability of Densities on Compact Gelfand Pairs
From now on, we will assume that (G, K) forms a compact Gelfand pair, so that G is a compact group, K is a closed subgroup, and the Banach algebra (with respect to convolution) L 1 (K\G/K) is commutative (see e.g. [35] for background on such structures, and for material that now follows). Haar measure m G will be normalised henceforth, so that m G (G) = 1. We denote by G the unitary dual of G, i.e. the set of all equivalence classes of irreducible representations of G, with respect to unitary conjugation. If π ∈ G, its representation space V π is finite-dimensional, and we will write d π :=dim(V π ). Fix an orthonormal basis {e π 1 , . . . , e π d π } for V π , and define the co-ordinate functions on G by π ij (·) := π(·)e π i , e π j for i, j = 1, . . . , d π . The celebrated Peter-Weyl theorem tells us that be the subspaces of L 2 (G) comprising functions that are almost-everywhere K-left-invariant, K-right-invariant, and K-bi-invariant (respectively). The orthogonal projections from L 2 (G) onto these spaces will be denoted, respectively P K , P K and Q K = P K P K = P K P K , so that for all f ∈ L 2 (G), g ∈ G, We can and will use natural isomorphisms between these spaces to identify L 2 lk (G) with L 2 (X , σ ), L 2 rk (G) with L 2 (X, σ ) and L 2 bk (G) with L 2 (E, α). This last space will often just be written as L 2 (K\G/K), in line with standard usage. Now recall that a representation π of G is said to be spherical if there exists a nonzero spherical vector u π ∈ V π , i.e. π(k)u π = u π for all k ∈ K. If this is the case, then u π is unique up to scalar multiplication, and we define V K π := {λu π ; λ ∈ C}. We find it convenient to define V K π = {0} if π is not spherical. In either case, let E K π = K π(k)dk be the orthogonal projection from V π to V K π . When π is spherical, we will, for convenience, assume that u π has norm one, and we always select our orthonormal basis {e π 1 , . . . , e π d π } in V π so that e π 1 = u π . Let G s be the subset of G comprising spherical representations. Now let f π u,v := π(·)u, v , where u, v ∈ V π , π ∈ G. Then f π u,v ∈ C(G), and easy algebra yields We have the following consequences of the Peter-Weyl theorem: Proof This follows easily from the Peter-Weyl theorem and Eq. 3.1. Note that at least (3)  In relation to Proposition 3.1(3), observe that the prescription φ π (g) := e π 1 , π(g)e π 1 , for g ∈ G defines a (positive-definite) spherical function on G, i.e. a non-trivial continuous function from G to C so that for all g, h ∈ G, K φ π (gkh)dk = φ π (g)φ π (h), (3.2) and all spherical functions on G arise in this way (see [21] pp.414-7 or [35] pp.204-5). Since the conjugate representation to π is both irreducible and spherical whenever π is, we can rewrite the result stated in the more familiar form that { √ d π φ π , π ∈ G s } is a complete orthonormal basis for L 2 bK (G). If μ ∈ M F (G), its Fourier transform is the matrix-valued function where π ∈ G. Properties of the Fourier transform are developed in section 4.2 of [6]. In particular, μ uniquely determines the measure μ.
If μ is K-bi-invariant, its spherical transform is the complex-valued mapping: where φ is a spherical function on G, and this also uniquely determines μ (see e.g. [22]).
In [3] (see also Theorem 4.5.1 in [6]), it is shown that μ ∈ M F (G) has a squareintegrable density if and only if where || · || H S denotes the matrix Hilbert-Schmidt norm, so that || μ(π )|| 2 H S := tr( μ(π ) * μ(π )). Furthermore, if Eq. 3.3 holds, then the density f μ := dμ/dg has the L 2 -Fourier expansion: We will need the following useful characterisations of K-invariant measures by means of their Fourier transforms, in relation to which it's worth noting that the mapping μ → μ is a bijection between K-left-invariant and K-right-invariant measures on G.
1. The following are equivalent: ∈ G s , or π ∈ G s and j = 1.
Proof We just prove (2) as (1) is similar and (3) follows from these two assertions. The equivalence of (b) and (c) is straightforward linear algebra and is left to the reader. To show that (a) implies (b), let π ∈ G and u, v ∈ V π . If μ is K-right-invariant, then π v and the result follows.

in [6], p.33), μ is determined by its integrals against functions in C rc (G), and so it is K-right-invariant.
Remark If π / ∈ G s , then E K π = 0 and so μ(π) = 0 in all three cases of Proposition 3.2. When we combine the main result of [3] (see also Theorem 4.5.1 in [6]) with that of Proposition 3.2 we get Theorem 3.3 Let μ ∈ M F (G).

If μ is K-bi-invariant, then it has an L 2 -density if and only if
Proof We just prove (1) as the others are similar. If μ has an L 2 -density, then the result follows from Eq. 3.3 and Proposition 3.2(1). For the converse direction, note that by Kleft-invariance of μ and Proposition 3.2(1), we have and then the result again follows by Eq. 3.3.
In all three cases, the Fourier expansion of the density is given by Eq. 3.4. Any (left, right or bi)-K-invariance of the measure is inherited by the density (almost everywhere). This is a consequence of Proposition 2.2 and its generalisation to the K-left-invariant case; it can also be deduced by uniqueness of Fourier transforms, using Proposition 3.2. If μ is K-bi-invariant, it is easy to check that (in the L 2 -sense), (3.5)

Remarks
1. The advantage of these results over (3.3) is the reduction in summing over the whole of G to summing over the subset G s , and then summing over a smaller number of matrix elements; in Theorem 3.3(3) there is the additional advantage of having a single matrix element. 2. The results of this section generalise beyond the category of Gelfand pairs, to arbitrary (G, K), where G is compact and K is closed; but the set-up presented here is convenient for the sequel.

K-Invariant Densities and Kernels for Convolution Semigroups
Let (μ t , t ≥ 0) be a convolution semigroup of probability measures on the compact group G. By this we mean that • μ s+t = μ s * μ t for all s, t ≥ 0, • weak-lim t→0 μ t = μ 0 .
It then follows that μ 0 is Haar measure on a compact subgroup of G (see Theorem 4.6.1 in [6], p.104). We say that the convolution semigroup is standard if μ 0 = δ e . In this case, (see e.g. Proposition 5.1.2 in [6] and the discussion that follows) for each π ∈ G, ( μ t (π ), t ≥ 0) is a strongly continuous one-parameter contraction semigroup on V π . Furthermore (P t , t ≥ 0) is a contraction semigroup of linear operators on L 2 (G) defined for each t ≥ 0, f ∈ L 2 (G), g ∈ G by and we have μ t (π ) ij = P t π ji (e) for each π ∈ G, 1 ≤ i, j ≤ d π .
It is shown in [26]

that a convolution semigroup is K-left-invariant if and only if it is K-right-invariant if and only if it is K-bi-invariant.
Then μ 0 = m K , (P t , t ≥ 0) as defined above 1 is a contraction semigroup on L 2 (K\G/K), and for each π ∈ G s , ( μ t (φ π ), t ≥ 0) is a strongly continuous one-parameter contraction semigroup of complex numbers.
For each μ ∈ M(G), using the Riesz representation theorem, we define μ (K) to be the unique measure in M(K\G/K) for which for all f ∈ C c (G).

Theorem 4.1 Let μ ∈ M F (G).
1. If μ has a square-integrable or continuous density then so does μ (K) . 2. If G is a connected Lie group having Lie algebra g and μ has a C p -density for p ∈ N, then so does μ (K) .
In all cases, if f is the density of μ, then that of μ (K) is Q K f . Proof 1. Both follow easily since Q K is an orthogonal projection from L 2 (G) to L 2 (K\G/K) which preserves continuity. 2. For all X 1 , . . . , X p ∈ g, g ∈ G the mapping g → X 1 . . . X p Q K f (g) is well defined and continuous, indeed standard arguments yield and the result follows by a theorem of Sugiura [32] pp. 42-3 (see also Theorem 1.3.5 on p.20 of [6]). Now suppose that (μ t , t ≥ 0) is a standard convolution semigroup, and consider the associated set of K-bi-invariant probability measures (μ (K) t , t ≥ 0).

Proposition 4.2 (μ (K) t , t ≥ 0) is a K-bi-invariant convolution semigroup on G.
Proof For all f ∈ C(G), s, t ≥ 0, by Eq. 4.1, Here we have used the bi-invariance of Haar measure on K to first make a change of variable l → ll and then l → l l. The fact that μ The weak continuity follows easily from Eq. 4.1.
Many explicit examples of convolution semigroups that we consider in the next section fall under the aegis of Theorem 4.1 and Proposition 4.2.
Recall that μ ∈ M(G) is said to be central (or conjugate-invariant) if μ(gAg −1 ) = μ(A) for all g ∈ G, A ∈ B(G), and we let M F c (G) be the set of all finite central measures on G. It is shown in Theorem 4.2.2 of [6] that μ ∈ M F c (G) if and only if for each π ∈ G there exists c π ∈ C so that μ(π) = c π I π . It then follows that μ (K) (φ π ) = c π for all π ∈ G s . We will see important examples of central measures in the next section. If G is abelian, then all measures on G are central, and all irreducible representations of G are one-dimensional. The next proposition presents some evidence that if G is compact and non-abelian and μ ∈ M F c is non-trivial, then μ (K) is not central, in general. 2

Proposition 4.3
Let μ ∈ M F c (G), so that μ(π ) = c π I π for all π ∈ G, and assume that there exists π ∈ G s with dim(V π ) > 1. If μ is K-bi-invariant, then c π = 0. Proof By Proposition 3.2(3), μ(π ) = c π I π = E K π μ(π )E K π , from which we deduce that c π I π = c π E K π . Assume c π = 0; since the range of E K π is one-dimensional, we can find a non-zero vector in (E K π ) ⊥ and this yields the desired contradiction.
We return to the study of convolution semigroups (μ t , t ≥ 0). We are interested in the case where μ t has a continuous density f t for all t > 0. The following theorem is essentially due to Liao [25], Theorem 4.8. (μ t , t ≥ 0) be a convolution semigroup of probability measures on the compact group G. The following are equivalent:
then f t is the uniform limit of a sequence of continuous functions on G and so is continuous.
Using the matrix inequality |tr(A * B)| ≤ ||A|| H S ||B|| H S for A, B ∈ M d π (C), and the Cauchy-Schwarz inequality, we have for all for all g ∈ G, t > r/2, For the remainder of this paper, we assume that X is a compact (globally Riemannian) symmetric space, so that G is a compact Lie group with Lie algebra having dimension d. Let L = (L(t), t ≥ 0) be a (left) Lévy process on G so that L has stationary and independent increments and is stochastically continuous (see e.g. [24] for relevant background). For each t ≥ 0, let μ t denote the law of L(t), so that μ t (A) = P (L(t) ∈ A) for all A ∈ B(G). We say that the process L is K-bi-invariant, if μ t is K-bi-invariant for all t ≥ 0. From now on we assume that L is a K-bi-invariant Lévy process with μ 0 = m K . Then for all t ≥ 0. Then as is shown in [25] (see also Theorem 3.2 in [9] o (a.s.) 3 The G-invariance is manifest as follows: for each t ≥ 0, x ∈ X, A ∈ B(X), let q t (x, A) = P (Y (t) ∈ A|Y (0) = x) be the usual transition probability, then for all g ∈ G: and as is easily verified (see also Proposition 1.16 of [25]) (4.2)

Theorem 4.5 Let (L(t), t ≥ 0) be a K-bi-invariant Lévy process on G, and (Y (t), t ≥ 0)
be the projected Feller process on X. If for all t > 0, L(t) has a continuous density ρ t , then Y (t) has a continuous transition density k t (·, ·), and for all g, h ∈ G we have k t (gK, hK) = ρ t (g −1 h).
where ρ t is the unique function in C(X) so that ρ t = ρ t • ξ . So the required transition density exists and for all g, h ∈ G, we have by Eq. 2.2,

Eigenfunction Expansions for the Transition Kernel
Let the processes L and Y be as in the previous section, so that (μ t , t ≥ 0) is a K-biinvariant convolution semigroup on G with μ 0 = m K . We continue to assume that μ t has a continuous density ρ t for all t > 0. We equip G with an Ad-invariant Riemannian metric which is compatible with the Riemannian structure on X, and let be the associated Laplace-Beltrami operator on G. Then {κ π , π ∈ G} will denote the Casimir spectrum for G so that κ π ≥ 0 (with κ π = 0 if and only if π is trivial) and φ π = −κ π φ π for all π ∈ G s . Assume that the symmetric space X is irreducible, in that the action of Ad(K) on p is irreducible, where p := g k, and k is the Lie algebra of K. A sufficient condition for this to hold is that G is semisimple (see Proposition 5.12 in [25]). Then Gangolli's Lévy Khinchine formula (see e.g. [2,19,27]) tells us that for all t ≥ 0, π ∈ G s μ t (φ π ) = e −tχ π , (5.1) where with a ≥ 0 and ν a K-bi-invariant Lévy measure on (G, B(G)). It follows easily from Proposition 4.3 that if G is non-abelian, then μ t cannot be central for t > 0.
If μ t has a continuous density for all t ≥ 0, then P t is trace-class in L 2 (G), and its trace is given by Proof 1. We argue as in the proof of Theorem 5.3 in [5]. First observe that since for each 1 ≤ i, j ≤ d π , g, h ∈ G, π ij (g) := π(g)e π i , e π j and π(gh) = π(g)π(h), then π ij (gh) = d π k=1 π ik (h)π kj (g). Hence π kj (g) μ t (π ) ki , and the result follows by Proposition 3.2 (3) and Eq. 5.1. 2. If μ t has a continuous density, P t is trace-class by Theorem 5.4.4 in [6]. From (1), we have Tr(P t ) = π∈ G s d π e −tχ π , but for each π ∈ G s , we have π ∈ G s , where π is the conjugate representation, and the result follows when we observe that χ π = χ π .
In the last theorem, we calculated the spectrum of P t in the space L 2 (G). In the next result, we restrict to the closed subspace L 2 (K\G/K).
The result then follows from Theorem 4.5. (b) As L g ρ t is K-right-invariant for all g ∈ G, we may use Proposition 3.1(2) to write, L g ρ t , π 1j π 1j , but for each j = 1, . . . , d π , since by K-bi-invariance of ρ t , ρ t , π 1j = 0 for all j = 1, and the result follows easily from here.
Remark Let A be the infinitesimal generator of the contraction semigroup (Q t , t ≥ 0) acting in L p (X), for 1 ≤ p < ∞. Then for f ∈ L p (X), it is well-known that the unique solution to the initial value problem ∂u(t) ∂t = Au, with u(0, ·) = f is given by for t ≥ 0, x ∈ X. However if f ∈ L p K (X), which is the closed subspace of L p (X) comprising functions that are invariant under the action of τ (K), then it follows directly from Theorem 5.2 (2)(a) or (b), that for g, h ∈ G is the "effective" transition kernel. It is interesting to compare Theorem 5.2 (2) (a) with results obtained by Bochner for spheres (see [10] p.1146). In the case of the heat kernel (so ν = 0 in Eq. 5.2), a formula of this type on general compact homogeneous spaces is presented in [8].
We now easily deduce the following trace formula: If μ t is K-bi-invariant and has a square-integrable density for all t > 0, then Proof This follows on putting g = h = e in Theorem 5.2 (2). density of μ t , is the fundamental solution of the pde ∂u(t) ∂t = 1 2 G u(t). The spherical heat kernel ρ t , which is the density of μ (K) t , is the fundamental solution of ∂u(t) ∂t = 1 2 p u(t), where the "horizontal Laplacian" p := m i=1 X 2 i . For further details and discussion, see [2] and section 3.4 of [25].
We close this section by giving a brief account of subordination and short time asymptotics. For background on subordination in compact Lie groups, we refer the reader to section 5.7 of [6], and to [1]. Let (S(t), t ≥ 0) be a subordinator having law λ t for t ≥ 0, that is independent of the Lévy process (L(t), t ≥ 0). Then for all u > 0, [0,∞) e −us λ t (ds) = e −tψ(u) , where ψ is a Bernstein function such that lim u→0+ ψ(u) = 0, so that for all u > 0, . It is clear that if L is K-bi-invariant, then so is L S , and we make this assumption henceforth. Then for all π ∈ G s , t ≥ 0 μ S t (φ π ) = e −tψ(χ π ) . (5.5) The subordinated semigroup (P S t , t ≥ 0), which is the transition semigroup of the process L S , is defined as for all f ∈ L 2 (G), g ∈ G, t ≥ 0, and by Eqs. 5.3 and 5.5 we deduce that for all t ≥ 0, π ∈ G, 1 ≤ i, j ≤ d π , P S t π ij = e −tψ(χ π ) π ij if i = 1, π ∈ G s 0 o t h e r w i s e (5.6) If L(t) has a density ρ t for all t > 0, then L S (t) has a density ρ S t given by ρ S t (g) = t 0 ρ s (g)λ t (ds), for each g ∈ G. From now on, let L be K-bi-invariant Brownian motion on G (denoted ψ above), so that k t is the heat kernel on X: k t (gK, hK) = π∈ G s d π e −tκ π φ π (g −1 h), for t > 0, g, h ∈ G. Then as t → 0, we have the well-known asymptotic behaviour (see e.g. [16]): Theorem 5.4 If b > 0 or γ := inf y>0 (0,1) ue −uy τ (du) > 0 then ρ S t is continuous for all t > 0.
If b > 0, we have Tr(P S t ) ≤ π∈ G s d π e −tbκ π < ∞, since the right hand side is the trace of a heat kernel semigroup (with variance b) which we know to be finite. The other case is similar.

y)f (y)σ (dy).
Then as shown in [15] pp.204-5 for more general integral operators having this form, the SO(d)-invariance of the kernel determines the existence of a continuous (non-negative) real-valued function a t on [−1, 1] so that k t (x, y) = a t (x · y), for all t > 0, x, y ∈ S, where · denotes the usual scalar product in R d . The Chapman-Kolmogorov equations take the form a s+t (x · y) = S a s (x · z)a t (z · y)σ (dz), for each s, t ≥ 0.
From the results of the previous section, we know that for each t > 0, n ∈ Z + , H n is an eigenspace for the operator Q t , and that the eigenvalue e −tχ n has multiplicity d n . But by the Funk-Hecke theorem (see Theorem 9.5.3 in [15] p.205-6), we have for all t > 0, n ∈ Z + , e −tχ n =