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Spectral asymptotics for kinetic Brownian motion on locally symmetric spaces

We prove the strong convergence of the spectrum of the kinetic Brownian motion to the spectrum of base Laplacian for a large class of compact locally Riemannian homogeneous spaces, in particular all compact locally symmetric spaces. This generalizes recent work of Kolb--Weich--Wolf [arXiv:2011.06434] on constant curvature surfaces.


Introduction
Kinetic Brownian motion is a stochastic process that interpolates between the geodesic flow and Brownian motion. It was studied by several authors [FL07] [GS13] [ABT15] [Li16] recently from the stochastic point of view. In this paper we focus on the spectral theory and prove the convergence of spectrum of the kinetic Brownian motion to the base Laplacian for a family of locally Riemannian homogeneous spaces. For additional references and background, see [KWW22].
More precisely, let (M, g) be a closed Riemannian manifold of dimension n ≥ 2, and let X be the geodesic vector field on the unit tangent bundle SM. Let ∆ M be the Laplacian on M, and ∆ V be the vertical Laplacian on SM, namely (∆ V f )| SpM = ∆ SpM (f | SpM ) for every p ∈ M (Our sign convention is that Laplacians have nonnegative eigenvalues). For γ ∈ (0, ∞), define an operator on SM, where c n = 1/n(n − 1).
In fact, later we will prove the theorem for a more general class of Riemannian manifolds (see Theorem 2).
Here, the resolvant (∆ M − λ) −1 in the statement is a priori only defined on L 2 (M), but L 2 (M) ⊂ L 2 (SM) naturally by pullback and we extend (∆ M − λ) −1 to an operator on L 2 (SM) by assigning zero on L 2 (M) ⊥ . This will be explained again in Section 3.2.
Drouot [Dro17] and Smith [Smi20] studied the limit of P γ when γ → 0 for Anosov geodesic flows using semiclassical analysis. The operator P γ is an analogy to the hypoelliptic Laplacian L γ introduced by Bismut [Bis05]. The main analytic difference is that P γ is defined on the unit tangent bundle SM (or the unit cotangent bundle if we want) while L γ is defined on the cotangent bundle T * M. Bismut-Lebeau [BL08] proved a statement similar to Theorem 1 for L γ in the limit γ → ∞ for arbitrary closed manifolds M. Bismut [Bis11] also studied the limit of L γ 1 when γ → 0 for locally symmetric spaces M of noncompact type and obtained formulas for orbit integrals. This was used by Shen [She17] to give a proof of Fried's conjecture for locally symmetric spaces. See Shen [She21] for a survey on Fried's conjecture. As noticed in Kolb-Weich-Wolf [KWW22], it seems possible to adapt the method of [BL08] to P γ to study the γ → ∞ limit. In this paper, however, instead of attacking the general case following this line, we give a simple proof in the locally symmetric case by exploiting the symmetry following the idea of [KWW19] [KWW22].
γ → 0 γ → ∞ P γ negatively curved (slightly more general than) locally symmetric L γ locally symmetric no condition Table 1. Condition on M for which we have a control of the limit Kolb-Weich-Wolf [KWW22] showed that the spectrum of P γ converges to the spectrum of the base Laplacian ∆ M in a weak sense, when M is a constant curvature compact surface. While we follow the same idea of decomposing L 2 (SM) into Casimir eigenspaces, our work is a generalization of theirs in two aspects. First, we extend the statement to one where M is in a large family of Riemannian homogeneous spaces including closed locally symmetric spaces. This is achieved by a general construction of a Casimir operator in Proposition 2.1. Second, we improve the weak convergence of spectrum to the convergence of spectrum in locally compact topology, and we also prove the resolvent convergence. In particular, we use a Grushin problem instead of Kato's perturbation theory, which gives us more precise information about the spectrum and resolvant.
The structure of the paper is as follows. In Section 2 we exploit the symmetry on M to construct an operator Ω on SM, called a Casimir operator, which commutes with all operators on SM we will be concerned with. In Section 3, we prove Theorem 1 using a Grushin problem on each eigenspace of the Casimir operator Ω. The stronger convergence is proved by showing that there are only finitely many eigenspaces of the Casimir operator that would matter. This is proved by a contradiction argument, with a careful analysis of projections to spherical harmonics.

A Casimir operator on SM
The goal of this section is to construct a certain second order differential operator Ω on the unit tangent bundle SM of M where M is Riemmannian homogeneous with some extra properties. The exact statement is Proposition 2.1. All constructions and proofs relevant to this paper will be given in Section 2.1. Section 2.2 exploits the construction in more detail in the special case when M is locally isotropic by giving an alternative more explicit construction.
2.1. The construction. We first recall and fix the use of some terminologies. A Riemannian homogeneous space is a Riemannian manifold N together with the action of a Lie group G that is effective, transitive, and isometric. We usually write N = G/K where K is the isotopy group of a chosen point on N. A (complete) locally Riemannian homogeneous space is a smooth quotient M = Γ\N where N = G/K is a Riemannian homogeneous space, and Γ < G is a discrete subgroup acting freely and properly discontinuously on N.
An importance class of Riemannian homogeneous spaces is given by symmetric spaces. A symmetric space is a Riemannian manifold N such that for any p ∈ N, there exists an isometry of N that fixes p and induces the negation map on T p N. A symmetric space is Riemannian homogeneous, thus can be written N = G/K, where G is the isometry group of N. A locally symmetric space is a Riemannian manifold M that is locally modelled on some symmetric space N = G/K; if it is complete, then M = Γ\G/K for some Γ < G. References on Riemannian homogeneous spaces and symmetric spaces include [Arv03] [Hel79] [Wol11].
As usual, we use script lowercase letters to denote the Lie algebra of a Lie group. For example g denotes the Lie algebra of G.
Proposition 2.1. Let M = Γ\G/K be a locally Riemannian homogeneous space. Suppose there exists a Γ-invariant element in Sym 2 g whose image in Sym 2 (g/Ad g k) is negative definite for every g ∈ G, then there is a differential operator Ω on SM satisfying the following: (1) Ω is a second order symmetric operator, and its symbol in the horizontal part is negative definite.
(2) Ω commutes with the vertical Laplacian ∆ V , the total Laplacian ∆, and the geodesic vector field X on SM.
In the sequel, the operator Ω so constructed will be called a Casimir operator on SM. Here, our convention for symbols of differential operators is the one used in differential geometry. Namely, at a local level, the symbol of the k-th order operator P = |α|≤k c α ∂ α is |α|=k c α ξ α . When k = 2, this differs from the microlocal analysis convention by a sign.
The condition on M in Proposition 2.1 is not too restrictive. For example, it holds for all finite Γ (thus all compact G), as well as all (complete) locally symmetric M as we will show in a moment.
Remark 2.2. Assuming the element in Sym 2 g in Proposition 2.1 is actually G-invariant, our argument will show the same for the slightly wider class of (G, N)-manifolds M (i.e. manifolds that is locally modelled on the G-manifold N = G/K, with transition functions in G). Since for our analytic application we only care about compact (in particular complete) M, in which case all (G, N)-manifolds are of the form Γ\G/K, we don't bother to define the notion of (G, N)-manifolds.
Corollary 2.3. If M is a locally symmetric space (in particular, if it has constant curvature), then there exists an operator Ω on SM satisfying the conditions above.
In particular, there exists a Casimir operator Ω as above when M has constant curvature. In fact, an explicit choice of Ω in this special case is given in Corollary 2.8.
Proof. We only give the proof when M is complete. The general case (which we won't need) follows from Remark 2.2.
Let N be the universal cover of M, which is a symmetric space. It splits into N = N 0 × N 1 × N 2 , where N 0 = R r is euclidean, N 1 is of compact type and N 2 is of noncompact type. The isometry group of N is G = G 0 × G 1 × G 2 where G i is the isometry group of N i (see e.g. Wolf [Wol11, Theorem 8.3.8, 8.3.9] for these statements). We claim there is a G-invariant element in Sym 2 g whose image is negative definite in each Sym 2 (g/Ad g k), which is sufficient for applying Proposition 2.1. It suffices to check the claim for each factor When i = 1, the Killing form of g 1 is negative definite, thus its dual is a G 1 -invariant negative definite element in Sym 2 g 1 .
When i = 2, the Killing form of g 2 is nondegenerate on g 2 , negative definite on each Ad g k 2 and positive definite on the orthogonal complement of each Ad g k 2 . Therefore its negative dual is a G-invariant element in Sym 2 g 2 whose image in each Sym 2 (g 2 /Ad g k 2 ) is negative definite.
vector) satisfies the desired conditions.
Proof of Theorem 2.1. If M is any manifold with a left G-action, the negative associated vector field construction gives a Lie algebra homomorphism g → X(M), which induces a Gequivariant filtered-algebra homomorphism U(g) → Diff * (M) where U(g) is the universal envelopping algebra of g, and Diff * (M) is the algebra of C ∞ -differential operators on M.
The induced map on the associated graded algebra is Sym(g) → Γ(M; Sym(T M)), where the right hand side is recognized as the domain of symbols of C ∞ -differential operators on M, and the left hand side is by the Poincaré-Birkhoff-Witt theorem.
In our situation, the isometric G-action on N = G/K lifts to an isometric G-action on the unit tangent bundle SN. We have the following G-equivariant commutative diagram.
Here, the maps from the first column to the second are symbol maps, Sym 2 g ⊂ g ⊗ g → U ≤2 (g) is a section of U ≤2 (g) → Sym 2 g, π : SN → N is the projection, and T V (SN) is the vertical subbundle of T (SN). Now pass to the Γ-quotient M = Γ\N. Taking Γ-invariant elements gives (still using π : SM → M for the projection) The operators ∆ V , ∆, X on SN are defined only in terms of metric properties of N, therefore they are G-invariant. It follows that every vector field in the image of g → X(SN) commutes with them. Since g generates U(g), the same is true for U(g) → Diff * (SN). Also, since the G-action on SN is symmetric, the image of g → X(SN) consists of Killing fields, which are anti-symmetric as operators on SN. Therefore every operator in the image of Sym 2 g → Diff * ≤2 (SN) is symmetric. Passing to the quotient then implies every operator in the image of (Sym 2 g) Γ → Diff * (SM) is symmetric and commutes with ∆ V , ∆, X on SM.
By assumption, there is an element ω ∈ (Sym 2 g) Γ whose image in each Sym 2 (g/Ad g k) is negative definite. We show its image Ω ∈ Diff * ≤2 (SM) in the above diagram satisfies our requirements. Condition (2) and the symmetric property in condition (1) has just been justified. For the negative definite property in (1), note the horizontal part of the symbol of a second order differential operator on SM sits in the upper-right corner of the diagram, thus it suffices to show the image of ω in Γ(M; Sym 2 (T M)) is pointwisely negative definite. We include ω back to Sym 2 g and prove the equivalent statement that its image in Γ(N; Sym 2 (T N)) is pointwisely negative definite.
There is a natural identification between the tangent bundle T N = T (G/K) and the associated bundle G × K (g/k) where K acts on g/k via the adjoint action, which on each fiber T gK N is given by Thus X(N) = Γ(G; g/k) K . The negative associated vector field map is given by Φ : g → Γ(G; g/k) K , ξ → (g → −[Ad g −1 ξ]). The map Sym 2 g → Γ(N; Sym 2 (T N)) is then identified with the second symmetric power of Φ, namely the map Sym 2 g → Γ(G; Sym 2 (g/k)) K , The assumption on ω exactly says its image under this map is fiberwisely negative definite.
In Lie theory, the Casimir element of a Lie algebra g with respect to an Ad-invariant nondegenerate symmetric bilinear form is the image of the dual of this bilinear form in U(g). We see in the previous proof that our choice of Ω exactly arises from (the negative of) a Casimir element in U(g) when M is locally symmetric without euclidean factors. This clarifies the name "Casimir operator" in those special cases. Finally, in the locally symmetric case, it is not hard to see if we break M into irreducible pieces and use appropriate multiples of the Killing form in each piece in the construction, the operator Ω on SM will have horizontal symbol agreeing with the pullback of the symbol of the usual Laplacian ∆ M on the base M. In the next section we will see this more explicitly in a simpler case. The next section will not be relevant to our main goal, though.
Let N = G/K be a noneuclidean irreducible symmetric space where G is the isometry group of N. Then g is semisimple whose Killing form is nondegenerate and is negative definite on k. Let p be the orthogonal complement of k, which is Ad| K -invariant and satisfies [p, p] ⊂ k. We have an identification T N ∼ = G × K p as in the proof of Proposition 2.1. Thus, a G-invariant Riemannian metric on N is the same as an Ad| K -invariant inner product on p. In fact, let κ be an appropriate scalar times the Killing form on g, then κ| p gives back the Riemannian metric on N.
Another way to recover the Riemannian metric on N from κ is the following. Left translating κ on G equips G with a bi-invariant pseudo-Riemannian metric. Then the Riemannian metric on N = G/K is characterized by requiring the quotient map π 0 : G → N to be a pseudo-Riemannian submersion. Moreover, since the Lie exponential map on G agrees with the pseudo-Riemannian exponential map, we know π 0 has totally geodesic fibers, i.e. every fiber of π 0 is a totally geodesic submanifold of G.
It is a classical result that the negative Casimir element −c(g) ∈ U(g) with respect to κ acts as the Laplacian ∆ G on the pseudo-Riemannian manifold G, via the map U(g) → Diff * (G) defined as in the proof of Proposition 2.1. Since π 0 has totally geodesic fibers, classical results for (pseudo-)Riemannian submersion tell that ∆ G π * 0 = π * 0 ∆ N ( [BB82, (3.10)]). Therefore, −c(g) acts on N as the Laplacian ∆ N . Now assume furthermore N is isotropic. Then we can write SN = G/H where H is the isotopy group of a unit tangent vector [(e, X 0 )] ∈ eK × K S(p) ∼ = S eK N (S(p) is the unit sphere in p). Let s be the orthogonal complement of h in k, then like above, . Under this identification, the horizontal/vertical subbundles are (2.1) (To see the claim on the horizontal part, one has to note the connection on the principal K-bundle K ֒→ G → N given by the horizontal distribution p induces the Levi-Civita connection on N because it is metric-compatible and torsion-free ( [Kow07, Proposition I.13]).) Let g 1 be the Riemannian metric on SN induced from the one on N, namely the restriction of the Sasaki metric on T N. Let g 2 be the (pseudo-)Riemannian metric on SN given by κ| s⊕p , or alternatively the one such that the projection G → G/H = SN is a pseudo-Riemannian submersion. By the same arguments as before, in view of the construction in the proof of Proposition 2.1 we have the following.
Corollary 2.4. The negative Casimir element −c(g) acts on SN as the Laplacian with respect to the (pseudo-)Riemannian metric g 2 . In particular, a Casimir operator Ω as in Proposition 2.1 can be chosen to be the Laplacian on (SN, g 2 ).
However, from the Riemannian point of view the more natural metric to use is g 1 . Therefore we need to relate these two metrics.
In general g 1 , g 2 do not agree, but with respect to SN → N, they induce the same horizontal/vertical splitting (the one given by the Levi-Civita connection on N) and are equal in the horizontal part (both equal to the lift of the Riemannian metric on N). Therefore the difference of g 1 , g 2 is only in the vertical part. Since they are both G-invariant, each of them is determined by an Ad| H -invariant nondegenerate symmetric bilinear form on s. We have noted that g 2 is given by κ| s , which is positive/negative definite if N is of compact/noncompact type, respectively.

2)
Here X ⊥ is the orthogonal complement in T H (SN) of the line bundle spanned by the geodesic vector field X, and X ⊥ 0 is the κ-orthogonal complement in p of the unit vector X 0 .
Proof. At a point gH ∈ G/H ∼ = SN, a vertical tangent vector v given by the d dt | t=0 of a curve gk(t)H in gK/H for some curve k(t) ∈ K, t ∈ (−ǫ, ǫ), with k(0) = e and k ′ (0) ∈ s. Via T V gH (SN) ∼ = s we can write v = k ′ (0). On the other hand, write gH = [(g, X 0 )] = (gK, [L g, * X 0 ]) ∈ gK × K p ∼ = T gK N, then the tautological map is given by Corollary 2.6. The vertical part of the Riemannian metric g 1 on SN is given by ι * κ| X ⊥ 0 on κ| s where ι is defined as in (2.2).
We can also translate the Lie-theoretic Corollary 2.6 entirely into the language of Riemannian geometry.
Lemma 2.7. At a point (x, v) ∈ SN, the vertical part of g 2 (as a bilinear form on T V (x,v) (SN)) is given by the inverse of R x (·, v, v, ·) with respect to the vertical part of g 1 . Here R denotes the Riemannian curvature tensor of N.
Proof. Write (x, v) = gH ∈ G/H ∼ = SN, g ∈ G, and use this to make the identification . On the other hand, the Riemannian curvature tensor is given by R Comparing the three expressions gives the desired statement.
Note that the discussions above carry to the local case. From the Lie point of view, g 1 , g 2 , Ω are G-invariant, thus local constructions of Ω on a locally isotropic space glue to a global Casimir operator Ω (since transition functions take values in G).  Proof. If K = 0, the statement is clear from our construction in Corollary 2.3. If K = 0, Ω = ∆ ′ , where (·) ′ means operators on SM with respect to the metric g 2 instead of g 1 . Since SM → M is a Riemannian submersion with totally geodesic fibers for both g 1 and g 2 on SM, we know ∆ ′ H = ∆ H (see e.g. [BB82, (1.9)]). On the other hand, Proposition 2.7 tells us the vertical part of g 2 is K −1 times that of g 1 , thus ∆ ′ V = K∆ V . The statement follows.

Convergence of spectrum
In this section we prove the following slightly refined (c.f. Corollary 2.3) version of Theorem 1.
Theorem 2. Suppose M = Γ\G/K is a locally Riemannian homogeneous space and assume there exists a Γ-invariant element in Sym 2 g whose image in Sym 2 (g/Ad g h) is define for every g ∈ G, then the conclusion of Theorem 1 holds.
The strategy is to decompose L 2 (SM) into eigenspaces of a Casimir operator constructed in Proposition 2.1 and then use the Schur complement formula on each eigenspace. For readers' convenience we give a brief review of the Schur complement formula.
3.1. Schur Complement formula. Schur's complement formula is a convenient tool to analyze the degeneracy of an operator.
are bounded operators on Banach spaces, then P is invertible if and only if E −+ is invertible.
Moreover, in such case we have Proof. The proof is direct. If E −+ is invertible, then from We conclude that P is invertible and The proof for the other direction is similar. This lemma will be used for the R +− = 0 case, which we call a Grushin problem for P .
3.2. Decomposition of L 2 (SM). Fix a Casimir operator Ω on SM constructed in Proposition 2.1, and as before let ∆ V be the vertical Laplacian on SM. Since Ω, ∆ V and the total Laplacian ∆ on SM commute with each other, we can do spectral decomposition on each eigenspace of ∆. Thus we get the following orthogonal decomposition: where V η = {u ∈ L 2 (SM) : Ωu = ηu} is the η-eigenspace of Ω and V η,k = {u ∈ V η : ∆ V u = λ k u} is the k-th eigenspace of ∆ V in V η . Here λ k = k(k + n − 2), k = 0, 1, 2, · · · are the eigenvalues of the spherical Laplacian. For sufficiently large constant C > 0, Ω + C∆ V is elliptic thanks to Proposition 2.1, so each V η,k is finite dimensional and consists of smooth functions.
We will do analysis for P γ = −γX + c n γ 2 ∆ V (where c n = 1/n(n − 1)) on each eigenspace V η . The advantage is that P γ = P γ + (Ω − η)/C : V η ∩ H 2 → V η is an elliptic operator. Let Π η,k : V η → V η,k denote the orthogonal projection with the abbreviated notation Lemma 3.2. The geodesic vector field X satisfies the following properties.
• X is anti-self-adjoint with respect to the natural L 2 (SM) norm defined via the metric; • X sends V η,k into V η,k+1 ⊕ V η,k−1 with the convention that V η,−1 = 0; Proof.
• We identify SM with the unit cotangent bundle S * M via the metric on M. The geodesic vector field on S * M is the Reeb vector field with respect to the natural contact 1-form α. Therefore X preserves both α and dα, thus preserves the volume form on S * M. It follows that X is anti-self-adjoint.
• We compute in local coordinates. Suppose we are in normal coordinates {x i } at p ∈ M, so that g ij (p) = δ ij and ∂ k g ij (p) = 0. Then at p, X = v j ∂ x j where v j 's are the induced coordinates on T M, which restrict to linear functionals on each fiber. The claim follows from the fact that multiplying spherical harmonics of degree k by linear functionals gives a combination of spherical harmonics in degree k − 1 and k + 1. • Again we compute in normal coordinates at p ∈ M. Then X 2 = v i v j ∂ x i ∂ x j . ΠX 2 Π corresponds to projecting v i v j to 0-th spherical harmonics. If i = j, then the projection is zero. If i = j, then the projection is given by 3.3. Invertibility lemma. We first prove an invertibility lemma which will be useful in our analysis later. We will use the semiclassical notation h = γ −1 andP h = c n ∆ V − hX. Note that P γ = γ 2P h .
Lemma 3.3. For |λ| ≤ C 0 , 2C 0 h 2 ≤ c n , the operator and hence we know Q 0 is injective, and the image is closed by ellipticity.
Now suppose Q 0 is not surjective, then there exists a nonzero v ∈ Π ⊥ V η such that By ellipticity, v ∈ C ∞ and this is a contradiction to injectivity.
3.4. Spectral convergence. In this section we prove the convergence of the spectrum in Theorem 2.
Let i 0 : V η,0 → V η be the inclusion. We will consider the following Grushin problem for P γ − λ.
We would like to use Lemma 3.1, so let us check the operator is invertible. In other words, we want to solve the equations Thanks to Lemma 3.3, we can explicitly solve the equation. From the second equation of (3.5) one knows the Π part of u. The Π ⊥ part of the first equation of (3.5) then gives the Π ⊥ part of u. Plugging the solution u back gives u − . Hence the unique solution of (3.5) is Now we can apply Lemma 3.1 and it follows that λ ∈ σ(P γ | Vη ) if and only if E −+ = γ −2 (λ + ΠX(Π ⊥ (P h − h 2 λ)Π ⊥ ) −1 XΠ) is not invertible. Let us analyze the asymptotic behavior of E −+ .
Note L 2 (SM) = V η and L 2 (M) = V η,0 , we would like to conclude that σ(P γ ) → σ(∆ M ). However, it is unclear a priori whether the convergence is uniform in η. We now prove the stronger convergence in Theorem 2 using contradiction. It suffices to prove the following proposition.