Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature

The kinetic Brownian motion on the sphere bundle of a Riemannian manifold $M$ is a stochastic process that models a random perturbation of the geodesic flow. If $M$ is a orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $L^2$-spectrum of the infinitesimal generator of a time rescaled version of the process converges to the Laplace spectrum of the base manifold.


Introduction
Kinetic Brownian motion is a stochastic process that describes a stochastic perturbation of the geodesic flow and has the property that the perturbation affects only the direction of the velocity but preserves its absolute value. It has been studied in the past years by several authors in pure mathematics [FL07,ABT15,Li16,Dro17,BT18] but versions of this diffusion process have been developed independently as surrogate models for certain textile production processes (see e.g. [GKMW07,GS13,KSW13]).
Kinetic Brownian motion (Y γ t ) t≥0 in the setting of a compact Riemannian manifold (M, g) can be informally described in the following way: (Y γ t ) t≥0 is a stochastic process with continuous paths described by a stochastic perturbation of the geodesic flow on the sphere bundle SM = {ξ ∈ T M, ξ g = 1}. More precisely, if we denote the geodesic flow vector field by X and the (non-negative) Laplace operator on the fibers of SM by ∆ S , then the kinetic Brownian motion is generated by the differential operator −X + 1 2 γ∆ S : L 2 (SM) → L 2 (SM).
The connection to the stochastic process (Y γ t ) t≥0 is given via with f ∈ L 2 (SM), x ∈ SM. Observe that the parameter γ > 0 controls the strength of the stochastic perturbation and it is a natural question to study the behavior of −X + 1 2 γ∆ S and Y γ t in the regimes γ → 0 as well as γ → ∞. Drouot [Dro17] has studied the convergence of the discrete spectrum of −X + 1 2 γ∆ S in the limit γ → 0 for negatively curved manifolds and has shown that it converges to the Pollicott-Ruelle resonances of the geodesic flow. These resonances are a replacement of the spectrum of X since its L 2 -spectrum is equal to iR and they can be defined in various generalities of hyperbolic flows as pole of the meromorphically continued resolvent [Liv04,FS11,DZ16,DG16,DR16,BW17]. A more general framework of semiclassical subelliptic operators that includes the kinetic Brownian motion for γ → 0 has been established by Smith [Smi18]. In the limit of large random noise Li [Li16] and Angst-Bailleul-Tardif [ABT15] proved that π(Y γ γt ) converges weakly to the Brownian motion on M with speed 2 as γ → ∞ where π : SM → M is the projection. This rescaled kinetic Brownian motion is generated by P γ = −γX + 1 2 γ 2 ∆ S whereas the Brownian motion on the base manifold is generated by the Laplace operator 1 2 ∆ M . Therefore, 1 one may conjecture that the discrete spectrum of P γ converges to the Laplace spectrum. We will give a proof of this fact in the case of constant curvature surfaces: Theorem 1. Let (M, g) be an orientable compact surface of constant curvature. For every η ∈ σ(∆ M ) with multiplicity n there is an analytic function λ η : ]r η , ∞[→ C such that λ η (γ) is an eigenvalue of P γ with multiplicity at least n and λ η (γ) → η as γ → ∞.
Note that this theorem does not imply that in a compact set all eigenvalues of P γ are close to eigenvalues of the Laplacian (see Remark 3.2 for a discussion of the problems that prevent us from proving this stronger statement).
Another question to ask is whether the kinetic Brownian motion converges to equilibrium, i.e.
We should point out that the given rate C γ converges to 0 as γ → ∞ but they conjecture that the optimal rate converges to the spectral gap of ∆ M which is the smallest non-zero Laplace eigenvalue η 1 (see [BT18, Section 3.1]). A direct consequence of Theorem 1 shows that the optimal rate C γ is less than Re λ η1 (γ) for surfaces of constant curvature. Hence lim sup γ→∞ C γ ≤ η 1 . For a more explicit study of the convergence towards equilibrium we prove a spectral expansion and explicit error estimates in the case of constant negative curvature in [KWW19].
Note that a problem related to the kinetic Brownian motion in SM is the study of the hypoelliptic Laplacian on T M introduced by Bismut [Bis05]. Like the kinetic Brownian motion the hypoelliptic Laplacian interpolates between the geodesic flow and the Brownian motion. In [BL08, Chapter 17] Bismut and Lebeau prove the convergence of the spectrum of the hypoelliptic Laplacian to the spectrum of the Laplacian on M using semiclassical analysis. It seems plausible that their techniques can also be transferred to the setting of kinetic Brownian motion and might give the spectral convergence without any curvature restriction. The purpose of this article is however not to attack this general setting but show that under the assumption of constant curvature allows to drastically reduce the analytical difficulties. In fact we are able to reduce the problem to standard perturbation theory. This is also the reason why we are able to obtain the explicit error estimates in [KWW19].
Let us give a short outline of the proof of Theorem 1: By the assumption of constant curvature we have a three-dimensional Lie algebra g = X, X ⊥ , V C of vector field on SM. Denoting the Gaussian curvature by K, the operator Ω = −X 2 − X 2 ⊥ − KV 2 commutes with g and P γ and has discrete spectrum. Hence, we can decompose the corresponding L 2 (SM) into eigenspaces of Ω. The generator P γ preserves this decomposition of L 2 (SM) and we can study the restriction of P γ on each occurring eigenspace separately. In each of these eigenspaces the spectral asymptotics of P γ can then be handled by standard perturbation theory of an operator family of type (A) in the sense of Kato. For the calculations it will be important that each eigenspace of Ω can be further split into the eigenspaces of the vector field V which correspond to the Fourier modes in the fibers of SM → M.
The article is organized as follows: We will give a short overview over the kinetic Brownian motion and the connection between constant curvature surfaces and the global analysis of sphere bundles of constant curvature surfaces in Sections 2.1 and 2.2. After that we will recall a few results of perturbation theory for unbounded linear operators (Section 2.3) which are mostly taken from [Kat76]. In the limit γ → ∞ one would like to consider the geodesic vector field as a perturbation of the spherical Laplacian. The major difficulty is that 1 γ X is no small perturbation in comparison with ∆ S . After the spectral decomposition with respect to Ω there is a precise way to consider X as small operator in any eigenspace of Ω. Afterwards we will give the proof of the convergence of the spectra (Section 3).

Preliminaries
We introduce the spherical Laplacian ∆ S as follows: for every x ∈ M the tangent space T x M is a Euclidean vector space via the Riemannian metric and S We now obtain the spherical Laplace operator ∆ S by where π : SM → M is the projection and g is the Riemannian metric on M. Then θ is a 1-form on SM and ν = θ ∧ (dθ) d−1 defines the Liouville measure on SM which is invariant under the geodesic flow φ t . The vector field X = d dt t=0 φ * t is called the geodesic vector field. Let us consider the operator P γ = −γX + 1 2 γ 2 ∆ S with domain dom(P γ ) = {u ∈ L 2 (SM) | P γ u ∈ L 2 (SM)} for γ > 0. Note that the action of P γ has to be interpreted in the sense of distributions. We first want to collect some properties of P γ .
Proposition 2.1. P γ is a hypoelliptic operator with Re P γ f, f ≥ 0) and coincides with the closure of P γ | C ∞ . Therefore, P γ has compact resolvent on L 2 (SM), discrete spectrum with eigenspaces of finite dimension, and the spectrum is contained in the right half plane. P γ generates a positive strongly continuous contraction semigroup e −tPγ .
Proof. See Appendix.
2.2. Surfaces of Constant Curvature. Let M be a orientable compact Riemannian manifold of dimension 2 and constant curvature and let K be the Gaussian curvature. Since M has finitely many connected components, let us assume without loss of generality that M is connected. We follow the notation of [PSU14]. Let X be the geodesic vector field on SM , and let V be the vertical vector field so that ∆ S = −V 2 . We define X ⊥ = [X, V ]. We then have the commutator relations X = [V, X ⊥ ] and [X, X ⊥ ] = −KV . In particular, g := CX ⊕ CX ⊥ ⊕ CV is a Lie algebra. The Casimir operator Ω is defined as Ω = −X 2 − X 2 ⊥ − KV 2 and it is routine to check that [Ω, X] = [Ω, X ⊥ ] = [Ω, V ] = 0 using the above commutator relations. The Laplace operator ∆ SM of SM for the metric which is declared by the requirement that the frame {X, X ⊥ , V } is an orthonormal basis (i.e. is an elliptic operator on the compact manifold SM and hence it as discrete spectrum with eigenvalues of finite multiplicity. Since ∆ SM is non-negative, each eigenvalue is non-negative and the eigenspaces are orthogonal. Both operators Ω and V leave these eigenspaces invariant. Thus we have the following decomposition: Remark 2.2. Note that g is isomorphic to the complexification of so(3), R 2 ⋊ so(2), sl 2 (R) if K > 1, K = 0, K < 0 respectively. We are essentially decomposing the representation of g on L 2 (SM) into irreducible ones. In fact in all three cases SM can be written as Γ\G for some torsion free, discrete, cocompact subgroup Γ The decomposition L 2 (SM) = V η can be seen as a Plancherel decomposition of this space and V η = V η,k as the decomposition into K-types or weights respectively. In all three cases the irreducible representations have explicit realizations on certain L 2 -spaces (see e.g. [Tay86, Ch. 8] for sl 2 ) and one could go on by analyzing those but they do not contain more information than the abstract decomposition we provided here for all three cases at once. We would like to note that this harmonic analysis point of view was our original approach motivated by previous works that used similar techniques for geodesic flows [FF03, DFG15, GHW18a, GHW18b, KW19].
Hence, X ± : V η,k → V η,k±1 . Moreover, X * ± = −X ∓ and Ω = −2X The next lemma is crucial for our main result as it connects the spectral values η of Ω to the spectrum of the Laplace operator of the base manifold M.
Pick f ∈ V η,0 and define H f as closure of the g-invariant subspace generated by f . Since X ± X ∓ are scalar on V η,k we see that H f = span{X k + f, X k − f | k ∈ N 0 } and it follows that H f ∩ V η,k is at most one-dimensional. Moreover, H f for different f ∈ V η,0 are isomorphic as g-spaces and we claim H g ⊥ H f for g ⊥ f . Since V η,k are orthogonal for different k we only have to verify (H f ∩ V η,k ) ⊥ (H g ∩ V η,k ). But as this subspace is given by CX |k| sign k f and CX |k| sign k g respectively we need to show X l ± f, X ± lg >= 0 for l ≥ 0. As X l ± f, X l ± g >= (−1) l X l ∓ X l ± f, g and X ± X ∓ is scalar on V η,k the claim follows.

Perturbation Theory.
We want to collect some basic results from perturbation theory for linear operators that can be found in [Kat76]. First, we introduce families of operators we want to deal with. Without loss of generality let us assume that 0 is contained in the domain D. We call T = T (0) the unperturbed operator and A(x) = T (x) − T the perturbation. Furthermore, let R(ζ, x) = (T (x) − ζ) −1 be the resolvent of T (x) and R(ζ) = R(ζ, 0). If ζ / ∈ σ(T ) and 1 + A(x)R(ζ) is invertible then ζ / ∈ σ(T (x)) and the following identity holds: Let us assume that σ(T ) splits into two parts by a closed simple C 1 -curve Γ. Then there is r > 0 such that R(ζ, x) exists for ζ ∈ Γ and |x| < r (see [Kat76,Ch. VII Thm. 1.7]). If the perturbation is linear (i.e. T (x) = T + xA) then a possible choice for r is given by min ζ∈Γ AR(ζ) −1 . Note that AR(ζ) is automatically bounded by the closed graph theorem. In particular, we obtain that Γ ⊆ C \ σ(T (x)) for |x| < r, i.e. the spectrum of T (x) still splits into two parts by Γ. Let us define σ int (x) as the part of σ(T (x)) lying inside Γ and σ ext (x) = σ(T (x)) \ σ int (x). The decomposition of the spectrum gives a T (x)-invariant decomposition of the space X = M int (x) ⊕ M ext (x) where M int (x) = P (x)X and M ext (x) = ker P (x) with the bounded-holomorphic projection To get rid of the dependence of x in the space M int (x) we will use the following proposition. Denoting since U (x) is an isomorphism. Here we denote the interior of Γ by int(Γ).
Let us from now on suppose that Γ encloses an eigenvalue µ of T with finite multiplicity and no other eigenvalues of T . Then σ int (0) = {µ} and M int (0) is finite dimensional. Hence, T (x)| Mint(0) is a holomorphic family of operators on a finite dimensional vector space. It follows that the eigenvalues of T (x) are continuous as a function in x. In addition to the previous assumptions, let us suppose that the eigenvalue µ is simple. Then M int (0) is one-dimensional and T (x)| Mint(0) is a scalar operator. We obtain that there is a holomorphic function µ : B r → C (with r = min ζ∈Γ AR(ζ) −1 as above) such that µ(x) is an eigenvalue of T (x), µ(x) is inside Γ and µ(x) is the only part of σ(T (x)) inside Γ since σ int (x) = σ( T (x)| Mint(0) ).
We now want to calculate the Taylor coefficients of µ(x) in order to get an approximation of µ(x) in the case where X = H is a Hilbert space and T (x) is a holomorphic family of type (A) with symmetric T but not necessarily symmetric T (x) for x = 0. To this end let ϕ(x) be a normalized holomorphic family of eigenvectors (obtained from P (x)). Consider the Taylor series µ(x) = x n µ (n) , ϕ(x) = x n ϕ (n) and T (x)u = x n T (n) u for every u ∈ dom(T ) which converges on a disc of positive radius independent of u. This is due to the fact that Taylor series of holomorphic functions converge on every disc that is contained in the domain.
We compare the Taylor coefficients in and A fortiori, where ϕ (1) fulfils For ϕ (1) being uniquely determined we can use the additional assumption that ϕ(x) is normalized. However µ (2) can be calculated without this consideration in our setting. Here Therefore, µ (2) depends only on v and not on c.

Perturbation Theory of the Kinetic Brownian Motion
We want to establish the limit γ → ∞ of the spectrum of P γ . To do so we write P γ = where T (x) = ∆ S + xX and we want to use the methods established in Chapter 2.3.
In order to have finite dimensional eigenspaces and holomorphic families of type (A) we will use the orthogonal eigenspace decomposition of L 2 (SM) derived in Section 2.2: Proposition 3.1. The family of operators T (x), x ∈ C, restricted to V η defines a holomorphic family of type (A) with domain H 2 (SM) ∩ V η . The same is true for V ′ η .
Proof. Since ∆ SM is a second-order elliptic differential operator, we have H 2 (SM) = {u ∈ L 2 (SM) | (∆ SM +xX)u ∈ L 2 (SM)} for each x ∈ C. The space V η is invariant under ∆ SM so that for K = 1. For K = 1 the same argument works if we replace ∆ SM by ∆ SM + ∆ S . Since T (x)| Vη is closed as a restriction of a closed operator the proposition is proven. The proof for V ′ η is identically.
We denote the restriction of T (x) to V η by T η (x).
The eigenspaces of the unperturbed operator ∆ S | Vη are V η,0 and V η,k ⊕ V η,−k which are finite dimensional. As we have seen in Section 2.3 the eigenvalues of a holomorphic family of type (A) are continuous as a function of x in this case. We deduce that for the eigenvalues µ(x) of T η (x) that arise from non-zero eigenvalues µ = µ(0) of ∆ S | Vη the limit γ → ∞ of γ 2 2 µ(2γ −1 ), which is an eigenvalue of P γ , is ∞. Therefore, we are only interested in eigenvalues of T η (x) which arise from the unperturbed eigenvalue 0. In order to have that 0 is an eigenvalue of T η (0) we must have V η,0 = 0, i.e. η ∈ σ(∆ M ) by Lemma 2.3. Let us first deal with η = 0. Here g acts trivially on V 0,0 by Lemma 2.3 and therefore the eigenvalue of T 0 (x) that arises from the eigenvalue 0 is 0. If η > 0 we restrict T η (x) to V ′ η which defines a holomorphic family of type (A) as well. In this case the eigenspace of the unperturbed eigenvalue is V ′ η,0 which is one-dimensional. Hence, we are in the precise setting of Section 2.3. We obtain that there is a holomorphic function µ defined on a neighbourhood of 0 (depending on η) such that µ(x) is an eigenvalue of T (x)| V ′ η with µ(0) = 0.
Remark 3.2. In order to obtain uniform convergence of the eigenvalues in compact sets we would like to deal with all η ∈ σ(∆ M ) simultaneously in a uniform way. More precisely, we want to separate converging eigenvalues (which arise from 0) from non-converging eigenvalues. For this to happen we must have that 1 + xX(∆ S − ζ) −1 on V ′ η is invertible for small |x| and for ζ in some closed curve enclosing 0 but no other element of σ(∆ S ) = {k 2 | k ∈ Z}. In particular, 1 + xX(∆ S − ζ) −1 has to be invertible for some ζ ∈ (0, 1) but we can only ensure this for this is impossible for all η ∈ σ(∆ M ) at once.
Appendix A. Proof of Proposition 2.1 The proof that P γ is hypoelliptic with the subelliptic estimate can be found in [Dro17, Chapter 2.2]. There exist vector fields X j on SM such that ∆ S = − d j=1 X 2 j and div X j = 0 (see [Dro17, §2.2.6]). Hence, the X j as well as X are skew-symmetric with respect to the inner product of L 2 (SM). It follows that Re P γ f, f = 1 2 γ 2 X j f, X j f − γ Re Xf, f ≥ 0, i.e. P γ | C ∞ is accretive since Xf, f ∈ iR.
For the positivity of the generated contraction semigroup we have to check if (sign f )P γ f, u ≥ |f |, (P γ ) * u for all real f ∈ C ∞ and a strictly positive subeigenvector u of (P γ ) * (see [AGG + 86, C-II Cor. 3.9]). Note that 1 is a strictly positive eigenvector of (P γ ) * and 1 2 ∆ S (x) as well as −X generate stochastic Feller processes on S x M and SM respectively (namely the Brownian motion on S x M and the geodesic flow). Hence, e −t∆ S (x) and e tX define positive semigroups so that (sign f )∆ S (x)f, 1 SxM ≥ 0 for f ∈ C ∞ (S x M) and (sign f )(−X)f, 1 ≥ 0 for f ∈ C ∞ (SM) (see [AGG + 86, C-II Thm.2.4]). Combining both statements completes the proof.