$L^2$ Properties of Lévy Generators on Compact Riemannian Manifolds

Abstract

We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an ${\mathbb {R}}^d$-valued Lévy process through rolling without slipping. We prove that the Feller semigroups associated with these processes extend to strongly continuous contraction semigroups on $L^p$, for $1\le p<\infty$. When $p=2$, we show that these semigroups are self-adjoint. If, in addition, the motion has a non-trivial Brownian part, we prove that the generator has a discrete spectrum of eigenvalues and that the semigroup is trace-class.

1 Introduction

The investigation of Lévy processes (essentially processes with stationary and independent increments) has become a rich theme in modern probability theory. These processes have a deep and interesting structure, there are wide-ranging applications, and they have been found to be a useful class of driving noises for stochastic differential equations. Usually Lévy processes are considered as having Euclidean space (or even the real line) as their state space, although Banach and Hilbert space valued processes are important for stochastic partial differential equations (see e.g. [22]). The most intensively studied class of nonlinear state spaces for Lévy processes has been Lie groups. Here, the notion of increment is defined using the group law and inverse operation, so if $0 \le s < t$, where the increment of the process X in Euclidean space is $X(t) - X(s)$, on a Lie group we instead consider $X(s)^{-1}X(t)$. For a dedicated monograph treatment of such processes, see Liao [18].

Having considered Lévy processes on Lie groups, the next obvious step is to move to a Riemannian manifold. Indeed, arguably the most important Lévy process in Euclidean space is Brownian motion, and Brownian motion on manifolds is a very well developed theory in its own right (see e.g. Elworthy [10] and Hsu [15]). As its generator is the Laplace–Beltrami operator and transition density is the heat kernel, it is clear that this process is a natural expression of the geometry of the manifold. For development of a more general theory of Lévy processes, we are hampered by the fact that there is no obvious notion of increment on a manifold. When the space is a symmetric space, there is an alternative point of view pioneered by Gangolli [11, 12], who used spherical functions to establish an analogue of the Lévy–Khintchine formula. Using the fact that every symmetric space is a homogeneous space G/K of a Lie group G, quotiented by a compact subgroup K, Applebaum [1] was able to realise these processes as images of Lévy processes on G through the natural map (see [19] for more recent developments).

Motivated in part by the results of [1], and also the Eels–Elworthy construction of Brownian motion on a manifold by projection from the frame bundle, Applebaum and Estrade ( [4]) constructed a process that they called a “Lévy process on a manifold” and proved that it was a Feller–Markov process. The generic such process has the structure of a Brownian motion that is interlaced with jumps along geodesics, which are controlled by an isotropic Lévy measure. It is now more than twenty years since this paper was published, and to the authors’ knowledge, there has been no further work carried out on such processes since that time, other than the ergodic theoretic analysis in Mohari [21]. In the current paper, we hope to start the process of resurrecting this neglected area.

Since every Lévy process on a manifold is a Feller process, there is an associated semigroup on the space of continuous functions vanishing at infinity, and the generator is the sum of the Laplace–Beltrami operator and an integral superposition of jumps along geodesics. Our first new result is to show that the semigroup also preserves the $L^p$ space of the Riemannian volume measure. We are particularly interested in the case $p=2$, and here, we show that the semigroup (and hence its generator) is self-adjoint. When there is a non-trivial Brownian motion component to the process, we find that the generator has a discrete spectrum of non-positive eigenvalues. Under the latter condition, we also prove that the semigroup is trace-class, and so the process has a transition density/heat kernel.

2 Lévy Processes on Manifolds

Let M be a compact, connected, d-dimensional Riemannian manifold of dimension $d\ge 1$, with Riemannian measure $\mu$. Let OM denote the corresponding bundle of orthonormal frames, a principle O(d)-bundle over M. The Levi-Civita connection induces a decomposition

$$\begin{aligned} T_rOM\cong H_rOM\oplus V_rOM, \forall r\in OM \end{aligned}$$

of each tangent space of OM into vertical and horizontal subspaces, and, writing $\pi :OM\rightarrow M$ for the projection map, we have that for each $p\in M$ and $r\in OM_p$, $d\pi _r$ vanishes on $V_rOM$ and defines a linear isomorphism $H_rOM\cong T_pM$, for each $p\in M$ and $r\in OM_p$. Moreover, OM may be given the structure of a Riemannian manifold in such a way that $d\pi _r:H_rOM\rightarrow T_pM$ is an isometric isomorphism.¹ The corresponding Riemannian measure ${\tilde{\mu }}$ is sometimes referred as Liouville measure. Observe that

$$\begin{aligned} \mu ={{\tilde{\mu }}}\circ \pi ^{-1}. \end{aligned}$$

(2.1)

One can easily show that OM is compact, using the fact that both its base manifold and structure group are compact. As such, both $\mu$ and ${{\tilde{\mu }}}$ are finite measures.

The canonical horizontal fields consist of a family of vector fields $\{H_x:x\in {\mathbb {R}}^d\}$ on OM, defined by

$$\begin{aligned} H_x(r)=r(x)^*, \forall r\in OM,x\in {\mathbb {R}}^d, \end{aligned}$$

where $^*$ denotes horizontal lift. Since OM is compact, these vector fields are complete; we write ${{\,\mathrm{Exp}\,}}(tH_x)$ for the associated flows of diffeomorphisms. These flows are related to the Riemann exponential map by

$$\begin{aligned} \pi ({{\,\mathrm{Exp}\,}}(H_x)(r))=\exp _pr(x), \forall x\in {\mathbb {R}}^d, \; p\in M, \; r\in OM_p. \end{aligned}$$

(2.2)

For standard basis vectors $e_i$ of ${\mathbb {R}}^d$, we use the abbreviation $H_{e_i}=H_i$. The horizontal Laplacian

$$\begin{aligned} \Delta _H = \sum _{i=1}^dH_i^2, \end{aligned}$$

on OM is related to the Laplace–Beltrami operator $\Delta$ on M by

$$\begin{aligned} \Delta f(p)=\Delta _H(f\circ \pi )(r), \forall f\in C^\infty (M), \; p\in M, \; r\in OM_p; \end{aligned}$$

for more details see Hsu [15] Chapter 3.

Let Y be an ${\mathbb {R}}^d$-valued Lévy process. It is shown in Applebaum and Estrade [4] that the Marcus canonical SDE on OM

$$\begin{aligned} dR(t)= \sum _{j=1}^{d} H_j(R(t-))\diamond dY_j(t), \; t\ge 0; R(0)=r \text { (a.s.)}, \end{aligned}$$

(2.3)

has a unique, càdlàg solution R that is a Feller process. As in [4], we will call processes obtained this way horizontal Lévy processes. We also impose the assumption from [4] that the Lévy process Y is isotropic, in that its law is O(d)-invariant. By Corollary 2.4.22 on page 128 of [2], the Lévy characteristics of Y then take the form $(0,aI,\nu )$, where $a\ge 0$, and $\nu$ is O(d)-invariant. Then, by Theorem 3.1 of [4], the process $X=\pi (R)$ obtained by projection onto the base manifold is also a Feller process. The infinitesimal generators of R and X are

$$\begin{aligned} {\mathscr {L}}=\frac{1}{2}a\Delta _H+{\mathscr {L}}_J \end{aligned}$$

(2.4)

and

$$\begin{aligned} {\mathscr {A}} = \frac{1}{2}a\Delta + {\mathscr {A}}_J, \end{aligned}$$

(2.5)

respectively, where the jump parts ${\mathscr {L}}_J$ and ${\mathscr {A}}_J$ are given by

$$\begin{aligned} {\mathscr {L}}_Jf(r)= & {} \int _{{\mathbb {R}}^d\backslash \{0\}}\left\{ f({{\,\mathrm{Exp}\,}}(H_x)(r))-f(r)-{\varvec{1}}_{|x|<1}H_xf(r)\right\} \nu (\mathrm{d}x), \nonumber \\&\quad \forall f\in C^\infty (OM), \; r\in OM \end{aligned}$$

(2.6)

and

$$\begin{aligned} {\mathscr {A}}_Jf(p)= & {} \int _{T_pM\backslash \{0\}}\left\{ f(\exp _py)-f(p)-{\varvec{1}}_{|y|<1}yf(p)\right\} \nu _p(\mathrm{d}x),\\&\quad \forall f\in C^\infty (M), \; p\in M. \end{aligned}$$

Here, the family of Lévy measures $\{\nu _p:p\in M\}$ act on each tangent space $T_pM$ and are defined by $\nu _p=\nu \circ r^{-1}$ for any frame $r\in OM_p$. The two generators also satisfy

$$\begin{aligned} {\mathscr {A}}f(p)={\mathscr {L}}(f\circ \pi )(r), \forall f\in C^\infty (M), \;p\in M, \;r\in OM_p. \end{aligned}$$

Observe that since $\nu$ is O(d)-invariant, the right hand side of (2.6) is invariant under the change of variable $x\mapsto -x$, and hence for all $f\in C^\infty (OM)$ and $r\in OM$,

$$\begin{aligned} {\mathscr {L}}_Jf(r) = \int _{{\mathbb {R}}^d\backslash \{0\}}\left\{ f({{\,\mathrm{Exp}\,}}(H_{-x})(r))-f(r)+{\varvec{1}}_{|x|<1}H_xf(r)\right\} \nu (\mathrm{d}x), \end{aligned}$$

(2.7)

where we have used the fact that $H_{-x}f=-H_xf$. Summing (2.6) and (2.7) and dividing by two,

$$\begin{aligned}&{\mathscr {L}}_Jf(r)=\frac{1}{2}\int _{{\mathbb {R}}^d\backslash \{0\}}\left\{ f\big ({{\,\mathrm{Exp}\,}}(H_{x})(r)\big )-2f(r)+f({{\,\mathrm{Exp}\,}}(H_{-x})(r))\right\} \nu (\mathrm{d}x)\nonumber \\ \end{aligned}$$

(2.8)

for all $f\in C^\infty (OM)$ and $r\in OM$. It follows that

$$\begin{aligned} {\mathscr {A}}_Jf(p) = \frac{1}{2}\int _{T_pM\backslash \{0\}}\left[ f(\exp _py)-2f(p)+f(\exp _p(-y))\right] \nu _p(\mathrm{d}y) \end{aligned}$$

(2.9)

for each $f\in C^\infty (M)$ and $p\in M$. Note the analogous expression (5.4.16) in [3] for symmetric Lévy motion on a Lie group.

3 $L^p$ Properties of the Semigroup

We now turn our attention to the Feller semigroups associated with R and X. These consist of families of contraction operators $(S_t,t\ge 0)$ and $(T_t,t\ge 0)$ acting on the Banach spaces C(OM) and C(M) by

$$\begin{aligned} S_tf(r)={\mathbb {E}}\big (f(R(t))|R(0)=r\big ), \forall f\in C(OM),\; r\in OM,\; t\ge 0, \end{aligned}$$

and

$$\begin{aligned} T_tf(p)={\mathbb {E}}\big (f(X(t))|X(0)=p\big ), \forall f\in C(M),\; p\in M,\; t\ge 0. \end{aligned}$$

Since $X=\pi (R)$, it is immediate that

$$\begin{aligned} T_tf(p)=S_t(f\circ \pi )(r) \end{aligned}$$

(3.1)

for all $f\in C(M)$, $p=\pi (r)\in M$, and $t\ge 0$.

Any Riemannian manifold has a natural $L^p$ structure arising from its Riemannian measure, and so we may consider the spaces $L^p(OM){:}{=}L^p(OM,{\tilde{\mu }},{\mathbb {R}})$ and $L^p(M){:}{=}L^p(M,\mu ,{\mathbb {R}})$ for $1\le p\le \infty$. For $p<\infty$, we prove that $(S_t,t\ge 0)$ and $(T_t, t\ge 0)$ extend to strongly continuous contraction semigroups on these spaces, and that they are self-adjoint when $p=2$.

We begin by considering the semigroup $(S_t,t\ge 0)$ associated with the horizontal process R; analogous results for $(T_t,t\ge 0)$ are then obtained by projection down onto M.

Theorem 3.1

For all $1\le p<\infty$, $(S_t,t\ge 0)$ extends to a $C_0$-semigroup of contractions on $L^p(OM)$.

Proof

Let $q_t(\cdot ,\cdot )$ denote the transition measure of $(S_t,t\ge 0)$, so that

$$\begin{aligned} S_tf(r) = \int _{OM}f(u)q_t(r,\mathrm{d}u), \forall t\ge 0, \; f\in C(OM), \; r\in OM. \end{aligned}$$

The horizontal fields $H_x$ are divergence free for all $x\in {\mathbb {R}}^d$ (Proposition 4.1 of [20]), and so by Theorem 3.1 of [5], ${\tilde{\mu }}$ is invariant for $S_t$, in the sense that

$$\begin{aligned} \int _{OM}(S_tf)(r){\tilde{\mu }}(\mathrm{d}r) = \int _{OM}f(r){\tilde{\mu }}(\mathrm{d}r), \forall t\ge 0, \; f\in C(OM). \end{aligned}$$

Therefore, by Jensen’s inequality,

$$\begin{aligned} \Vert S_tf\Vert _p^{p}&=\int _{OM}\left| \int _{OM}f(u)q_t(r,\mathrm{d}u)\right| ^p{{\tilde{\mu }}}(\mathrm{d}r) \le \int _{OM}\int _{OM}|f(u)|^pq_t(r,\mathrm{d}u){{\tilde{\mu }}}(\mathrm{d}r) \\&= \int _{OM}(S_t|f|^p)(r){{\tilde{\mu }}}(\mathrm{d}r) = \int _{OM}|f(r)|^p{{\tilde{\mu }}}(\mathrm{d}r) = \Vert f\Vert _p^{p} \end{aligned}$$

for all $t\ge 0$ and $f\in C(OM)$. Each $S_t$ has domain C(OM), which is a dense subspace of $L^p(OM)$ for all $1\le p<\infty$. It follows that each $S_t$ extends to a unique contraction defined on the whole of $L^p(OM)$, which we also denote by $S_t$. By continuity, the semigroup property

$$\begin{aligned} S_tS_s=S_{t+s}, \forall s,t\ge 0 \end{aligned}$$

continues to hold in this larger domain. It remains to prove strong continuity, i.e. that

$$\begin{aligned} \lim _{t\rightarrow 0}\Vert S_tf-f\Vert _p = 0 \end{aligned}$$

(3.2)

for all $f\in L^p(OM)$. By density of C(OM) in $L^p(OM)$, it is sufficient to verify this for $f\in C(OM)$. The map $t\mapsto S_t$ is strongly continuous in C(OM), and so $\lim _{t\rightarrow 0}\Vert S_tf-f\Vert _\infty =0$ for all $f\in C(OM)$. Since $(OM,{\mathcal {B}}(OM),{{\tilde{\mu }}})$ is a finite measure space,

$$\begin{aligned} \Vert S_tf-f\Vert _p\le {{\tilde{\mu }}}(OM)^\frac{1}{p}\Vert S_tf-f\Vert _\infty , \end{aligned}$$

for all $f\in C(OM)$. Equation (3.2) now follows. $\square$

Remark 3.2

The final part of the above proof applies more generally, in that if X is a compact space equipped with a finite measure m, and if $(P_t,t\ge 0)$ is a Feller semigroup on X, then $(P_t,t\ge 0)$ is strongly continuous in $L^p(X, m)$.

Projection down onto the base manifold yields the following.

Theorem 3.3

For all $1\le p<\infty$, $(T_t, t\ge 0)$ extends to a strongly continuous semigroup of contractions on $L^p(M)$.

Proof

Let $1\le p<\infty$. By Eqs. (2.1) and (3.1), many of the conditions we must check follow from their analogues on the frame bundle. Indeed, for all $f\in C^\infty (M)$ and $t\ge 0$, we have

$$\begin{aligned} \Vert T_tf\Vert _{L^p(M)}^p = \int _M|T_tf(p)|^p\mathrm{d}\mu = \int _{OM}|S_t(f\circ \pi )(r)|^p\mathrm{d}{{\tilde{\mu }}} = \Vert S_t(f\circ \pi )\Vert _{L^p(OM)}^p, \end{aligned}$$

and so, using the fact that $S_t$ is a contraction of $L^p(OM)$,

$$\begin{aligned} \Vert T_tf\Vert _{L^p(M)}=\Vert S_t(f\circ \pi )\Vert _{L^p(OM)}\le \Vert f\circ \pi \Vert _{L^p(OM)}=\Vert f\Vert _{L^p(M)}. \end{aligned}$$

Hence, $T_t$ extends to a contraction of $L^p(M)$ for all $t\ge 0$. It is clear by continuity that the semigroup property continues to hold on this larger domain, as does Eq. (3.1). Strong continuity follows by Remark 3.2, or alternatively can be seen by the observation

$$\begin{aligned} \Vert T_tf-f\Vert _{L^p(M)} = \Vert S_t(f\circ \pi )-f\circ \pi \Vert _{L^p(OM)} \forall f\in L^p(M). \end{aligned}$$

Thus, $(T_t, t\ge 0)$ extends to a contraction semigroup on $L^p(M)$ for all $1\le p<\infty$. $\square$

We continue to denote the generators of $(S_t,t\ge 0)$ and $(T_t,t\ge 0)$ by ${\mathscr {L}}$ and ${\mathscr {A}}$, respectively. Note that by Lemma 6.1.14 of [9], ${\mathscr {L}}$ and ${\mathscr {A}}$ are both closed operators on $L^p(OM)$.

4 The Case $p=2$

For the remainder of this paper, we focus on the case $p=2$. Our aim in this section is to prove that the semigroups $(S_t,t\ge 0)$ and $(T_t,t\ge 0)$ are self-adjoint semigroups on $L^2(OM)$ and $L^2(M)$, respectively. By a standard result from semigroup theory, it will follow that ${\mathscr {L}}$ and ${\mathscr {A}}$ are self-adjoint linear operators.

Let us first impose the assumption that the Lévy measure $\nu$ is finite. In this case, ${\mathscr {A}}_J$ is the generator of a compound Poisson process on M (see [4]).

Lemma 4.1

If $\nu$ is finite, then ${\mathscr {L}}$ is a self-adjoint operator on $L^2(OM)$.

Proof

Since $\nu$ is finite, ${\mathscr {L}}_J$ is a bounded linear operator on $L^2(OM)$, and so Eq. (2.8) extends by continuity to the whole of $L^2(OM)$. It follows that ${\mathscr {L}}$ is a bounded perturbation of the horizontal Laplacian, and so its domain is Dom$(\Delta _{H})$. Clearly, ${\mathscr {L}}$ is symmetric on this domain.

Since ${\mathscr {L}}$ is a closed, symmetric operator, by Theorem X.1 on page 136 of Reed and Simon [23], the spectrum $\sigma ({\mathscr {L}})$ of ${\mathscr {L}}$ is equal to one of the following:

• 1. The closed upper-half plane.

• 2. The closed lower-half plane.

• 3. The entire complex plane.

• 4. A subset of ${\mathbb {R}}$.

Moreover, ${\mathscr {L}}$ is self-adjoint if and only if Case 4 holds. By Theorem 8.2.1 of [9],

$$\begin{aligned} \sigma ({\mathscr {L}})\subseteq (-\infty ,0], \end{aligned}$$

(4.1)

from which we see that Case 4 is the only option. $\square$

We now drop the assumption that $\nu$ is finite.

Theorem 4.2

$(S_t, t\ge 0)$ and $(T_t,t\ge 0)$ are self-adjoint semigroups of operators on $L^2(OM)$ and $L^2(M)$, respectively.

Proof

We will find it convenient to rewrite the process R(t) (with initial condition $R(0) = r$ (a.s.)) as the action of a stochastic flow $\eta _t$ on the point r, as in [4]. Then, as shown in Section 4 of [4], there is a sequence $(\eta ^{(n)}_t)$ of stochastic flows on OM such that each $\eta ^{(n)}_t$ is the flow of a horizontal Lévy process with finite Lévy measure, and

$$\begin{aligned} \lim _{n \rightarrow \infty }\eta ^{(n)}_t(r) = \eta _t(r)~~~\text{(a.s.) }, \end{aligned}$$

for all $r \in OM$ and $t\ge 0$.

Let $(S_{t}^{(n)}, t \ge 0)$ be the transition semigroup corresponding to the flow $(\eta ^{(n)}_t, t \ge 0)$, for each $n\in {\mathbb {N}}$. It is a standard result in semigroup theory that a semigroup of operators on a Hilbert space is self-adjoint if and only if its generator is self-adjoint.² By Lemma 4.1, $(S_t^{(n)},t\ge 0)$ is a self-adjoint semigroup on $L^2(OM)$, for all $n\in {\mathbb {N}}$.

By dominated convergence, for each $f \in C(OM)$ and $t\ge 0$, we have

$$\begin{aligned} \lim _{n \rightarrow \infty }\left\| S_{t}f - S_{t}^{(n)}f\right\| _{L^2(OM)}^{2} = \lim _{n \rightarrow \infty }\int _{OM}\left| {\mathbb {E}}\left( f\left( \eta _{t}(r)\right) -f\left( \eta _{t}^{(n)}(r)\right) \right) \right| ^{2}{\tilde{\mu }}(\mathrm{d}r) = 0. \end{aligned}$$

Then, by the density of C(OM) in $L^{2}(OM)$, and a standard $\epsilon /3$ argument (using the fact that $S_{t}^{(n)}$ is an $L^{2}$-contraction), we deduce that for all $f\in L^{2}(OM)$,

$$\begin{aligned} \lim _{n \rightarrow \infty }\left\| S_{t}f-S_t^{(n)}f\right\| _{L^2(OM)}=0 \end{aligned}$$

So, $S_t$ is the strong limit of a sequence of bounded self-adjoint operators, and hence is itself self-adjoint. To see that $(T_t,t\ge 0)$ is also self-adjoint, let $t\ge 0$ and $f,g\in L^2(M)$, and observe that by (2.1) and (3.1),

$$\begin{aligned} \langle T_tf,g\rangle _{L^2(M)}= & {} \langle S_t(f\circ \pi ),g\circ \pi \rangle _{L^2(OM)} = \langle f\circ \pi ,S_t(g\circ \pi )\rangle _{L^2(OM)} \\= & {} \langle f,T_tg\rangle _{L^2(M)}. \end{aligned}$$

$\square$

By Theorem 4.6 of Davies [8], $-{\mathscr {L}}$ and $-{\mathscr {A}}$ are positive self-adjoint operators on $L^2(OM)$ and $L^2(M)$, respectively.

5 Spectral Properties of the Generator

For this final section, we restrict attention to the case in which X has non-trivial Brownian part (that is, when $a>0$) and prove some spectral results that are already well-established for the case of Brownian motion and the Laplace–Beltrami operator $\Delta$. For example, it is well known that $\Delta$ has a discrete spectrum of eigenvalues. Each eigenspace is finite dimensional, and the eigenvectors may be normalised so as to form an orthonormal basis of $L^2(M)$ (see for example Lablée [17] Theorem 4.3.1). Moreover, such an eigenbasis $(\psi _n)$ can be ordered so that the corresponding sequence of eigenvalues decreases to $-\infty$. For each $n\in {\mathbb {N}}$, write $-\mu _n$ for the eigenvalue associated with $\psi _n$, so that the real sequence $(\mu _n)$ satisfies

$$\begin{aligned} 0\le \mu _1\le \mu _2\le \cdots \le \mu _n\rightarrow \infty \text { as } n\rightarrow \infty . \end{aligned}$$

(5.1)

We prove an analogous result for ${\mathscr {A}}$, using a generalisation of the approach used by Lablée [17].

Theorem 5.1

Let X be an isotropic Lévy process on M with non-trivial Brownian part. Then, its generator ${\mathscr {A}}$ has a discrete spectrum

$$\begin{aligned} \sigma ({\mathscr {A}}) = \{-\lambda _n:n\in {\mathbb {N}}\}, \end{aligned}$$

where $(\lambda _n)$ is a sequence of real numbers satisfying

$$\begin{aligned} 0\le \lambda _1\le \lambda _2\le \cdots \le \lambda _n\rightarrow \infty \text { as } n\rightarrow \infty . \end{aligned}$$

(5.2)

Moreover, each of the associated eigenspaces is finite dimensional, and there is a corresponding sequence $(\phi _n)$ of eigenvectors that forms an orthonormal basis of $L^2(M)$.

Remark 5.2

We will generally assume that (5.2) is listed with multiplicity, so that for all $n\in {\mathbb {N}}$, $-\lambda _n$ is the eigenvalue associated with $\phi _n$.

Proof

Without loss of generality, assume that $a=2$ so that

$$\begin{aligned} {\mathscr {A}} = \Delta + {\mathscr {A}}_J, \end{aligned}$$

(5.3)

where ${\mathscr {A}}_J$ is given by (2.9). Both ${\mathscr {A}}$ and ${\mathscr {A}}_J$ are generators of self-adjoint contraction semigroups, and hence, $-{\mathscr {A}}$ and $-{\mathscr {A}}_J$ are positive, self-adjoint operators.³ For $f,g\in {{\,\mathrm{Dom}\,}}{\mathscr {A}}$, define

$$\begin{aligned} \langle f,g\rangle _{{\mathscr {A}}} = \langle f,g\rangle _2 -\langle {\mathscr {A}}f,g\rangle _2. \end{aligned}$$

(5.4)

The operator $I-{\mathscr {A}}$ is also positive and self-adjoint, and so by Theorem 11 of [7], there is a unique positive self-adjoint operator B such that $B^2=I-{\mathscr {A}}$. By (4.1), $I-{\mathscr {A}}$ is invertible, and hence B is injective. Moreover,

$$\begin{aligned} \langle f,g\rangle _{{\mathscr {A}}} = \langle Bf,Bg\rangle _2, \forall f,g\in {{\,\mathrm{Dom}\,}}{\mathscr {A}}, \end{aligned}$$

from which it is easy⁴ to see that $\langle \cdot ,\cdot \rangle _{{\mathscr {A}}}$ defines an inner product on ${{\,\mathrm{Dom}\,}}{\mathscr {A}}$.

Let V denote the completion of $C^\infty (M)$ with respect to $\langle \cdot ,\cdot \rangle _{{\mathscr {A}}}$. This is a “Lévy analogue” of the Sobolev space $H^1(M)$ considered in Lablée [17] or Grigor’yan [14], where the completion is instead taken with respect to the Sobolev inner product

$$\begin{aligned} \langle f,g\rangle _{H^1} = \langle f,g\rangle _2 - \langle \Delta f,g\rangle _2, \forall f,g\in C^\infty (M). \end{aligned}$$

(5.5)

In the case when $M = {{\mathbb {R}}}^{d}$, spaces of this type are discussed in Section 3.10 of Jacob [16], who refers to them as anisotropic Sobolev spaces. By (5.3), we have

$$\begin{aligned} \langle f,g \rangle _{{\mathscr {A}}} = \langle f,g\rangle _{H^1} - \langle {\mathscr {A}}_J f,g\rangle _2 \forall f,g\in C^\infty (M), \end{aligned}$$

and since $-{\mathscr {A}}_J$ is a positive operator, it follows that $\Vert f\Vert _{{\mathscr {A}}} \ge \Vert f\Vert _{H^1}$ for all $f\in C^\infty (M)$. Similarly, (5.5) implies $\Vert f\Vert _{H^1}\ge \Vert f\Vert _2$ for all $f\in C^\infty (M)$. Hence,

$$\begin{aligned} V\subseteq H^1(M)\subseteq L^2(M), \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _2\le \Vert f\Vert _{H^1}\le \Vert f\Vert _{{\mathscr {A}}}, \forall f\in V. \end{aligned}$$

(5.6)

In particular, the inclusion $V\hookrightarrow H^1(M)$ is bounded. By Rellich’s theorem⁵, the inclusion $H^1(M)\hookrightarrow L^2(M)$ is compact, and hence so is the inclusion $i:V\hookrightarrow L^2(M)$ (it is a composition of a compact operator with a bounded operator).

Let $f\in L^2(M)$ and consider $l\in V^*$ given by

$$\begin{aligned} l(g) = \langle f,g\rangle _2 \forall g\in V. \end{aligned}$$

For all $g\in V$, we have by the Cauchy–Schwarz inequality

$$\begin{aligned} |l(g)|\le \Vert f\Vert _2\Vert g\Vert _2\le \Vert f\Vert _2\Vert g\Vert _{{\mathscr {A}}}. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert l\Vert _{V^*}\le \Vert f\Vert _2, \end{aligned}$$

(5.7)

where $\Vert \cdot \Vert _{V^*}$ denotes the norm of $V^*$. By the Riesz representation theorem, there is a unique $v_f\in V$ for which

$$\begin{aligned} \langle v_f,g\rangle _{{\mathscr {A}}} = l(g), \forall g\in V. \end{aligned}$$

Moreover,

$$\begin{aligned} \Vert v_f\Vert _{{\mathscr {A}}} = \sup _{g\in V\backslash \{0\}}\frac{|\langle v_f,g\rangle _{{\mathscr {A}}}|}{\Vert g\Vert _{{\mathscr {A}}}} = \Vert l\Vert _{V^*}. \end{aligned}$$

Define $T:L^2(M)\rightarrow V$ by $Tf=v_f$ for all $f\in L^2(M)$. Then,

$$\begin{aligned} \langle Tf,g\rangle _{{\mathscr {A}}} = \langle f,g\rangle _2 \forall f\in L^2(M),g\in V, \end{aligned}$$

(5.8)

and T is bounded, since by (5.7),

$$\begin{aligned} \Vert Tf\Vert _{{\mathscr {A}}} = \Vert l\Vert _{V^*}\le \Vert f\Vert _2, \end{aligned}$$

for all $f\in L^2(M)$. By (5.6),

$$\begin{aligned} \Vert Tf\Vert _{{\mathscr {A}}}\le \Vert f\Vert _{{\mathscr {A}}} \forall f\in V, \end{aligned}$$

and so $T|_V$ is a bounded operator on V. We also have

$$\begin{aligned} T|_V=T\circ i, \end{aligned}$$

and, since i is compact, so too is $T|_V$. By symmetry of inner products and Eq. (5.8), T is self-adjoint. Equation (5.8) also implies that $T|_V$ is a positive operator, and that 0 is not an eigenvalue of $T|_V$ (indeed, if $Tf=0$, then $\Vert f\Vert _2^2=\langle Tf,f\rangle _{{\mathscr {A}}}=0$).

By the Hilbert-Schmidt theorem⁶, the spectrum of $T|_V$ consists of a sequence $(\alpha _n)$ of positive eigenvalues that decrease to 0. Each eigenspace is finite dimensional, and the corresponding eigenvectors can be normalised so as to form an orthonormal basis $(v_n)$ of $(V,\langle \cdot ,\rangle _{{\mathscr {A}}})$.

In fact, it is easy to see from the definition of $\langle \cdot ,\cdot \rangle _{{\mathscr {A}}}$ that

$$\begin{aligned} T=(I-{\mathscr {A}})^{-1}, \end{aligned}$$

(5.9)

and hence, the spectrum of ${\mathscr {A}}$ is just $\{1-\alpha _n^{-1}:n\in {\mathbb {N}}\}$, with corresponding eigenvectors still given by the $v_n$. Moreover, we may scale these eigenvectors so that they are $L^2$-orthonormal. Indeed, for each $n\in {\mathbb {N}}$, let

$$\begin{aligned} \phi _n = \frac{1}{\sqrt{\alpha _n}} v_n. \end{aligned}$$

Then, for all $m,n\in {\mathbb {N}}$,

$$\begin{aligned} \langle \phi _n,\phi _m\rangle _2 = \frac{1}{\sqrt{\alpha _n\alpha _m}}\langle Tv_n,v_m\rangle _{{\mathscr {A}}} = \sqrt{\frac{\alpha _n}{\alpha _m}}\langle v_n,v_m\rangle _{{\mathscr {A}}} = \delta _{m,n}. \end{aligned}$$

By denseness of V in $L^2(M)$, the $\phi _n$ form an orthonormal basis of $L^2(M)$.

Finally, let $\lambda _n=\alpha _n^{-1}-1$ for each $n\in {\mathbb {N}}$. Then, $(\lambda _n)$ satisfies Eq. (5.2), since $-{\mathscr {A}}$ is a positive operator, and $(\alpha _n)$ is a positive sequence that decreases to 0. $\square$

It is well known that the heat semigroup $(K_t,t\ge 0)$ associated with Brownian motion on a compact manifold is trace-class and possesses an integral kernel. The final two results of this section extend this to the Lévy semigroup $(T_t,t\ge 0)$, subject to the assumption that $a>0$.

Theorem 5.3

Let X be an isotropic Lévy process on M with non-trivial Brownian part. Then, the transition semigroup operator $T_t$ is trace-class for all $t>0$.

Proof

We again assume $a=2$, so that ${\mathscr {A}}$ has the form (5.3). The case for general $a>0$ is very similar.

Let $(\lambda _n)$ and $(\phi _n)$ be as in the statement of Theorem 5.1, and let $(\mu _n)$ and $(\psi _n)$ be the analogous sequences for $\Delta$, so that $\psi _n$ is the $n^{\text {th}}$ eigenvector of $\Delta$, with associated eigenvalue $-\mu _n$.

Let $(K_t,t\ge 0)$ denote the heat semigroup associated with Brownian motion on M. This operator semigroup is known to possess many wonderful properties, including being trace-class. It follows that

$$\begin{aligned} {{\,\mathrm{tr}\,}}K_t = \sum _{n=1}^\infty \langle K_t\psi _n,\psi _n\rangle = \sum _{n=1}^\infty \mathrm{e}^{-t\mu _n}<\infty \end{aligned}$$

(5.10)

for all $t>0$. As an element of $[0,\infty ]$, the trace of each $T_t$ is given by

$$\begin{aligned} {{\,\mathrm{tr}\,}}T_t = \sum _{n=1}^\infty \langle T_t\phi _n,\phi _n\rangle = \sum _{n=1}^\infty \mathrm{e}^{-t\lambda _n}. \end{aligned}$$

(5.11)

By the min-max principle for self-adjoint semibounded operators,⁷ we have for all $n\in {\mathbb {N}}$,

$$\begin{aligned} \lambda _n = -\sup _{f_1,\ldots ,f_{n-1}\in C^\infty (M)}\left[ \inf _{f\in \{f_1,\ldots ,f_{n-1}\}^{\perp }, \;\Vert f\Vert =1}\langle {\mathscr {A}}f,f\rangle \right] , \end{aligned}$$

and

$$\begin{aligned} \mu _n = -\sup _{f_1,\ldots ,f_{n-1}\in C^\infty (M)}\left[ \inf _{f\in \{f_1,\ldots ,f_{n-1}\}^{\perp }, \;\Vert f\Vert =1}\langle \Delta f,f\rangle \right] . \end{aligned}$$

As noted in the proof of Theorem 5.1, for all $f\in C^\infty (M)$,

$$\begin{aligned} -\langle {\mathscr {A}}f,f\rangle \ge -\langle \Delta f,f\rangle \ge 0, \end{aligned}$$

and hence $\lambda _n\ge \mu _n$ for all $n\in {\mathbb {N}}$. But then, $\mathrm{e}^{-t\lambda _n}\le \mathrm{e}^{-t\mu _n}$ for all $t>0$ and $n\in {\mathbb {N}}$, and so, comparing (5.11) with (5.10),

$$\begin{aligned} {{\,\mathrm{tr}\,}}T_t<{{\,\mathrm{tr}\,}}K_t<\infty \end{aligned}$$

for all $t>0$. $\square$

By Lemma 7.2.1 of Davies [9], we immediately obtain the following.

Corollary 5.4

Let X be an isotropic Lévy process on M with non-trivial Brownian part. Then, its semigroup $(T_t,t\ge 0)$ has a square-integrable kernel. That is, for all $t > 0$ there is a map $p_t\in L^2(M\times M)$ such that

$$\begin{aligned} T_tf(x) = \int _Mf(y)p_t(x,y)\mu (\mathrm{d}y) \end{aligned}$$

for all $f\in L^2(M)$ and $x\in M$. Moreover, we have the following $L^2$-convergent expansion:

$$\begin{aligned} p_t(x,y) = \sum _{n=1}^\infty \mathrm{e}^{-\lambda _nt}\phi _n(x)\phi _n(y), \forall x,y\in M,\; t\ge 0. \end{aligned}$$

It is natural to enquire as to whether similar results hold in the pure jump case when $a=0$ (perhaps under some further condition on $\nu$)? When M is a compact symmetric space, we can find an orthonormal basis of eigenfunctions that are common to the $L^{2}$–semigroups associated with all isotropic Lévy processes (see the results in section 5 of [6]) . The key tool here, which enables a precise description of the spectrum of eigenvalues, is Gangolli’s Lévy–Khinchine formula [11]. In the general case, as considered here, such methods are not available, and we are unable to make further progress at the present time.