Sub-Feller Semigroups Generated by Pseudodifferential Operators on Symmetric Spaces of Noncompact Type

For a thorough treatment of this topic, see Helgason (2001, 1984). Let ${\bf D}(G)$ denote the set of all left invariant differential operators on $G$, and let ${\bf D}_{K}(G)$ denote the subspace of those operators that are also $K$-right-invariant. A mapping $\phi:G\to\mathbb{C}$ is called a spherical if it is $K$-bi-invariant, satisfies $\phi(e)=1$, and is a simultaneous eigenfunction of every element of ${\bf D}_{K}(G)$.

Fix an Iwasawa decomposition $G=NAK$, where $N$ is a nilpotent Lie subgroup of $G$, and $A$ is Abelian. Let $\mathfrak{n}$ and $\mathfrak{a}$ denote respectively the Lie algebras of $N$ and $A$. For each $\sigma\in G$, let $A(\sigma)$ denote the unique element of $\operatorname{\mathfrak{a}}$ such that $\sigma\in Ne^{A(\sigma)}K$. Harish-Chandra’s integral formula states that every spherical function on $G$ takes the form

$$\phi_{\lambda}(\sigma)=\int_{K}e^{(\rho+i\lambda)(A(k\sigma))}dk,\hskip 20.0pt\forall\sigma\in G,$$

(2.2)

for some $\lambda\in\operatorname{\mathfrak{a}_{\mathbb{C}}^{\ast}}*$. Moreover, $\phi_{\lambda}=\phi_{\lambda^{\prime}}$ if an only if $s(\lambda)=\lambda^{\prime}$ for some element $s$ of the Weyl group $W$. A spherical function $\phi_{\lambda}$ is positive definite if and only if $\lambda\in\operatorname{\mathfrak{a}^{\ast}}*$.

The spherical transform of a function $f\in L^{1}(K|G|K)$ is the function $\hat{f}:\operatorname{\mathfrak{a}^{\ast}}*\to\mathbb{C}$ given by

$$\hat{f}(\lambda)=\int_{G}\phi_{-\lambda}(\sigma)f(\sigma)d\sigma,\hskip 20.0pt\forall\lambda\in\operatorname{\mathfrak{a}^{\ast}}*.$$

(2.3)

Similarly, given a finite Borel measure $\mu$ on $G$, the spherical transform of $\mu$ is the mapping $\hat{\mu}:\operatorname{\mathfrak{a}^{\ast}}*\to\mathbb{C}$ given by

$$\hat{\mu}(\lambda)=\int_{G}\phi_{-\lambda}(\sigma)\mu(d\sigma).$$

The spherical transform enjoys many useful properties, the most powerful being that it defines an isomorphism of the Banach convolution algebra $L^{1}(K|G|K)$ with the space $L^{1}(\operatorname{\mathfrak{a}^{\ast}}*,\omega)^{W}$ of Weyl group invariant elements of $L^{1}(\operatorname{\mathfrak{a}^{\ast}}*,\omega)$. The Borel measure $\omega$ is called Plancherel measure, and is given by

$$\omega(d\lambda)=|\operatorname{\bf c}(\lambda)|^{-2}d\lambda,$$

where $\operatorname{\bf c}$ denotes Harish-Chandra’s $\operatorname{\bf c}$-function. According to the spherical inversion formula, for all $f\in C_{c}^{\infty}(K|G|K)$ and all $\sigma\in G$,

$$f(\sigma)=\int_{\operatorname{\mathfrak{a}^{\ast}}*}\phi_{\lambda}(\sigma)\hat{f}(\lambda)\omega(d\lambda).$$

(2.4)

There is also a version of Plancherel’s identity for the spherical transform, namely

$$\|f\|_{L^{2}(K|G|K)}=\|\hat{f}\|_{L^{2}(\operatorname{\mathfrak{a}^{\ast}}*,\omega)},\hskip 20.0pt\forall f\in C_{c}^{\infty}(K|G|K).$$

(2.5)

Let $L^{2}(\operatorname{\mathfrak{a}^{\ast}}*,\omega)^{W}$ denote the subspace of $L^{2}(\operatorname{\mathfrak{a}^{\ast}}*,\omega)$ consisting of $W$-invariants. Then the image of $C_{c}^{\infty}(K|G|K)$ under spherical transformation is a dense subspace of $L^{2}(\operatorname{\mathfrak{a}^{\ast}}*,\omega)^{W}$, and as such the spherical transform extends to an isometric isomorphism between the Hilbert spaces $L^{2}(K|G|K)$ and $L^{2}(\operatorname{\mathfrak{a}^{\ast}}*,\omega)^{W}$. For more details, see for example Helgason (1984) Chapter IV § 7.3, pp. 454.

Similarly to classical Fourier theory, the most natural setting for the spherical transform is Schwarz space. A function $f\in C^{\infty}(G)$ is called rapidly decreasing if

$$\sup_{\sigma\in G}(1+|\sigma|)^{q}\phi_{0}(\sigma)^{-1}(Df)(\sigma)<\infty,\hskip 20.0pt\forall D\in{\bf D}(G),\;q\in\mathbb{N}\cup\{0\},$$

(2.6)

where $|\sigma|$ denotes the geodesic distance on $G/K$ from $o:=eK$ to $\sigma K$. Equivalently, if $\sigma=e^{X}k$, where $X\in\mathfrak{p}$, then $|\sigma|=\|X\|$ — see Gangolli and Varadarajan (1980) pp.167 for more details.

The ($K$-bi-invariant) Schwarz space $\mathcal{S}(K|G|K)$ is the Fréchet space comprising of all rapidly decreasing, $K$-bi-invariant functions $f\in C^{\infty}(G)$, together with the family of seminorms given by the left-hand side of (2.6). By viewing the spaces $\operatorname{\mathfrak{a}}$ and $\operatorname{\mathfrak{a}^{\ast}}*$ as finite dimensional vector spaces, we also consider the classical Schwartz spaces $\mathcal{S}(\operatorname{\mathfrak{a}^{\ast}}*)$ and $\mathcal{S}(\operatorname{\mathfrak{a}})$, as well as $W$-invariant subspaces $\mathcal{S}(\operatorname{\mathfrak{a}})^{W}$ and $\mathcal{S}(\operatorname{\mathfrak{a}^{\ast}}*)^{W}$. The Euclidean Fourier transform

$$\mathscr{F}(f)(\lambda)=\int_{\operatorname{\mathfrak{a}}}e^{-i\lambda(H)}f(H)dH,\hskip 20.0pt\forall f\in\mathcal{S}(\operatorname{\mathfrak{a}}),\lambda\in\operatorname{\mathfrak{a}^{\ast}}*$$

(2.7)

defines a topological isomorphism between the spaces $\mathcal{S}(\operatorname{\mathfrak{a}})^{W}$ and $\mathcal{S}(\operatorname{\mathfrak{a}^{\ast}}*)^{W}$ in the usual way. Given $f\in\mathcal{S}(K|G|K)$ and $H\in\operatorname{\mathfrak{a}}$, the Abel transform is defined by

$$\mathscr{A}f(H)=e^{\rho(H)}\int_{N}f((\exp H)n)dn.$$

The Abel transform is fascinating in its own right, and we refer to Sawyer (2003) for more information. However, for our purposes we are mainly interested in its role in the following:

This result will be extremely useful in later sections, especially when proving Theorem 4.17. For more details, see Proposition 3 in Anker (1990), Gangolli and Varadarajan (1980) page 265, and Helgason (1984) pp. 450.

We summarise a few key notions from probability theory on Lie groups and symmetric spaces. Sources for this material include Liao and Wang (2007) and Liao (2004, 2018).

Fix a probability space $(\Omega,\mathcal{F},P)$. Just as with functions on $G$ and $G/K$, we may view stochastic processes on $G/K$ as projections of processes on $G$ whose laws are $K$-right invariant. Let $Y=(Y(t),t\geq 0)$ a stochastic process taking values on $G$. The random variables

$$Y(s)^{-1}Y(t),\hskip 20.0pt0\leq s\leq t,$$

are called the increments of $Y$. Equipped with its natural filtration $\{\mathcal{F}^{Y}_{t},t\geq 0\}$, $Y$ is said to have independent increments if for all $t>s\geq 0$, $Y(s)^{-1}Y(t)$ is independent of $\mathcal{F}^{X}_{s}$, and stationary increments if

$$Y(s)^{-1}Y(t)\sim Y(0)^{-1}Y(t-s)\hskip 20.0pt\forall t>s\geq 0.$$

A process $Y=(Y(t),t\geq 0)$ on $G$ is stochastically continuous if, for all $s\geq 0$ and all $B\in\mathcal{B}(G)$ with $e\notin B$,

$$\lim_{t\rightarrow s}P(Y(s)^{-1}Y(t)\in B)=0.$$

A stochastically continuous process $Y$ on $G$ with stationary and independent increments is called a Lévy process on $G$. A process on $G/K$ is called a Lévy process if it is the projection of a Lévy process on $G$, under the canonical surjection $\pi:G\mapsto G/K$. Lévy processes on $G/K$ correspond precisely to the $G$-invariant Feller processes on $G/K$. The proof of this is similar to the well-known result for $\mathbb{R}^{d}$-valued Lévy processes.

The convolution product of two Borel measures $\mu_{1},\mu_{2}$ on $G$ is defined for each $B\in\mathcal{B}(G)$ by

$$(\mu_{1}\ast\mu_{2})(B)=\int_{G}\int_{G}\operatorname{\bf 1}_{B}(\sigma\tau)\mu_{1}(d\sigma)\mu_{2}(d\tau).$$

(2.8)

Note that since $G$ is semisimple, it is unimodular, and hence this operation is commutative. It is also clear from the definition that $\mu_{1}\ast\mu_{2}$ is $K$-bi-invariant whenever $\mu_{1}$ and $\mu_{2}$ are.

Note that $\mu_{0}$ must be an idempotent measure, in the sense that $\mu_{0}\ast\mu_{0}=\mu_{0}$. By Theorem 1.2.10 on page 34 of Heyer (1977), $\mu_{0}$ must coincide with Haar measure on a compact subgroup of $G$. We we will frequently take $\mu_{0}$ to be normalised Haar measure on $K$, so that the image of $\mu_{0}$ after projecting onto $G/K$ is $\delta_{o}$, the delta mass at $o:=eK$.

One may also define convolution of measures on $G/K$, and convolution semigroups on $G/K$ are defined analogously — see Liao (2018) Section 1.3 for more details. In fact, the projection map $\pi:G\to G/K$ induces a bijection between the set of all convolution semigroups on $G/K$ and the set of all $K$-bi-invariant convolution semigroups on $G$ — see Liao (2018) Propositions 1.9 and 1.12, pp. 11–13. We henceforth identify these two sets, but generally opt to perform calculations using objects defined on $G$, for simplicity.

Let $Y$ be a Lévy process on $G$, and for each $t\geq 0$, let $\mu_{t}$ denote the law of $Y(0)^{-1}Y(t)$. By Liao (2018) Theorem 1.7, pp. 8, $(\mu_{t},t\geq 0)$ is a convolution semigroup of probability measures on $G$.

Let $Y$ be a Lévy process on $G/K$, and $X$ a Lévy process on $G$ for which $Y=\pi(X)$. Let $(p_{t},t\geq 0)$ and $(q_{t},t\geq 0)$ denote the transition probabilities of $Y$ and $X$, respectively. Then for all $t\geq 0$, $\sigma\in G$ and $A\in\mathcal{B}(G/K)$,

$$p_{t}(\sigma K,A)=\mathbb{P}\left.\left(\pi\big{(}X(t)\big{)}\in A\right|\pi\big{(}X\big{)}=\sigma K\right)=q_{t}\left(\sigma,\pi^{-1}(A)\right).$$

In particular, the prescription

$$\nu_{t}:=p_{t}(o,\cdot),\hskip 20.0pt\forall t\geq 0$$

defines a convolution semigroup $(\nu_{t},t\geq 0)$ on $G/K$. By Liao (2018) Proposition 1.12, pp. 13, $(\nu_{t},t\geq 0)$ is $K$-invariant, and there is a $K$-bi-invariant convolution semigroup $(\mu_{t},t\geq 0)$ on $G$ for which

$$\nu_{t}=\mu_{t}\circ\pi^{-1},\hskip 20.0pt\forall t\geq 0.$$

(2.9)

It may be tempting to think that $(\mu_{t},t\geq 0)$ should be the convolution semigroup of $X$. In fact, this is not the case: if it were, then we would have $\mu_{0}=\delta_{0}$, which is not a $K$-bi-invariant measure on $G$. However, if we denote the convolution semigroup of $X$ by $(\mu^{e}_{t},t\geq 0)$, and normalised Haar measure on $K$ by $\rho_{K}$, then by Liao (2018) Theorem 3.14, pp. 88,

$$\mu_{t}:=\rho_{K}\ast\mu^{e}_{t},\hskip 20.0pt\forall t\geq 0$$

is a suitable choice for the $K$-bi-invariant convolution semigroup $(\mu_{t},t\geq 0)$ on $G$, for which (2.9) is satisfied. In particular, $\mu_{0}=\rho_{K}$.

In this way, Lévy processes on $G/K$ may be understood through the study of $K$-bi-invariant convolution semigroups on $G$. The corresponding Lévy processes on $G$ are called $K$-bi-invariant Lévy processes. For such a process $X$, with $K$-bi-invariant convolution semigroup $(\mu_{t},t\geq 0)$, the Hunt semigroup $(T_{t},t\geq 0)$ of $(\mu_{t},t\geq 0)$) is given by

$$T_{t}f(\sigma)=\int_{G}f(\sigma\tau)\mu_{t}(d\tau)\hskip 20.0pt\forall f\in B_{b}(G),\;\sigma\in G.$$

(2.10)

Note that since $\mu_{0}=\rho_{K}$, we have $T_{0}=I$. In fact, $(T_{t},t\geq 0)$ forms a strongly continuous operator semigroup on $C_{0}(G/K)$, and the restriction of each $T_{t}$ to $C_{0}(K|G|K)$ yields a strongly continuous semigroup on $C_{0}(K|G|K)$. $(T_{t},t\geq 0)$ is a left invariant Feller semigroup in each of these cases (see Ngan (2019) pp. 82–83).

Restricting to the $K$-bi-invariant functions in this way will be advantageous, as we have the spherical transform at our disposal. As an early application of this, we prove the following useful eigenvalue relation for the Hunt semigroup of a $K$-bi-invariant convolution semigroup.

The infinitesimal generator of a Lévy process $Y$ on $G$ is given by the celebrated Hunt formula (Hunt (1956) Theorem 5.1). We describe a version of this next, specialising to the $K$-bi-invariant case most relevant to our work on symmetric spaces. We first introduce a local coordinate system on $G$, defined in terms of the orthogonal decomposition (2.1).

The $x_{i}$ may be chosen so as to be $K$-right-invariant for $i=1,\ldots,m$, and such that

$$\sum_{i=1}^{d}x_{i}(k\sigma)X_{i}=\sum_{i=1}^{d}x_{i}(\sigma)\operatorname{Ad}(k)X_{i}\hskip 20.0pt\forall k\in K.$$

For more details, see Liao (2018) pp.36–37, 83.

The choice of basis of $\mathfrak{p}$ enables us to view $\operatorname{Ad}(k)$ as a $d\times d$ matrix, for each $k\in K$. A vector $b\in\mathbb{R}^{m}$ is said to be $\operatorname{Ad}(K)$-invariant if

$$b=\operatorname{Ad}(k)^{T}b,\hskip 20.0pt\forall k\in K.$$

Similarly, a $d\times d$ real-valued matrix $a=(a_{ij})$ is $\operatorname{Ad}(K)$-invariant if

$$a=\operatorname{Ad}(k)^{T}a\operatorname{Ad}(k)\hskip 20.0pt\forall k\in K.$$

A Borel measure $\nu$ on $G$ is called a Lévy measure if $\nu(\{e\})=0$, $\nu(U^{c})<\infty$, and $\int_{G}\sum_{i=1}^{l}x_{i}(\sigma)^{2}\nu(d\sigma)$.

We state a useful corollary of the famous Hunt formula. For more details, including a proof, see Section 3.2 of Liao (2018), pp. 78.

Since $G$ is semisimple, $\mathfrak{p}$ has no non-zero $\operatorname{Ad}(K)$-invariant elements. This means that for the class of manifold we are considering, $K$-bi-invariant Lévy generators will take the form

$$\mathcal{A}f(\sigma)=\sum_{i,j=1}^{d}a_{ij}X_{i}X_{j}f(\sigma)+\int_{G}\left(f(\sigma\tau)-f(\sigma)-\sum_{i=1}^{d}x_{i}(\tau)X_{i}f(\sigma)\right)\nu(d\sigma),$$

(2.13)

Given such a Lévy generator, we write $\mathcal{A}_{D}=\sum_{i,j=1}^{d}a_{ij}X_{i}X_{j}$ for the diffusion part of $\mathcal{A}$. By the discussion surrounding (3.3) in Liao (2018), pp. 75, $\mathcal{A}_{D}\in{\bf D}_{K}(G)$, and so for each $\lambda\in\operatorname{\mathfrak{a}^{\ast}}*$ there is $\beta(\mathcal{A}_{D},\lambda)\in\mathbb{C}$ such that

$$\mathcal{A}_{D}\phi_{\lambda}=\beta(\mathcal{A}_{D},\lambda)\phi_{\lambda}.$$

(2.14)

Moreover, $\lambda\mapsto\beta(\mathcal{A}_{D},\lambda)$ is a $W$-invariant quadratic polynomial function on $\operatorname{\mathfrak{a}^{\ast}}*$.

This result was first proven in Gangolli (1964), see also Liao and Wang (2007). For a proof of the specific statement above, see page 139 of Liao (2018).

The function $\psi$ given by (2.15) will be called the Gangolli exponent of the process $X$.

By viewing $\operatorname{\mathfrak{a}^{\ast}}*$ as a finite-dimensional real vector space, we may consider positive and negative definite functions on $\operatorname{\mathfrak{a}^{\ast}}*$, defined in the usual way.

By choosing a basis of $\operatorname{\mathfrak{a}^{\ast}}*$, we may identify it with $\mathbb{R}^{m}$, and apply classical results about positive (resp. negative) definite functions on Euclidean space to functions on $\operatorname{\mathfrak{a}^{\ast}}*$, to obtain results about positive (resp. negative) definite functions in this new setting.

One useful application of this is the Schoenberg correspondence, which states that a map $\psi:\operatorname{\mathfrak{a}^{\ast}}*\to\mathbb{C}$ is negative definite if and only if $\psi(0)\geq 0$ and $e^{-t\psi}$ is positive definite for all $t>0$. This is immediate by the Schoenberg correspondence on $\mathbb{R}^{m}$ — see Berg and Forst. (1975) page 41 for a proof.

We finish this subsection with a collection of results about negative definite functions, which will be useful in later sections.

Suppose $\psi$ is a real-valued continuous negative definite function, and let $s\in\mathbb{R}$. We define the (spherical) anisotropic Sobolev space associated with $\psi$ and $s$ to be

$$H^{\psi,s}:=\left\{u\in\mathcal{S}^{\prime}(K|G|K):\int_{G}(1+\psi(\lambda))^{s}|\hat{u}(\lambda)|^{2}\omega(d\lambda)<\infty\right\},$$

where $\mathcal{S}^{\prime}(K|G|K)$ denotes the space of $K$-bi-invariant tempered distributions. One can check that each $H^{\psi,s}$ is a Hilbert space with respect to the inner product

$$\langle u,v\rangle_{\psi,s}:=\int_{\operatorname{\mathfrak{a}^{\ast}}*}(1+\psi(\lambda))^{s}\hat{u}(\lambda)\overline{\hat{v}(\lambda)}\omega(d\lambda),\hskip 20.0pt\forall u,v\in H^{\psi,s}.$$

These spaces are a generalisation of the anisotropic Sobolev spaces first introduced by Niels Jacob, see Jacob (1993), and developed further by Hoh, see Hoh (1998). For the special case $\psi(\lambda)=|\rho|^{2}+|\lambda|^{2}$, we will write $H^{\psi,s}=H^{s}$. Note also that $H^{\psi,0}=L^{2}(K|G|K)$, by the Plancherel theorem. In this case, we will omit subscripts and just write $\langle\cdot,\cdot\rangle$ for the $L^{2}$ inner product.

Note that $\psi$ is a non-negative function, since it is negative definite and real-valued. We impose an additional assumption, namely that there exist constants $r,c>0$ such that

$$\psi(\lambda)\geq c|\lambda|^{2r}\hskip 20.0pt\forall\lambda\in\operatorname{\mathfrak{a}^{\ast}}*,\;|\lambda|\geq 1.$$

(2.18)

Analogous assumptions are made in Jacob (1994) (1.5) and Hoh (1998) (4.2), and the role of (2.18) will be very similar.