Aspects of Micro-Local Analysis and Geometry in the Study of Lévy-Type Generators

Abstract

Generators of Feller processes are pseudo-differential operators with negative definite symbols, thus they are objects of micro-local analysis. Continuous negative definite functions (and symbols) give often raise to metrics and these metrics are important to understand, for example, transition functions of certain Feller processes. In this survey we outline some of the more recent results and ideas while at the same time we long to introduce into the field.

The author Niels Jacob wrote the appendix jointly with James Harris.

Appendix: On the Metric Balls \(B^{d_\psi }(0, r)\)

Continuous negative definite functions are in general not smooth, in fact their smoothness is determined by the moments of their Lévy measure. Moreover for n ≥ 2 they can have rather anisotropic behaviour, i.e. in the case that $\psi ^{\frac {1}{2}}$ gives rise to a metric, the metric balls can be rather anisotropic. Both facts must be taken into account in the analysis of the operator ψ(D) and the corresponding operator semigroups $\left (T_t^\psi \right )_{t \geq 0}$.

The lack of smoothness has the effect that some “nice” looking estimates are not suitable for our analysis. The following example is taken from [44]: In $\mathbb {R}^2$ the two functions ψ ₁(ξ, η) = |ξ| + |η| and $\psi _2(\xi , \eta ) = \sqrt {\psi ^2 + \eta ^2}$ are continuous negative definite functions and the estimates

$$\displaystyle \begin{aligned} \frac{1}{\sqrt{2}} \left(|\xi| + |\eta| \right) \leq \sqrt{|\xi|{}^2 + |\eta|{}^2} \leq |\xi| + |\eta| \end{aligned} $$

(A.1)

hold. The corresponding densities of $\left (T_t^{\psi _j} \right )_{t \geq 0}$ are given by

$$\displaystyle \begin{aligned} p_t^{\psi_1} (x, y) = \frac{1}{\pi^2} \frac{t^2}{(x^2 + t^2)(y^2 + t^2)} \end{aligned} $$

(A.2)

and

$$\displaystyle \begin{aligned} p_t^{\psi_2}(x, y) = \frac{1}{2\pi} \frac{t}{\left( (x^2 + y^2) + t^2 \right)^{\frac{3}{2}}}. \end{aligned} $$

(A.3)

If we choose x = 0 and consider the limit |y|→∞ we find for t = 1

$$\displaystyle \begin{aligned} p_1^{\psi_1}(0, y) \asymp |y|{}^{-2} \end{aligned}$$

and

$$\displaystyle \begin{aligned} p_1^{\psi_2}(0, y) \asymp |y|{}^{-3}. \end{aligned}$$

Thus, although we have symbols which are comparable, the decay of the corresponding semigroups is not. The different degrees of smoothness leads to a different decay of the Fourier transforms of $e^{-t\psi _j}$, j = 1, 2, as we do expect. However the diagonal terms can be compared once we have an estimate such as (A.1). Indeed, ψ ₁ ≍ ψ ₂ leads to similar lower bounds of the corresponding Dirichlet forms:

$$\displaystyle \begin{aligned} \mathcal{E}^{\psi_1}(u, u) &= \int_{\mathbb{R}^n} \psi_1(\xi) \left| \hat{u}(\xi) \right|{}^2 \, \mathrm{d}\xi \\ &\leq c_1 \int_{\mathbb{R}^n} \psi_2(\xi) \left| \hat{u}(\xi) \right|{}^2 \, \mathrm{d}\xi \\ &= c_1 \mathcal{E}^{\psi_2} (u, u), \end{aligned} $$

and similarly we find $\mathcal {E}^{\psi _2}(u, u) \leq c_2 \mathcal {E}^{\psi _1}(u, u)$. Therefore, for example in the transient case, we obtain with the same q > 2 the estimates

$$\displaystyle \begin{aligned} \| u \|{}_{L^q}^2 \leq \tilde{c}_1 \mathcal{E}^{\psi_1} (u, u) \hspace{0.3cm} \text{and} \hspace{0.3cm} \| u \|{}_{L^q}^2 \leq \tilde{c}_2 \mathcal{E}^{\psi_2} (u, u). \end{aligned}$$

This observation implies also that the diagonal behaviour of p _t(⋅) alone cannot determine the off-diagonal behaviour.

There are three classes of examples of continuous negative definite functions which we often use and each is requiring some different considerations when investigating the corresponding operator semigroups:

• (i) On $\mathbb {R}^n$ we may look at the sum ψ = ψ ₁ + ψ ₂ of two continuous negative definite functions ψ ₁ and ψ ₂. Our running example ψ _ER is of this type. The convolution theorem yields $p_t^\psi = p_t^{\psi _1} \ast p_t^{\psi _2}$ and from this we derive using Young’s inequality that

  $$\displaystyle \begin{aligned} p_t^\psi(0) = \left\| p_t^\psi(0) \right\|{}_\infty \leq \left\| p_t^{\psi_1} \right\|{}_\infty \wedge \left\| p_t^{\psi_2} \right\|{}_\infty = p_t^{\psi_1}(0) \wedge p_t^{\psi_2}(0). \end{aligned} $$

  (A.4)

  Now special properties of ψ ₁ and ψ ₂ are needed to get further results. In some cases one can determine explicitly a time T > 0 such that for t < T we have $p_t^{\psi _1}(0) \wedge p_t^{\psi _2}(0) = p_t^{\psi _1}(0)$ and for t > T it follows that $p_t^{\psi _1}(0) \wedge p_t^{\psi _2}(0) = p_t^{\psi _2}(0)$. This is, for example, possible for ψ(ξ) = ∥ξ∥^α + ∥ξ∥^β. This observation shows now that making use of the full symbol and not only the principal symbol gives more detailed information.

• (ii) We may have a decomposition of $\mathbb {R}^n$, $\mathbb {R}^n = \mathbb {R}^{n_1} \times \mathbb {R}^{n_2}$, and ψ(ξ, η) = ψ ₁(ξ) + ψ ₂(η). In this case we have of course $p_t^\psi (x, y) = p_t^{\psi _1}(x) p_t^{\psi _2}(y)$ and we can reduce the study of $p_t^\psi$ directly to investigations on $p_t^{\psi _j}$. In this case we should work with $B^{d_{\psi _1}}(0, r) \times B^{d_{\psi _2}}(0, r)$ rather than with $B^{d_\psi }(0, r)$.

• (iii) The final class of examples is obtained by subordination, i.e. by considering f ∘ ψ where f is a Bernstein function and $\psi : \mathbb {R}^n \to \mathbb {R}$ is a given continuous negative definite function. For some questions, compare with Theorem 3.15 and Corollary 3.16, a type of (operator) functional calculus is available. However, in general, subordination may destroy some structural properties: It may happen that for the symbol ψ(ξ) = ψ ₁(ξ) + ψ ₂(ξ) we can consider ψ ₁(ξ) as a type of principal symbol, however $f\left (\psi _1(\xi ) + \psi _2(\xi ) \right )$ need not allow a decomposition into a principal symbol and a “lower order” term with both being continuous negative definite functions.

The first geometric question we want to discuss is that of the convexity of metric balls. Note that two notions of convexity are possible, we may consider convexity in the vector space $\mathbb {R}^n$, and this notion is the important one for us, but we remind the reader on

Definition A.1

A subset G of a metric space (X, d) is called metrically convex if for every pair p, q ∈ G, p ≠ q, there exists a point r ∈ G such that d(p, q) = d(p, r) + d(r, q).

Using the Lévy-Khinchine representation of ψ, Harris and Rhind could prove, see [30], that in general the metric space $\left ( \mathbb {R}^n, d_\psi \right )$ is not metrically convex.

The following result is natural and we refer to [48] for a proof:

Proposition A.2

Let $\psi : \mathbb {R}^n \to \mathbb {R}$ be a continuous negative definite function generating a metric on $\mathbb {R}^n$ and let f be a Bernstein function such that f ∘ ψ also generates a metric on $\mathbb {R}^n$ . Then the balls $B^{d_\psi }(0, r)$ are convex if and only if the balls $B^{d_{f \circ \psi }}(0, r)$ are convex. In particular metric balls related to subordinate Brownian motion are for appropriate Bernstein functions convex.

In general the metric balls $B^{d_\psi }(0, r)$ will not be convex, examples are easily constructed with the help of

$$\displaystyle \begin{aligned} \psi(\xi, \eta) = \left( \| \xi \|{}^\alpha + \| \eta \|{}^\beta \right)^{\frac{1}{2}}, \hspace{0.5cm} 0 < \alpha < 1 \hspace{0.3cm} \text{or} \hspace{0.3cm} 0 < \beta < 1, \end{aligned}$$

where $\xi \in \mathbb {R}^{n_1}$ and $\eta \in \mathbb {R}^{n_2}$, see [48], or by looking at

$$\displaystyle \begin{aligned} \psi(\xi_1, \xi_2, \xi_3) = \operatorname{\mathrm{arcosh}} \left( |\xi_1|{}^2 + 1 \right) + \operatorname{\mathrm{arsinh}} \left( |\xi_2|{}^2 \right) + |\xi_3|{}^\alpha, \hspace{0.3cm} 0 < \alpha < 2, \end{aligned}$$

see [30].

The following two examples do not only illustrate the failure of convexity of metric balls $B^{d_\psi }(0, r)$, they also illustrate the anisotropic behaviour of the balls. These examples are taken from [30].

The first example is the continuous negative definite function defined on $\mathbb {R}^3$ by

$$\displaystyle \begin{aligned} \psi_\alpha (\xi) = |\xi_1|{}^{\frac{3}{4}} + \operatorname{\mathrm{arcosh}} \left( |\xi_2|{}^2 + 1 \right) + |\xi_3|{}^{2\alpha}. \end{aligned}$$

The following graphic shows the corresponding metric balls $B^{d_{\psi _\alpha }}(0, r)$ for r = 1 and α = 0.35, α = 0.5, α = 0.75 and α = 0.9, respectively.

The second example is the continuous negative definite function

$$\displaystyle \begin{aligned} \psi(\xi) = \operatorname{\mathrm{arcosh}} \left( |\xi_1|{}^2 + 1 \right) + \operatorname{\mathrm{arsinh}} \left(|\xi_2|{}^2 \right) + |\xi_3|{}^{\frac{3}{5}}, \end{aligned}$$

and we consider the different radii r = 0.5, r = 1, r = 1.5 and r = 2.

In Sect. 4 we have introduced the doubling property for the metric balls $B^{d_\psi }(0, r)$ and in the case where $\left (\mathbb {R}^n, d_\psi , \lambda ^{(n)} \right )$ is a metric measure space in which the doubling property holds we could derive better estimates for $p_t^\psi (0)$ and therefore we want to study $\left (\mathbb {R}^n, d_\psi , \lambda ^{(n)} \right )$ in relation to the doubling property.

First we note that the metric $d_\psi = \psi ^{\frac {1}{2}} (\xi - \eta )$ is translation invariant which allows us to reduce all studies to metric balls with centre $0 \in \mathbb {R}^n$. Thus the conditions

$$\displaystyle \begin{aligned} \lambda^{(n)} \left( B^{d_\psi}(0, 2r) \right) \leq c_0 \lambda^{(n)} \left(B^{d_\psi} (0, r) \right) \hspace{0.5cm} \text{for all}\ r > 0, \end{aligned} $$

(A.5)

and

$$\displaystyle \begin{aligned} \lambda^{(n)} \left(B^{d_\psi} (x, 2r) \right) \leq c_0 \lambda^{(n)} \left(B^{d_\psi} (x, r)\right) \hspace{0.5cm} \text{for all}\ r > 0, x \in \mathbb{R}^n, \end{aligned} $$

(A.6)

with c ₀ independent of r and x are equivalent. The doubling property implies power growth for $R \mapsto \lambda ^{(n)} \left ( B^{d_\psi }(0, R) \right )$, i.e. we have

$$\displaystyle \begin{aligned} \lambda^{(n)} \left(B^{d_\psi} (x, R) \right) \leq \kappa R^{\ln c_0}, \hspace{0.5cm} \kappa = \lambda^{(n)} \left( B^{d_\psi} (0, 1) \right). \end{aligned} $$

(A.7)

We say that $\left ( \mathbb {R}^n, d_\psi , \lambda ^{(n)} \right )$ has the local volume doubling property if (A.6) holds for all r, 0 < r < r ₀. We often say d _ψ has the volume doubling property when we mean that $\left (\mathbb {R}^n, d_\psi , \lambda ^{(n)} \right )$ has this property.

Example A.3

• A. The metric measure space $\left (\mathbb {R}^n, d_{\psi _\alpha }, \lambda ^{(n)} \right )$ with ψ _α(ξ) = |ξ|^α, 0 < α ≤ 2, has the volume doubling property since $\lambda ^{(n)} \left (B^{d_{\psi _\alpha }}(0, r) \right ) = c_{n, \alpha } r^{\frac {2n}{\alpha }}$.

• B. The metric measure space $\left (\mathbb {R}^n, d_\psi , \lambda ^{(n)} \right )$ with

  $$\displaystyle \begin{aligned} \psi(\xi) = 1 - e^{-\gamma |\xi|{}^2}, \hspace{0.3cm} \gamma > 0, \end{aligned} $$

  (A.8)

  has the local volume doubling property for 0 < r < 1, but not the volume doubling property.

From this observation we deduce that in general, if d _ψ has the volume doubling property, d _f∘ψ, where f is a Bernstein function, need not have the doubling property. In [44] some conditions on f and ψ are discussed for d _f∘ψ having the doubling property, we just quote as one result Corollary 3.11 from [44].

Corollary A.4

If d _ψ has the volume doubling property and f is a Bernstein function such that for some C > 1

$$\displaystyle \begin{aligned} \liminf_{r \to 0} \frac{f(Cr)}{f(r)} > 1 \hspace{0.3cm} \mathit{\text{and}} \hspace{0.3cm} \liminf_{r \to \infty} \frac{f(Cr)}{f(r)} > 1, \end{aligned}$$

then d _{f∘ ψ} has the volume doubling property too.

Many of our examples are of type $\psi (\xi ) = \sum _{j = 1}^N \psi _j (\xi _j)$ with ξ = (ξ ₁, …, ξ _N), $\xi _j \in \mathbb {R}^{n_j}$. Suppose that each of the continuous negative definite functions ψ _j generates a metric $d_{\psi _j}$ on $\mathbb {R}^{n_j}$. On $\mathbb {R}^n$, n = n ₁ + … + n _N, the natural choice of a metric is $d_\psi ^{(1)} = \sum _{j = 1}^N d_{\psi _j}$ and the question arises whether $d_\psi ^{(1)}$ has the volume doubling property ( with respect to λ ⁽ⁿ⁾) if each $d_{\psi _j}$ has the doubling property with respect to $\lambda ^{(n_j)}$? The metric balls with respect to $d_\psi ^{(1)}$ are not as easy to treat as the metric balls with respect to $d_{\psi }^{(\infty )} = \max _{1 \leq j \leq N} d_{\psi _j}$. It is helpful to note the following result from [30].

Proposition A.5

Let $d_\psi ^{(p)} = \left ( \sum _{j = 1}^N d_{\psi _j}^p \right )^{\frac {1}{p}}$ , 1 ≤ p < ∞, and let $d_\psi ^{(\infty )}$ be defined as above. If one of the metrics $d_\psi ^{(p)}$ , 1 ≤ p ≤∞ has the volume doubling property, then they all have the volume doubling property.

Corollary A.6

Suppose that each metric measure space $\left ( \mathbb {R}^{n_j}, d_{\psi _j}, \lambda ^{(n_j)} \right )$ , 1 ≤ j ≤ N, has the volume doubling property. Then the metric measure spaces $\left (\mathbb {R}^n, d_\psi ^{(p)}, \lambda ^{(n)} \right )$ , $\psi = \sum _{j = 1}^N d_{\psi _j}$ , $n = \sum _{j = 1}^N n_j$ , has the volume doubling property for all 1 ≤ p ≤∞.

Proof

We prove the doubling property for p = ∞ and Proposition A.5 will imply the result. Now we observe

$$\displaystyle \begin{aligned} \lambda^{(n)} \left( B^{d_\psi^{(\infty)}} (0, 2r) \right) &= \prod_{j = 1}^N \lambda^{(n_j)} \left( B^{d_{\psi_j}} (0, 2r) \right) \\ &\leq \prod_{j = 1}^N c_j \lambda^{(n_j)} \left( B^{d_{\psi_j}} (0, r) \right) \leq c \lambda^{(n)} \left(B^{d_\psi^{(\infty)}}(0, r) \right). \end{aligned} $$

□

Now let $q : \mathbb {R}^n \times \mathbb {R}^n \to \mathbb {R}$ be a continuous negative definite symbol, i.e. for all $x \in \mathbb {R}^n$ the function $q(x, \cdot ) : \mathbb {R}^n \to \mathbb {R}$ is negative definite and q as a function on $\mathbb {R}^n \times \mathbb {R}^n$ is continuous. Assume that for a fixed continuous negative definite function ψ we have the estimates

$$\displaystyle \begin{aligned} \kappa_0 \psi(\xi) \leq q(x, \xi) \leq \kappa_1 \psi(\xi) \end{aligned} $$

(A.9)

for all $x \in \mathbb {R}^n$, $\xi \in \mathbb {R}^n$, and 0 < κ ₀ ≤ κ ₁ are independent of x and ξ. Suppose that ψ satisfies our standard assumptions and the corresponding metric d _ψ has the volume doubling property. The following result taken from [30] is a first step to enable us to use “freezing the coefficients techniques” to investigate the pseudo-differential operator − q(x, D) and in the case it generates a sub-Markovian or Feller semigroup to study associated transition densities with the help of the metric d _q(x,⋅).

Proposition A.7

Let $\psi : \mathbb {R}^n \to \mathbb {R}$ be a fixed continuous negative definite function satisfying our standard conditions and let $q : \mathbb {R}^n \times \mathbb {R}^n \to \mathbb {R}$ be a continuous negative definite symbol. Further assume that uniformly in x the estimates (A.9) hold. For every $x \in \mathbb {R}^n$ , now fixed, $\left (\mathbb {R}^n, \left (q(x, \cdot ) \right )^{\frac {1}{2}}, \lambda ^{(n)} \right )$ is a metric measure space and the metrics $(q(x, \xi - \eta ))^{\frac {1}{2}}$ and $d_\psi (\xi , \eta ) = \psi ^{\frac {1}{2}} (\xi - \eta )$ are equivalent. Moreover, if for some γ > 0 we can find two constants 0 < c ₀ < c ₁ such that

$$\displaystyle \begin{aligned} c_0 r^\gamma \leq \lambda^{(n)} \left( B^{d_\psi}(0, r) \right) \leq c_1 r^\gamma \end{aligned}$$

holds, then for every $x \in \mathbb {R}^n$ the metric $(q(x, \cdot ))^{\frac {1}{2}}$ has the volume doubling property.

Example A.8

Choose 0 < α, β < 2 and ψ(ξ ₁, ξ ₂) = ∥ξ ₁∥^α + ∥ξ ₂∥^β, $\xi _1 \in \mathbb {R}^{n_1}$, $\xi _2 \in \mathbb {R}^{n_2}$. Further let $q : \mathbb {R}^{n} \times \mathbb {R}^{n} \to \mathbb {R}$ be a continuous negative definite symbol satisfying (A.9) with ψ as defined above. Since $\lambda ^{(n)} \left (B^{d_\psi }(0, r) \right ) = c r^{2\left ( \frac {n_1}{\alpha } + \frac {n_2}{\beta } \right )}$ we may apply Proposition A.7 to q(x, D).

For further results on the metric measure space $\left ( \mathbb {R}^n, d_\psi , \lambda ^{(n)} \right )$, we refer to [30] and [48]. We would like to mention once more that the Appendix is co-authored by J. Harris.