Topological preliminaries

For the convenience of our readers, we summarize a few topological preliminaries with references. We will work in a Polish space M (i.e., a completely metrizable topological space such that the metric defines the topology, and the topology has a countable base (second countable)). Being a metric space, M is also first countable (each point has a countable neighborhood basis), and it is separable (i.e., it contains a countable dense subset); cf. [Reference Engelking18, Corollary 1.3.8, Theorem 4.1.15]. We call M locally compact if each $x\in M$ has a relatively compact open neighborhood $U(x)$ . If a Polish space is locally compact, it is $\sigma $ -compact (or countable at infinity) (i.e., there is an increasing sequence of compact sets $K_n$ such that $\bigcup _{n\in \mathbb {N}} K_n = M$ ); cf. [Reference Engelking18, Theorem 4.1.15.iii)]. In a metric space, compact sets are closed, and $\sigma $ -compactness allows us to write any closed set as a countable union of compact sets. Therefore, the Borel $\sigma $ -algebra $\mathcal B(M)$ of a locally compact Polish space is generated by the compact sets. A Borel measure $\mu $ is a positive measure defined on the Borel sets $\mathcal B(M)$ . It is locally finite if each point $x\in M$ has an open neighborhood with finite measure. This implies that $\mu $ is finite on compact sets. Thus, a locally finite measure on a locally compact Polish space is $\sigma $ -finite.

Setting and assumptions

Throughout, we assume that M is a locally compact Polish space and $\mu $ a positive, locally finite Borel measure with full topological support. We do not assume that the space M is compact or that the measure $\mu $ is finite. On $L^2(M,\mu )$ , we consider a strongly continuous semigroup of bounded linear operators, $\left \{U_t: t \geqslant 0 \right \}$ and its adjoint semigroup $\left \{U_t^*: t \geqslant 0 \right \}$ , which is given by

$$ \begin{gather*} \int U_t f(x) \cdot \overline{g(x)}\,\mu(dx) = \int f(x)\cdot \overline{U_t^* g(x)}\,\mu(dx), \quad f,g \in L^2(M,\mu),\; t>0. \end{gather*} $$

Moreover, we assume that $U_t$ and $U_t^*$ are integral operators, which are given by a real-valued measurable kernel $u_t(x,y)$ :

(2.1) $$ \begin{align} U_t f(x) = \int u_t(x,y) f(y)\,\mu(dy), \qquad U_t^* g(y) = \int u_t(x,y) g(x) \,\mu(dx), \quad t>0. \end{align} $$

Moreover, we use the following assumptions:

Assumptions.

(A0) $$\begin{align} &\text{The linear operators } U_t : L^2(M,\mu)\to L^2(M,\mu), t>0, \text{ are compact;} \end{align}$$

(A1) $$\begin{align} &\text{For some } t_0> 0, \text{ the operator } U_{t_0} \text{ is positivity improving: for every positive }\notag\\ & f \in L^2(M,\mu), f\not\equiv 0, \text{ we have } U_{t_0} f(x) > 0 \text{ for } \mu\text{-almost all } x\in M. \end{align}$$

(A2) $$\begin{align} &\text{The semigroups } \left\{U_t: t \geqslant 0 \right\} \text{ and } \left\{U_t^*: t \geqslant 0 \right\} \text{ have the strong Feller property:}\notag\\ &\text{ for every } t>0 \text{ and } f \in L^{\infty}(M,\mu), \text{ we have } U_t f, U_{t}^*f \in C_b(M); \end{align}$$

(A3) $$\begin{align} &\text{For some } t_0>0, \text{ the operators } U_{t_0}, U_{t_0}^*: L^2(M,\mu) \to L^{\infty}(M,\mu) \text{ are bounded }\notag\\ &\text{(i.e.,ultracontractive):}\\ &\qquad\qquad\displaystyle\int u^2_{t_0}(\cdot,y) \,\mu(dy) \in L^{\infty}(M,\mu) \quad \text{and} \quad \int u^2_{t_0}(x,\cdot) \,\mu(dx) \in L^{\infty}(M,\mu).\notag \end{align}$$

Our results in Sections 4 and 5 need, when considering non-compact spaces M, a further strong decay condition.

(A4)

Let us comment on the assumptions ( A0)–( A4) and how they are related.

Remark 2.1. ( A0) is equivalent to assuming that the adjoint operators $U_t^*$ are compact on $(L^2(M,\mu ))^* = L^2(M,\mu )$ .

( A1) is equivalent to assuming that $U_{t_0}^*$ is positivity improving for some $t_0>0$ . Moreover, $U_{t_0}$ and $U_{t_0}^*$ are positivity improving if, and only if, the integral kernel $u_{t_0}(x,y)$ is for $\mu \otimes \mu $ almost all $(x,y)$ strictly positive.

( A2) is equivalent to the condition that and are bounded continuous functions of x. These functions will play a central role in our investigations.

( A3) implies, because of the reflexivity of the space $L^2(M,\mu )$ , that (the restrictions of the second duals to $L^1(M,\mu ) \subset \big (L^{\infty }(M,\mu )\big )^*$ of) the operators $U_{t_0}$ and $U^*_{t_0}$ are bounded operators from $L^1(M,\mu )$ to $L^2(M,\mu )$ . Due to the semigroup property, the operators $U_{2t_0}$ and $U^*_{2t_0}$ are bounded operators from $L^1(M,\mu )$ to $L^{\infty }(M,\mu )$ ; this is equivalent to saying

(2.2) $$ \begin{align} u_{2t_0}(\cdot,\cdot) \in L^{\infty}(M \times M, \mu \otimes \mu). \end{align} $$

Using the semigroup property, these boundedness properties extend to all $t \geqslant t_0$ , resp. $t \geqslant 2 t_0$ . Conversely, if there is some $t_0>0$ , for which (2.2) holds, then the condition ( A3) is also satisfied.

Let $t>2t_0$ . From ( A2), we know that $U_t$ and $U_t^*$ are continuous operators from $L^\infty (M,\mu )\to L^\infty (M,\mu )$ , and ( A3) means that $U_t$ and $U_t^*$ are continuous operators from $L^2(M,\mu )\to L^\infty (M,\mu )$ ; by duality, we see that $U_t$ and $U_t^*$ map $L^1(M,\mu )$ continuously into $L^2(M,\mu )$ , and using again ( A3) and the semigroup property, we conclude that $U_t$ and $U_t^*$ are continuous from $L^1(M,\mu )$ into $L^\infty (M,\mu )$ . Therefore, a standard interpolation argument shows that $U_t$ and $U_t^*$ are continuous $L^p(M,\mu )\to L^\infty (M,\mu )$ for any $p\in (1,\infty )$ and, in view of ( A2), we also have

(2.3) $$ \begin{align} U_t \big(L^p(M,\mu)\big), U_t^*\big(L^p(M,\mu)\big) \subseteq C_b(M), \quad t>2t_0, \; p \in [1,\infty]. \end{align} $$

Let A, and $A^*$ , denote the generators of semigroups $\left \{U_t: t \in T \right \}$ , and $\left \{U_t^*: t \geqslant 0 \right \}$ , respectively. A version of the Jentzsch theorem (see [Reference Schaefer50, Theorem V.6.6]) shows that $\lambda _0:= \inf \mathop {\mathrm {Re}} \big ( \operatorname {\mathrm {Spec}} (-A)\big ) = \inf \mathop {\mathrm {Re}} \big (\operatorname {\mathrm {Spec}} (-A^*)\big )$ is common isolated and simple eigenvalue of $-A$ and $-A^*$ . Moreover, the corresponding normalized eigenfunctions $\phi _0, \psi _0 \in L^2(M,\mu )$ can be assumed to be strictly positive. In particular,

(2.4) $$ \begin{align} U_t \phi_0 = e^{-\lambda_0 t} \phi_0 \quad\text{and}\quad U_t^* \psi_0 = e^{-\lambda_0 t} \psi_0, \quad t>0. \end{align} $$

The number $\lambda _0$ is called the ground state eigenvalue, and the functions $\phi _0, \psi _0$ , are the ground states of $-A$ , and $-A^*$ , respectively.

By (2.3), we have $\phi _0, \psi _0 \in C_b(M)$ , and both eigenequations in (2.4) are also true in a pointwise sense. We will frequently use the following constant

$$ \begin{gather*} \Lambda := \int \phi_0(x) \psi_0(x)\,\mu(dx) \end{gather*} $$

and set

(2.5) $$ \begin{align} \mathcal{N} := \left\{x \in M: \phi_0(x) = 0 \right\} \cup \left\{x \in M: \psi_0(x) = 0 \right\}. \end{align} $$

As mentioned above, $\mu (\mathcal {N})=0$ . Due to the continuity of $\phi _0$ and $\psi _0$ , the set $\mathcal {N}$ is a closed set. If for some $t>0$ , the kernel $u_t(x,y)$ is defined pointwise (e.g., if the map $(x,y) \mapsto u_t(x,y)$ is continuous) and strictly positive for all $x,y \in M$ , then $\mathcal {N} = \emptyset $ . We are mainly interested in examples which come from the theory of stochastic processes, and for these examples, $\mathcal {N} = \emptyset $ is always satisfied. Since the original question comes from the $L^2$ -theory, we do not want to assume a priori $\mathcal {N} = \emptyset $ .

The following result is a straightforward adaptation of a Lemma by Zhang, Li and Song [Reference Zhang, Li and Song65, Lemma 2.1]; it is a variant of the estimate which was originally obtained for intrinsic semigroups by Kim and Song [Reference Kim and Song28, Theorem 2.7]. The very first result in this direction is due to Pinsky [Reference Pinsky48, Theorem 3] for diffusion semigroups on bounded domains in $\mathbb {R}^d$ . The result of Zhang et al. is valid for rather general nonsymmetric semigroups. These authors use sub-Markovian semigroups of standard Markov processes with strictly positive kernels of Hilbert-Schmidt type satisfying ( A3). A close inspection of their proof reveals that only the compactness of the operators $U_t$ and $U^*_t$ is needed and that the result is valid in the more general setting, which is described by our present conditions ( A0)–( A3). Note that Zhang et al. use $t_0=1$ and that their proof requires $t/2> 1 = t_0$ .

Lemma 2.2 [Reference Zhang, Li and Song65, Lemma 2.1]

If ( A0)–( A3) hold for some $t_0>0$ , then there exist $C, \gamma> 0$ such that

(2.6) $$ \begin{align} \left|e^{\lambda_0 t} u_t(x,y) - (1/\Lambda)\phi_0(x) \psi_0(y) \right| \leqslant C e^{-\gamma t}, \quad t> 2t_0, \quad\text{for } \mu\otimes\mu \text{ a.a.} \ x,y\in M. \end{align} $$

This estimate leads to a rate of convergence (as $t\to \infty $ ) for compact semigroups, which can be seen as a variant of the classical result of Chavel and Karp [Reference Chavel and Karp7]; see also [Reference Simon59, Reference Keller, Lenz, Vogt and Wojciechowski27].

Unless otherwise stated, we take the same in ( A1), ( A3) and ( A4).

We will now briefly discuss the relation between ( A4) and ( A0).

Proof. We consider only the operators $U_t$ . The proof for $U^*_t$ is similar. For a compact set $K \subset M$ , we define

(i.e., $U_t^K$ is an integral operator with the kernel ). Observe that by ( A3),

$$ \begin{align*} \int_M\int_M u^K_t(x,y)^2 \,\mu(dy)\,\mu(dx) & = \int_K\int_M u_t(x,y)^2 \,\mu(dy)\,\mu(dx) \\ & \leqslant \mu(K) \cdot {\mathrm{ess\ sup}}_{x \in M} \int_M u_t(x,y)^2 \,\mu(dy) < \infty, \quad t \geqslant t_0, \end{align*} $$

which means that $U^K_t$ , $t \geqslant t_0$ , are Hilbert–Schmidt operators. Moreover, for $f \in L^2(M,\mu )$ , by the Cauchy–Schwarz inequality and the Tonelli theorem,

The assumptions ( A2) and ( A4) yield that for every $t> t_0$ and $\epsilon>0$ , we can find a compact set K such that

$$ \begin{gather*} \left\|U_t f - U^K_t f\right\|_2 \leqslant \epsilon \left\|f\right\|_2. \end{gather*} $$

Letting $\epsilon \to 0$ , we see that every operator $U_t$ , $t \geqslant t_0$ , is a strong limit of compact operators, and hence compact. This completes the proof.

If the operators $U_t$ , $t>0$ , are self-adjoint, then compactness for some $t>0$ implies compactness for every $t>0$ – this is due to spectral theorem. In that case, ( A4) implies ( A0), even if ( A3), ( A4) are true for some $t_0>0$ only. For non-self-adjoint operators, this is not clear. Therefore, in Sections 4 and 5, we have to assume both ( A0) and ( A4).

In unbounded spaces, it is often true that ( A0) and ( A4) (holding for all $t_0>0$ ) are equivalent see, for example, [Reference Kwaśnicki38, Lemma 1] and [Reference Kaleta and Kulczycki23, Lemma 9]. Some general sufficient conditions for compactness in various settings, including Markov, Schrödinger and Feynman–Kac semigroups, and more general perturbations can be found in [Reference Wang and Wu63, Reference Takeda, Tawara and Tsuchida61, Reference Lenz, Stollmann and Wingert40].

7.1. Feynman–Kac semigroups of Feller processes

Suppose that $(M,d)$ is a locally compact Polish space, and $\mu $ is a positive, locally finite measure with full topological support on the Borel sets $\mathcal {B}(M)$ . We assume that the space $(M,d)$ is unbounded (in particular, it is not compact) and the measure $\mu $ is not finite. Let $(X_t)_{t \geqslant 0}$ and $(\widehat {X}_t)_{t \geqslant 0}$ be Markov processes with values in M and the following transition semigroups:

$$ \begin{gather*} P_t f(x) = \mathbb{E}^x f(X_t) \quad\text{and}\quad \widehat P_t f(y) = \widehat{\mathbb{E}}^y f(\widehat{X}_t), \quad f \in L^2(M,\mu); \end{gather*} $$

Throughout this section, we assume that $P_t$ and $\widehat P_t$ are contractions on $L^2(M, \mu )$ and that the semigroups $\{P_t:t \geqslant 0\}$ , $\{\widehat P_t :t \geqslant 0\}$ have both the Feller and the strong Feller property (i.e., they are ‘doubly Feller’ in the sense of Chung [Reference Chung11]). Moreover, we assume that the corresponding Markov resolvents are in a weak duality relation with respect to the measure $\mu $ ; see [Reference Chung and Walsh12, Definition 13.1]. By [Reference Chung and Walsh12, Proposition 13.6], we have

$$ \begin{gather*} \int f(x) \cdot P_t g(x)\,\mu(dx) = \int \widehat P_t f(x) \cdot g(x)\,\mu(dx), \quad f,g \in L^2(M,\mu),\; t>0. \end{gather*} $$

This means that $\widehat P_t$ is the (functional analytic) adjoint operator $P_t^*$ of $P_t$ in the space $L^2(M,\mu )$ . Finally, we assume that for every $t>0$ , the operators $P_{t}$ and $\widehat P_{t}$ are positivity improving and bounded from $L^2(M,\mu )$ to $L^{\infty }(M,\mu )$ .

In particular, the ‘free’ semigropus $\{P_t, t\geqslant 0\}$ and $\{\widehat P_t, t\geqslant 0\}$ are such that they satisfy our assumptions ( A1)–( A3). We are now going to give sufficient criteria such that the corresponding Feynman–Kac semigroups $\{U_t, t\geqslant 0\}$ and $\{U_t^*, t\geqslant 0\}$ satisfy ( A0)–( A4).

Let V be a locally bounded potential on M which is bounded below, that is, $V = V_{+}-V_{-}$ with $V_{+} \in L^{\infty }_{\mathrm {loc}}(M,\mu )$ and $V_{-} \in L^{\infty }(M,\mu )$ , for which we define the Feynman–Kac semigroup

$$ \begin{gather*} U_t f(x) = \mathbb{E}^x \left[e^{-\int_0^t V(X_s)\, ds} f(X_t) \right], \quad f \in L^2(M,\mu), \; t>0. \end{gather*} $$

By [Reference Chung and Walsh12, Theorem 13.25], the weak dual or adjoint semigroup consists of operators

$$ \begin{gather*} \widehat U_t f(x) = U^*_t f(x) = \widehat{\mathbb{E}}^x \left[e^{-\int_0^t V(\widehat X_s)\,ds} f(\widehat X_t) \right], \quad f \in L^2(M,\mu), \; t>0. \end{gather*} $$

We first verify the assumptions ( A1)–( A3) and show that $U_t$ and $U^*_t$ are integral operators. If $(X_t)_{t\geqslant 0}$ is a stochastic process, we denote by $\tau _B := \inf \left \{t>0 \,:\, X_t\in B^c\right \}$ the first exit time from the set B.

Proof. a) Let $0\not \equiv f \in L^2(M,\mu )$ be positive and $t, r>0$ . We have

The expectation appearing on the right-hand side tends to $\mathbb {E}^x f(X_t) = P_tf(x)$ as $r \to \infty $ . Since $P_t$ is positivity improving, and V is locally bounded, the expression on the right-hand side is strictly positive for $\mu $ -almost every x and sufficiently large $r>0$ . To see this, fix any x such that $P_t f(x)>0$ . Since $\mathbb {E}^x \left [f(X_t); t < \tau _{B_r(x)}\right ]$ increases to $P_t f(x) = \mathbb {E}\left [f(X_t)\right ]$ as $r\uparrow \infty $ , there is some $r=r(x)$ such that $\mathbb {E}^x \left [f(X_t); t < \tau _{B_r(x)}\right ]>0$ ; hence, $U_t f(x)>0$ . This gives the assertion.

Part b) follows directly from Chung [Reference Chung11, Theorem 2]. The assumptions (a)–(c) of Chung’s theorem can be easily verified as V is both bounded below and locally bounded.

We still have to prove c). First, observe that for every $t>0$ , the operators $U_t, U^*_t: L^2(M,\mu ) \to L^{\infty }(M,\mu )$ are bounded. Indeed, for $f \in L^2(M,\mu )$ and $t>0$ , we have

$$ \begin{gather*} U_t f(x) = \mathbb{E}^x \left[e^{-\int_0^t V(X_s) ds} f(X_t) \right] \leqslant e^{t \left\|V_{-}\right\|_{\infty}} \mathbb{E}^x f(X_t) = e^{t \left\|V_{-}\right\|_{\infty}} P_t f(x), \end{gather*} $$

implying that $\left \|U_t\right \|_{2,\infty } \leqslant e^{t \left \|V_{-}\right \|_{\infty }} \left \|P_t\right \|_{2,\infty }$ . In the same way, we get $\left \|U^*_t\right \|_{2,\infty } \leqslant e^{t \left \|V_{-}\right \|_{\infty }} \left \|P^*_t\right \|_{2,\infty }$ . Combining this with part b) and the semigroup property, we see that for every $t>0$ , the operators $U_t, U^*_t$ map $L^2(M,\mu )$ $\left \{P_t, t\geqslant 0\right \}$ , $\left \{\widehat P_t, t\geqslant 0\right \}$ into $L^2(M,\mu ) \cap C_b(M)$ , and we can apply [Reference Jacob and Schilling21, Theorem 1.6] to show that $U_t, U^*_t$ , $t>0$ , are integral operators. That is, there exist non-negative measurable kernels $u_t(x,y)$ and $u_t^*(x,y)$ such that

$$ \begin{gather*} U_t f(x) = \int_M f(y) u_t(x,y) \mu(dy), \quad U^*_t f(x) = \int_M f(y) u^*_t(x,y) \mu(dy), \quad f \in L^2(M,\mu), \; t>0. \end{gather*} $$

Since the operators $U_t$ and $U^*_t$ are adjoint, we have $u^*_t(x,y) = u_t(y,x)$ .

Now we give general sufficient conditions for the compactness of the operators $U_t$ , $U^*_t$ , $t>0$ (cf. assumption ( A0)) and for the finiteness of the heat content for large times. Recall that the heat content is defined as

Lemma 7.2. For $x \in M$ and $t>0$ , we have

(7.1)

as well as similar estimates for . In particular,

(7.2)

for $t>0$ . Consequently, the following assertions hold.

1. a) If

   (7.3) $$ \begin{align} \lim_{d(x,x_0) \to \infty} V(x) = \infty, \end{align} $$
   then the operators $U_t$ , $U^*_t$ , $t>0$ , are compact; in particular, ( A0) holds.

2. b) Fix $t>0$ . Then

   $$ \begin{gather*} \int_{M} e^{-t V(x)}\mu(dx) < \infty \implies Z(t)<\infty. \end{gather*} $$

   However, if $\inf _{x \in M} \mathbb {P}^x(t \leqslant \tau _{B_1(x)})>0$ , then

   

Proof. Fix $t>0$ . For the proof of the upper bound in (7.1), it is enough to use Jensen’s inequality and Tonelli’s theorem

and the upper estimate in (7.2) follows by integration with respect to $\mu (dx)$ and duality. The lower bounds in (7.1) and (7.2) follow from the definitions. Observe that Part b) follows directly from the latter inequality.

To get Part a), we show ( A4) for every $t_0>0$ , and then we use Lemma 2.3. By (7.3), there exists a function $g \in C_{\infty }(M)$ such that

(7.4) $$ \begin{align} e^{-tV(x)} \leqslant g(x),\quad\text{for } \mu \text{ almost every } x\in M. \end{align} $$

Footnote ²Hence, by (7.1) and (7.4),

By the Feller property, the function $x \mapsto (1/t)\int _0^t P_s g (x)$ ds belongs to $C_{\infty }(M)$ , which implies that ( A4) holds for any $t\geqslant t_0$ and any $t_0>0$ .

The proof of ( A4) for adjoint operators $U_t^*$ is similar.

In particular, Lemma 7.2.b) shows that the finiteness of the heat content is completely independent of the free process.

We will now discuss some concrete classes of Feller processes which are covered by the framework described above. In order to keep the exposition short, we give in each case standard references for the notation and further results.

7.1.1. Lévy processes

Standard references: Sato [Reference Sato49], Jacob [Reference Jacob20, Vol. 1], Schilling [Reference Schilling51]. Throughout this section, the reference measure $\mu $ is d-dimensional Lebesgue measure and $d(\cdot ,\cdot )$ is the Euclidean distance.

A Lévy process $(X_t)_{t\geqslant 0}$ is a stochastic process on $\mathbb {R}^d$ with càdlàg (right-continuous with finite left limits) paths and stationary and independent increments. A Lévy process can be described in terms of its characteristic exponent $\psi (\xi ) = -\log \mathbb {E} e^{i\xi \cdot X_1}$ . It is known that $\psi $ is uniquely determined by the Lévy–Khintchine formula

(7.5) $$ \begin{align} \psi(\xi) = - i \xi \cdot b + \frac 12 \xi \cdot Q \xi + \int_{\mathbb{R}^d \setminus \left\{0\right\}} \left(1 - e^{i \xi \cdot y} + i \xi \cdot y \mathbb{1}_{(0,1)}(|y|)\right)\nu(dy) , \quad \xi \in \mathbb{R}^d, \end{align} $$

or by the corresponding the Lévy triplet $(b,Q,\nu )$ , which is given by $b \in \mathbb {R}^d$ (drift term), Q is a symmetric, positive semidefinite $d \times d$ matrix (Gaussian covariance matrix) and $\nu $ is a measure on $\mathbb {R}^d \setminus \left \{0\right \}$ such that $\int _{\mathbb {R}^d \setminus \left \{0\right \}} (1 \wedge |x|^2) \nu (dx) < \infty $ (Lévy measure).

Lévy processes are Markov processes, which are invariant under translations; that is, the transition semigroup is a convolution semigroup of the form

$$ \begin{gather*} P_t f(x) = \mathbb{E}^x \left[f(X_t)\right] = \mathbb{E}^0\left[f(X_t+x)\right] = f*\widetilde\mu_t(x), \end{gather*} $$

where $\widetilde \mu _t(dy) = \mathbb {P}^0(X_t \in -dy)$ is the law of $-X_t$ . The dual process $(\widehat {X}_t)_{t \geqslant 0}$ of a Lévy process $(X_t)_{t \geqslant 0}$ is given by $\widehat {X}_t = - X_t$ , and its transition semigroup $\big \{ \widehat P_t:t \geqslant 0\big \}$ is defined accordingly. The convolution structure of $P_t$ shows that $(P_t)_{t\geqslant 0}$ is both a Feller semigroup on $C_\infty (\mathbb {R}^d)$ and an $L^p(\mathbb {R}^d,dx)$ Markov semigroup for any $p\in [1,\infty )$ .

The process $(X_t)_{t \geqslant 0}$ is a strong Feller process, that is, $P_t$ maps $L^{\infty }(\mathbb {R}^d,dx)$ into $C_b(\mathbb {R}^d)$ , if and only if $\mu _t(dy) = p_t(y)\,dy$ is absolutely continuous w.r.t. Lebesgue measure; cf. Jacob [Reference Jacob20, Vol. 1, Lemma 4.8.19, 4.8.20]. In particular, the dual process is strong Feller, too. A sufficient condition for the existence of the densities $p_t(x)$ , $t>0$ is that $Q \neq 0$ or that $\nu $ is infinite and absolutely continuous with respect to Lebesgue measure; see, for example, [Reference Sato49, Theorem 27.7], and see also the discussion in [Reference Knopova and Schilling32]. If, in addition, there is some $t_0>0$ such that $\int _{\mathbb {R}^d} e^{-t_0 \mathrm {Re} \psi (\xi )}\, d\xi < \infty $ , then $\sup _{t \geqslant t_0} \sup _{x \in \mathbb {R}^d} p_t(x) < \infty $ , thanks to the Fourier inversion formula. Consequently, the operators $P_t, \widehat P_t:L^2(\mathbb {R}^d,dx) \to L^{\infty }(\mathbb {R}^d,dx)$ are bounded for every $t \geqslant t_0$ . In general, if the integral $\int _{\mathbb {R}^d} e^{-t_0 \mathrm {Re} \psi (\xi )}\, d\xi $ is finite for every $t_0>0$ , then we get the existence of bounded and jointly (in $(t,x)$ ) continuous densities for every $t>0$ ; see [Reference Knopova and Schilling32, Lem. 2.1].

In order to see that $P_t$ and $\widehat P_t$ are positivity improving, it is enough to show that $\mathbb {P}^0(X_t \in B) = \mathbb {P}^0(\widehat {X}_t \in -B) = \int _B p_t(x)\,dx>0$ for every Borel set $B \subset \mathbb {R}^d$ with $\mathrm {Leb}(B)>0$ . Let us fix such a Borel set B. Using the Lévy–Khintchine formula, we can decompose a Lévy process into a sum of independent Lévy processes $X_t = W_t + Y_t$ whose triplets are $(b,Q,0)$ and $(0,0,\nu )$ . If $\mathrm {det}(Q)\neq 0$ , then we always have a nondegenerate Gaussian part, and it is clear that

$$ \begin{gather*} \mathbb{P}^0(X_t\in B) = \int \mathbb{P}^0(W_t + y\in B) \,\mathbb{P}^0(Y_t\in dy)> 0 \end{gather*} $$

since $\mathbb {P}^0(W_t + y \in B)>0$ for all $y\in \mathbb {R}^d$ .

Let us assume that $\mathrm {det}(Q)=0$ . If the Lévy measure $\nu $ satisfies $\nu (dy) \geqslant h(y)\,dy$ with a density $h(y)$ which is strictly positive in some neighborhood of $0\in \mathbb {R}^d$ , say $B_\epsilon (0)$ , then we can decompose $X_t = Z_t + C_t$ into two independent Lévy processes whose triplets are $(b,Q,\nu (dy)-h(y)\,dy)$ and $(0,0,h(y)\,dy)$ . Since $C_t$ is a compound Poisson process, we know from [Reference Schilling51, Cor. 3.5] that

$$ \begin{gather*} \mathbb{P}^0(C_t \in B) = e^{-h t}\delta_0(B) + e^{-h t} \sum_{k=1}^\infty \frac{t^k}{k!} \int_B h^{*k}(x)\,dx, \quad h = \int_{\mathbb{R}^d} h(x)\,dx. \end{gather*} $$

Since the k-fold convolution $h^{*k}(x)$ is strictly positive in $B_{\epsilon k}(0)$ , we conclude that ${\mathbb {P}^0(C_t \in B+z)>0}$ for any $z\in \mathbb {R}^d$ ; hence, $\mathbb {P}^0(X_t\in B) = \int \mathbb {P}^0(C_t + z\in B)\,\mathbb {P}^0(Z_t\in dz)>0$ . With a little more effort, the above argument even shows that the density of $X_t$ satisfies $p_t(x)>0$ for all $t>0$ and Lebesgue a.a. $x\in \mathbb {R}^d$ .

It remains to check the conditions for compactness and the finiteness of the heat content stated in Lemma 7.2. Let $x_0 = 0$ . Choosing the potential V in such a way that $V(x) \to \infty $ as $|x| \to \infty $ , we obtain that the Feynman–Kac semigroup operators $U_t, \widehat U_t$ , $t>0$ , are compact operators on $L^2(M,\mu )$ . Moreover, if we know that

$$ \begin{gather*} \liminf_{|x| \to \infty} \frac{V(x)}{\log|x|}> 0, \end{gather*} $$

then there is $t>0$ such that $Z(t)<\infty $ .

7.1.2. Levy-type processes

Standard references: Jacob [Reference Jacob20, Vol. 1], Böttcher et al. [Reference Böttcher, Schilling and Wang5]. A Lévy-type or Feller process is a Markov process $X_t$ with values in $\mathbb {R}^d$ which behaves locally like a Lévy process. In particular, the transition semigroup $P_tu(x) = \mathbb {E}^x u(X_t)$ is a Feller semigroup, and one can show that the infinitesimal generator is a pseudo-differential operator of the form $Au(x) = \mathcal -F^{-1}_{\xi \to x} \left (p(x,\xi )\mathcal Fu(\xi )\right )$ ( $\mathcal F$ denotes the Fourier transform, u is a sufficiently regular function – for example, a test function $u\in C_c^\infty (\mathbb {R}^d)$ ) whose symbol $p(x,\xi )$ is, for fixed x, the characteristic exponent of a Lévy process. Therefore, $p(x,\xi )$ is given by a Lévy–Khintchine formula (7.5) where the Lévy-triplet depends on x (i.e. $(b,Q,\nu (dy)) = (b(x),Q(x),\nu (x,dy))$ ). Thus, A is a ‘Lévy generator with variable coefficients’. A typical example are stable-like generators of the form $Au(x) = (-\Delta )^{\alpha (x)}u(x)$ where $\alpha :\mathbb {R}^d \to [0,2]$ is a variable order of differentiability and the symbol is of the form $p(x,\xi ) = |\xi |^{\alpha (x)}$

It is a major problem to construct a Lévy-type process given an admissible symbol $p(x,\xi )$ or, which is the same, an x-dependent Lévy triplet. A thorough discussion is in Jacob [Reference Jacob20, Vol. 3] and Böttcher et al. [Reference Böttcher, Schilling and Wang5]; probably the most general results to-date are due to Kühn [Reference Kühn34] and Knopova et al. [Reference Knopova, Kulik and Schilling33].

In what follows, we indicate some conditions, under which the assumptions needed in Section 7.1 are satisfied. Our main intention is not to cover the most general situation, but to illustrate that within the class of Lévy-type processes, there are many examples to be found. The following assumptions on the symbol are frequently used:

(7.6) $$ \begin{align} C_1 |\xi|^{\alpha_0} &\leqslant \mathrm{Re} p(x,\xi) &&\forall x\in\mathbb{R}^d,\; |\xi|\geqslant 1, &&\text{(minimal growth)} \end{align} $$

(7.7) $$ \begin{align} |p(x,\xi)-p(y,\xi)| &\leqslant C_2 |x-y|^\gamma |\xi|^\alpha &&\forall x\in\mathbb{R}^d,\; |\xi|\geqslant 1, &&\text{(H\"{o}lder condition)} \end{align} $$

(7.8) $$ \begin{align} |p(x,\xi)| &\leqslant C_3 (1+|x|^\gamma) |\xi|^\beta &&\forall x\in\mathbb{R}^d,\; |\xi|\leqslant 1, &&\text{(growth of coefficients)} \end{align} $$

where $C_1,C_2,C_3$ are constants and $\alpha _0,\alpha ,\beta \in (0,2]$ and $\gamma \in (0,\beta )$ are suitable exponents.

Assume that $p(x,\xi )$ is the symbol of a Lévy-type process. If either ${\alpha \kern1.3pt{\geqslant}\kern1.3pt \gamma\kern1.3pt{>}\kern1.3pt \frac {\alpha }{\alpha _0}\left (\alpha \kern1.3pt{-}\kern1.3pt\alpha _0\kern1.3pt{+}\kern1.3pt\frac 12\right )}$ or $\alpha \leqslant \gamma < \frac 2\alpha _0-\frac 12$ , then $X_t$ has a transition density $p_t(x,y)$ such that $\sup _x \int p_t^2(x,y)\,dy<\infty $ (i.e., ( A3) holds for the free semigroup $P_t$ ; see [Reference Kühn34, Thm. 2.14]).

The existence of a Lévy-type process with a given symbol $p(x,\xi )$ is a much more delicate issue. Here, we content ourselves to mentioning that in the stable-like case $p(x,\xi ) = |\xi |^{\alpha (x)}$ where $\alpha : \mathbb {R}^d \to [a_1,a_2] \subset (0,2)$ is Hölder continuous, the existence of a stable-like process with a continuous density $(t,x,y)\mapsto p_t(x,y)> 0$ has been established in [Reference Kühn34, Thm. 3.8, Thm. 3.11] as well as in [Reference Knopova, Kulik and Schilling33, Section 3]. In particular, we get Feller and strong Feller processes. It is also clear that stable-like symbols satisfy the assumptions (7.6)–(7.8). We would like to mention that the methods presented in these papers go way beyond the stable-like case; [Reference Kühn34] covers symbols of the type $p(x,\xi ) = \psi _{a(x)}(|\xi |)$ where $z\mapsto \psi _a(z)$ has a holomorphic extension on a bow-tie shaped domain of the complex plane (and satisfies certain polynomial boundedness assumptions), while [Reference Knopova, Kulik and Schilling33] are mainly interested in non-rotationally symmetric symbols. Therefore, ( A1) and ( A2) are also satisfied.

In general, it is very difficult to determine the symbol of the (formal) adjoint to A. Building on [Reference Schilling and Uemura53], the paper [Reference Schilling and Wang54] contains the general form of the symbol of the adjoint operator for a wide class of symbols (Section 3) and a fully-worked out discussion of the case $p(x,\xi )=|\xi |^{\alpha (x)}$ . If $\alpha (x)$ is Hölder, takes values in ${[a_1,a_2]\subset (0,2)}$ and satisfies (7.6)–(7.8), then so does the adjoint symbol $p^*(x,\xi )$ which is given by a Lévy–Khintchine formula with triplet $(0,0,w(x)|x-y|^{-d-\alpha (x)}\,dy)$ for the weight ${w(x) = \alpha (x)2^{\alpha (x)-1}\Gamma \left (\frac 12\alpha (x)+\frac 12d\right )\big /\Gamma \left (1-\frac 12\alpha (x)\right )}$ . This means that the assumptions ( A1)–( A3) also hold for the adjoint free semigroup $\widehat P_t$ in this case. Of course, other symbols require case-by-case investigations (e.g., based on the above-mentioned literature).

7.1.3. Continuous-time Markov chains on a countably infinite uniformly discrete metric space

Standard references: Murugan and Saloff–Coste [Reference Murugan and Saloff-Coste46, Reference Murugan and Saloff-Coste47] and Cygan, Kaleta and Sliwiński [Reference Cygan, Kaleta and Śliwiński14]. Let $(M,d)$ be a countably infinite, unbounded, complete metric space, which is also uniformly discrete; this means that there exists a number $a>0$ such for any two distinct points $x,y \in M$ , one has $d(x,y) \geqslant a$ ; see [Reference Murugan and Saloff-Coste46, Reference Murugan and Saloff-Coste47]. The underlying measure $\mu :M \to (0,\infty )$ is assumed to be a Frostman measure with exponent $d_M>0$ (i.e., there is a constant $c>0$ such that $\mu (B_r(x)) \leqslant c r^{d_M}$ for any $x \in M$ and $r>0$ ).

In particular, we have for all $\gamma>d_M$ ,

$$ \begin{align*} \int_{B_1(x_0)^c} d(x,x_0)^{-\gamma}\, \mu(dx) & = \sum_{n=0}^{\infty} \int_{2^n \leqslant d(x,x_0) < 2^{n+1}} d(x,x_0)^{-\gamma}\, \mu(dx) \\ & \leqslant \sum_{n=0}^{\infty} 2^{-\gamma n} \mu(B_{2^{n+1}}(x_0)) \leqslant c 2^{d_M}\sum_{n=0}^{\infty} 2^{-(\gamma-d_M) n} < \infty. \end{align*} $$

Under these assumptions, we can give a sufficient condition for the integrability of the function $x \mapsto e^{-tV(x)}$ which is needed for the finiteness of the heat content; see Lemma 7.2.b). Indeed, if there exist $c_0, R>0$ and $x_0 \in M$ such that

$$ \begin{gather*} \frac{V(x)}{\log d(x,x_0)} \geqslant c_0>0 , \quad d(x,x_0) \geqslant R, \end{gather*} $$

then for every $t> d_M/c_0$ , we have

$$ \begin{gather*} \int_M e^{-tV(x)} \mu(dx) \leqslant e^{\left\|V_-\right\|_{\infty}} \mu(B_R(x_0)) + \int_{B_R(x_0)^c} d(x,x_0)^{-c_0t} \mu(dx) < \infty. \end{gather*} $$

Consider two probability kernels $Q, \widehat Q:M \times M \to [0,1]$ , $\sum _{y \in M} Q(x,y) = 1=\sum _{y \in M} \widehat Q(x,y)$ , $x \in M$ , which are connected via the duality relation with respect to the measure $\mu $ – that is,

(7.9) $$ \begin{align} \mu(x) Q(x,y) = \mu(y) \widehat Q(y,x), \quad x,y \in M. \end{align} $$

Thus, there are two time-homogeneous Markov chains $\{Z_n: n \in \mathbb {N}_0\}$ , $\{\widehat Z_n: n \in \mathbb {N}_0\}$ with values in M and one-step transition probabilities given by Q and $\widehat Q$ , respectively. We denote by $\mathbb {P}^x$ , $\widehat {\mathbb {P}}^x$ the laws of the chains starting at $x \in M$ – that is,

$$ \begin{align*} \mathbb {P}^x (Z_n=y) = \mathbb {P}(Z_n=y \mid Z_0=x) \quad \text{and}\quad \widehat {\mathbb {P}}^x (\widehat Z_n=y) = \mathbb {P}(\widehat Z_n=y \mid \widehat Z_0=x) .\end{align*} $$

We have

$$ \begin{align*} \mathbb {P}^x (Z_{n}=y) = Q_n(x,y) \quad\text{and}\quad \widehat {\mathbb {P}}^x (\widehat Z_{n}=y) = \widehat Q_n(x,y), \end{align*} $$

where , $Q_1(x,y)=Q(x,y)$ and ${Q_{n+1}(x,y) = \sum _{z \in M} Q_n(x,z)Q_1(z,y)}$ , ${n \geqslant 1}$ , and $\widehat Q_n$ ’s are defined accordingly. The densities with respect to $\mu $ are given by

$$ \begin{gather*} q_n(x,y) = \frac{\mathbb{P}^x (Z_{n}=y)}{\mu(y)} = \frac{Q_n(x,y)}{\mu(y)} \quad \text{and} \quad \widehat q_n(x,y) = \frac{\widehat{\mathbb{P}}^x (\widehat Z_{n}=y)}{\mu(y)} = \frac{\widehat Q_n(x,y)}{\mu(y)}. \end{gather*} $$

Clearly, (7.9) extends to $\mu (x) Q_n(x,y) = \mu (y) \widehat Q_n(y,x)$ , $x,y \in M$ , $n \in \mathbb {N}$ , and we have $\widehat q_n(x,y) = q_n(y,x)$ . The corresponding n-step transition operator is

$$ \begin{gather*} Q_n f(x) = \sum_{y \in M} Q_n(x,y) f(y) = \sum_{y \in M} q_n(x,y) f(y) \mu(y) \end{gather*} $$

whenever the series is finite; $\widehat Q_n f$ is defined in a similar way.

The main objects are the continuous-time chains $\{X_t: t \geqslant 0\}$ and $\{\widehat X_t: t \geqslant 0\}$ defined by $X_t:= Z_{N_t}$ , $\widehat X_t:= \widehat Z_{N_t}$ , respectively, where $\{N_t: t \geqslant 0\}$ is an independent Poisson process with parameter $\lambda =1$ . Clearly, we have $\mathbb {P}^x(X_t \in A) = \sum _{y \in A} P_t(x,y)$ , $A \subset M$ , where

$$ \begin{gather*} P_t(x,y) = e^{-t}\sum_{n=0}^{\infty} \frac{t^n}{n!} Q_{n}(x,y), \quad t>0, \ x \in M. \end{gather*} $$

The transition semigroup $\{P_t: t \geqslant 0\}$ of this process is defined as

$$ \begin{gather*} P_t f(x) = \sum_{y \in M} f(y) P_t(x,y) = \sum_{y \in M} f(y) p_t(x,y) \mu(y), \quad \text{with} \quad p_t(x,y):=\frac{P_t(x,y)}{\mu(y)}, \end{gather*} $$

for all admissible functions f on M. The transition probabilities $\widehat P_t(x,y)$ , kernels $\widehat p_t(x,y)$ and the transition semigroup $\{\widehat P_t: t \geqslant 0\}$ of the process $\{\widehat {X}_t: t \geqslant 0\}$ are defined accordingly. By the Cauchy–Schwarz inequality, we can easily see that all operators $P_t$ and $\widehat {P}_t$ are contractions on $L^2(M,\mu )$ ; again, we have $\widehat p_t(x,y) = p_t(y,x)$ , which means that the weak duality relation, which was described at the beginning of Section 7.1, is in force. Moreover, it is straightforward to see that both transition semigroups are doubly Feller. In particular, the strong Feller property follows from the fact that $(M,d)$ is uniformly discrete. One can also check that the ultracontractivity of these semigroups is equivalent to the boundedness of the kernel $q_1(x,y)$ (e.g., if $\inf _{x \in M} \mu (x)>0$ ). The positivity improving property follows from the strict positivity of the kernels $P_t(x,y)$ and $\widehat P_t(x,y)$ . This is the case if the chains $\{Z_n: n \in \mathbb {N}_0\}$ , $\{\widehat Z_n: n \in \mathbb {N}_0\}$ are irreducible (i.e., for every $x,y \in M$ , there is $n=n(x,y) \in \mathbb {N}$ such that $P_n(x,y), \widehat P_n(x,y)>0$ ).

We can now explicitly estimate the space-rates of the quasi-ergodic behavior of the Feynman–Kac semigroups with confining potentials (see Corollary 3.5). These rates take the form and , for some $t_1>0$ . The functions and can be estimated by a constant or by using the upper bound in (7.1). Sharper estimates for a large class of processes will be provided in [Reference Cygan, Kaleta, Schilling and Śliwiński13]. However, sharp lower estimates for the ground states for a fairly general class of processes and rather general confining potentials can be found in [Reference Cygan, Kaleta and Śliwiński14].

If we have good lower estimates for the ground states, then we can apply Lemma 4.5 and Theorem 4.6 to derive the progressive GSD and progressive exponential quasi-ergodicity from the finiteness of the heat content for large times. Such estimates can, for example, be found in [Reference Cygan, Kaleta and Śliwiński14].

7.1.6. Sobordinate processes

Standard references: Sato [Reference Sato49], Schilling, Song and Vondraček [Reference Schilling, Song and Vondraček52]. Another class of processes which is covered by our results are subordinate processes. Recall that a subordinator is an increasing Lévy process with values in $[0,\infty )$ , starting at $0$ . It is uniquely determined by its Laplace transform: we have $\mathbb {E} e^{-\lambda S_t} = e^{- t \phi (\lambda )}$ , $t>0$ , where the exponent $\phi :(0,\infty ) \to [0,\infty )$ is a Bernstein function of the form

$$ \begin{gather*} \phi(\lambda) = b \lambda + \int_0^{\infty} (1- e^{-\lambda u}) \,\rho(du). \end{gather*} $$

Here, $b \geqslant 0$ is the drift term and $\rho $ is a measure on $(0,\infty )$ such that $\int _0^{\infty } (1 \wedge u) \rho (du) < \infty $ (Lévy measure).

Let $\{X_t: t \geqslant 0\}$ and $\{\widehat X_t: t \geqslant 0\}$ be Markov processes as in Section 7.1, and let $\{S_t: t \geqslant 0\}$ be a subordinator which is independent of $\{X_t: t \geqslant 0\}$ and $\{\widehat X_t: t \geqslant 0\}$ . We define the subordinate processes $\{X^{\phi }_t: t \geqslant 0\}$ , $\{\widehat X^{\phi }_t: t \geqslant 0\}$ by the formulas $X^{\phi }_t:= X_{S_t}$ , $\widehat X^{\phi }_t:= \widehat X_{S_t}$ , respectively. By independence, their transition semigroups $\{P^{\phi }_t: t \geqslant 0\}$ and $\{\widehat P^{\phi }_t: t \geqslant 0\}$ are given by

$$ \begin{gather*} P^{\phi}_t f(x) = \int_{[0,\infty)} P_s f(x) \,\mathbb{P}(S_t \in ds), \quad \widehat P^{\phi}_t f(x) = \int_{[0,\infty)} \widehat P_s f(x) \,\mathbb{P}(S_t \in ds), \end{gather*} $$

for $f \in L^2(M,\mu )$ or $f \in L^{\infty }(M,\mu )$ . It is obvious that these formulas define contractions on $f \in L^2(M,\mu )$ , and the weak duality relation described in Section 7.1 holds. Moreover, these semigroups inherit the Feller property, the strong Feller property (provided that $S_t$ is not a compound Poisson subordinator), and the positivity improving property from the semigroups $\{P_t: t \geqslant 0\}$ and $\{\widehat P_t: t \geqslant 0\}$ . The only regularity property that requires an additional assumption is ultracontractivity: Suppose that $P_t, \widehat P_t$ are bounded operators from $L^2(M,\mu )$ to $L^{\infty }(M,\mu )$ for every $t>0$ . By Remark 2.1, this is equivalent to the boundedness from $L^1(M,\mu )$ to $L^{\infty }(M,\mu )$ , for any $t>0$ . Therefore, ultracontractivity of the subordinate semigroup and its dual is ensured by the assumption that

$$ \begin{gather*} \left\|P^{\phi}_t\right\|_{1,\infty} \leqslant \int_{[0,\infty)} \left\|P_s\right\|_{1,\infty} \mathbb{P}(S_t \in ds) < \infty, \quad t>0. \end{gather*} $$

7.2. Progressive uniform exponential quasi-ergodicity for the harmonic oscillator

Let $H= -\Delta +|x|^2$ be the Schrödinger operator acting in $L^2(\mathbb {R}^d,dx)$ , and denote by $U_t = e^{-tH}$ , $t \geqslant 0$ , the operators of the corresponding Schrödinger semigroup. It is well known that, for every $t>0$ , $U_t$ is a compact, self-adjoint and positivity-improving integral operator with the kernel

(7.10) $$ \begin{align} u_t(x,y) = (2 \pi \sinh(2t))^{-d/2} \exp\left(-\frac{1}{4}\left(\tanh(t)|x+y|^2+ \coth(t)|x-y|^2\right)\right) \end{align} $$

(see, for example, [Reference Stempak and Torrea60, (1.4)]. In particular, the assumptions ( A0)–( A3) are satisfied. The ground state eigenfunction and eigenvalue are given by

(7.11) $$ \begin{align} \phi_0(x) = \pi^{-d/4} e^{-\frac{|x|^2}{2}} \quad \text{and} \quad \lambda_0 = d, \end{align} $$

and (see [Reference Stempak and Torrea60, Proposition 3.3]Footnote ³)

(7.12)

It is easy to check that the semigroup $\left \{U_t: t \geqslant 0 \right \}$ is not aGSD; cf. [Reference Kaleta, Kwaśnicki and Lőrinczi24, Example 4.4]; in particular, using Corollary 4.4, it is not uniformly quasi-ergodic (see also the discussion in [Reference Knobloch and Partzsch31, Section 4.3]). We will show that our present results can be directly used to describe the true quasi-ergodic properties of this semigroup.

First of all, observe that (7.12) shows that the heat content is finite for every $t>0$ , so that our results from Section 3 apply. In particular, Theorem 3.4 ensures that the semigroup $\left \{U_t: t \geqslant 0 \right \}$ is exponentially quasi-ergodic for every finite initial distribution, and the limiting measure

$$ \begin{gather*} m(dx) = \frac{\phi_0(x)}{\left\|\phi_0\right\|_1}\, dx = \frac{1}{(2\pi)^{d/2}} e^{-\frac{|x|^2}{2}}\, dx \end{gather*} $$

is the only quasi-stationary measure of $\left \{U_t: t \geqslant 0 \right \}$ . By Corollary 3.5, we also get a pointwise version with explicit space-rate

which is uniform on compact sets.

We will show that the quasi-ergodicity is progressively uniform. Observe that the semigroup $\big \{U_t:t \geqslant 0\big \}$ is pGSD; cf. Definition 4.2.b).

Proposition 7.3. Let $x \in \mathbb {R}^d$ , $t>0$ and $C>0$ . Then

In particular, if $C < (2\sqrt {\pi })^{d/2}$ , the right-hand side is void for large t, and if $C = (2\sqrt {\pi })^{d/2}$ , then the inequality on the right-hand side defines a bounded set of x for large t’s.

Proof. Because of (7.11) and (7.12), the inequality on the left-hand side reads

$$ \begin{gather*} \exp\left[{|x|^2 \left(\frac{e^{-2t}}{e^{2t}+e^{-2t}}\right)}\right] \leqslant C \pi^{-\frac{d}{4}} \left(\frac{e^{2t}+e^{-2t}}{2}\right)^{\frac{d}{2}} e^{-dt} = C (2 \sqrt{\pi})^{-\frac{d}{2}} (1+e^{-4t})^{\frac{d}{2}}, \end{gather*} $$

which can be equivalently rewritten as

$$ \begin{align*} |x|^2 \leqslant \big(1+e^{4t}\big) \left(\log C - \frac{d}{2}\log (2\sqrt{\pi}) + \frac{d}{2} \log (1+e^{-4t})\right).\\[-42pt] \end{align*} $$

Assume now that $C> (2\sqrt {\pi })^{d/2}$ and write

$$ \begin{gather*} \rho(t):= K e^{2t} \sqrt{\left(1+e^{-4t}\right)\left(1 + \frac{d}{2K^2} \log (1+e^{-4t})\right)}, \quad t \geqslant 1, \end{gather*} $$

where $K = \sqrt {\log \big (C/(2\sqrt {\pi })^{d/2}\big )}$ . Clearly, we have $\rho (t) = K e^{2t} (1+o(1))$ , as $t \to \infty $ . It follows from Proposition 7.3 that the semigroup $\left \{U_t: t \geqslant 0 \right \}$ is pGSD with the exhausting family $\left \{K_t: t \geqslant 1\right \}$ where $K_t = \overline {B}_{\rho (t)}(0)$ ; that is,

(7.13)

Therefore, we can apply Theorem 5.2 to show that the progressive exponential uniform quasi-ergodicity holds on any space $L^p$ ; it is easily seen from (7.12) that the assumption ( A4) holds for every $t_0>0$ . More precisely, for every $a \in (0,1)$ and $p \in [1,\infty ]$ , there are constants $C_1, C_2>0$ such that

(7.14)

for sufficiently large $t>1$ .

It is interesting to note that one can use (7.10) and (7.12) to verify the following: for a family of measures $\big \{\sigma _t\big \}_{t \geqslant 1}$ given by $\sigma _t = \delta _{x_t}$ with $x_t = (e^{2t}x_0)/2$ where $x_0 \in \mathbb {R}^d$ is fixed and for where K is a Borel set of finite Lebesgue measure, we have

(7.15)

This was first noticed in [Reference Knobloch and Partzsch31, p. 135] for $d=1$ ; the aim was to show that the quasi-ergodicity property for the harmonic oscillator is not uniform in $\mathbb {R}$ (note that the authors of that paper work with a Schrödinger operator with rescaled kinetic term; that is, ${H f = (1/2) f" + |x|^2 f}$ ).

We can now see more. Indeed, (7.15) shows that the convergence in (7.14) cannot be uniform in the sets $K_t$ that come from the pGSD property (7.13) – one can choose $x_0$ is such a way that $x_t \in K_t$ for sufficiently large t. This means that Theorem 5.2 is optimal in the sense that, in general, the pGSD property with the exhausting family $\left \{K_{t} : t \geqslant t_0\right \}$ implies the progressive exponential quasi-ergodicity with $\left \{K_{at} : t \geqslant t_0/\theta \right \}$ only for $a < 1$ ; cf. Remark 5.3.

7.3. Semigroups of non-local Schrödinger operators in the DJP setting

One of the main motivations for this paper was to describe the quasi-ergodic properties of the Schrödinger semigroups studied in a recent paper [Reference Kaleta and Schilling26]. These are evolution semigroups that belong to non-local Schrödinger operators of the form $H=-L+V$ , where L is the generator of a symmetric Lévy process $\{X_t: t \geqslant 0\}$ (so-called Lévy operator) with the direct jump property (DJP in short) in $\mathbb {R}^d$ , $d \geqslant 1$ , and V is a sufficiently regular confining potential. Working in that generality, we identified in [Reference Kaleta and Schilling26] a new regularity property of compact semigroups in $L^2$ , that we called progressive intrinsic ultracontractivity (pIUC in short). It means that the regularity of the semigroup improves as $t \to \infty $ . We now want to understand what is the impact of pIUC on the quasi-ergodic properties of the semigroup.

The class of processes that we consider in this section consists of symmetric (self-dual) and strong Feller Lévy processes (cf. Section 7.1.1, also for notation). In addition, we have to impose some extra regularity conditions on the density $p_t(x)$ and the Lévy measure $\nu $ (Assumptions (A1)–(A2) in [Reference Kaleta and Schilling26]; we do not give details here). The key property is the following: we assume that the Lévy measure $\nu (dx)$ has a density $\nu (x) \, dx$ and that there exists a decreasing ‘profile’ function $f:(0,\infty ) \to (0,\infty )$ such that $\nu (x) \asymp f(|x|)$ , $x \neq 0$ , and

(7.16) $$ \begin{align} f_1 * f_1(x) \leqslant c f_1(x), \quad x \in \mathbb{R}^d, \end{align} $$

for a constant $c>0$ , where $f_1:= f \wedge 1$ . This condition has a very suggestive probabilistic interpretation, and therefore, it is called the direct jump property (DJP). Note that this class contains many examples of jump Lévy processes and the corresponding Lévy operators which play an important role in various applications. For example, it includes the

• • fractional Laplace operator $L=-(-\Delta )^{\alpha /2}$ , $\alpha \in (0,2)$ (being the generator of the isotropic $\alpha $ -stable process); here, $\nu (x) = c_{d,\alpha } |x|^{-d-\alpha }$ ;

• • relativistic operator $L=-(-\Delta +m^{2/\alpha })^{\alpha /2}+m$ , $\alpha \in (0,2)$ , $m>0$ (being the generator of the isotropic relativistic $\alpha $ -stable process); here, ${\nu (x) \asymp e^{-m^{1/\alpha }|x|} (1 \wedge |x|)^{-\frac {d+\alpha -1}{2}} |x|^{-\frac {d+\alpha +1}{2}}}$ .

We assume that $V \in L^{\infty }_{\mathrm {loc}}(\mathbb {R}^d,dx)$ is a confining potential (i.e., $V(x) \to \infty $ as $|x| \to \infty $ ) such that there is a an increasing profile $g:[0,\infty ) \to (0,\infty )$ , growing at infinity not faster than an exponential function, and there are constants $c_1, c_2 \geqslant 1$ such that $c_1^{-1} g(|x|) \leqslant V(x) \leqslant c_1 g(|x|)$ , for all $|x| \geqslant c_2$ ; see the detailed statement in [Reference Kaleta and Schilling26, Assumption (A3)].

The Schrödinger operator $H=-L+V$ is well defined, bounded below and self-adjoint in $L^2(\mathbb {R}^d,dx)$ . We denote by $U_t = e^{-tH}$ , $t \geqslant 0$ , the operators of the corresponding evolution semigroup. They are known to have the following probabilistic Feynman–Kac representation:

(7.17) $$ \begin{align} U_tf(x) = \mathbb{E}^x \left[e^{-\int_0^t V(X_s)\,ds} f(X_t)\right], \quad f \in L^2(\mathbb{R}^d,dx), \ t>0. \end{align} $$

The operators $U_t$ are self-adjoint, and by [Reference Kaleta and Schilling26, Lemma 2.3] (or Lemma 7.1) and Lemma 7.2.a), all regularity assumptions ( A0)–( A3) are satisfied and ( A4) follows from the proof of Lemma 7.2.a). Our standard reference for self-adjoint Schrödinger operators and the Feynman–Kac semigroups is the monograph [Reference Demuth and van Casteren17].

We will now show how our present results apply to these semigroups. The discussion will be divided into three parts.

(1) Heat content and exponential quasi-ergodicity. First of all, due to Lemma 7.2.b), we know that

(7.18) $$ \begin{align} \liminf_{|x| \to \infty} \frac{V(x)}{\log|x|}>0 \qquad \Longleftrightarrow \qquad \exists \, t_1>0 \::\: Z(t_1)< \infty, \end{align} $$

and our results in Section 3 apply. More precisely, by Theorem 3.4, the semigroup $\left \{U_t: t \geqslant 0 \right \}$ is exponentially quasi-ergodic for every finite initial distribution, and the limiting measure

$$ \begin{gather*} m(dx) = \frac{\phi_0(x)}{\left\|\phi_0\right\|_1}\, dx, \end{gather*} $$

where $\phi _0(x)$ is the ground state of the operator H, is the only quasi-stationary measure of $\left \{U_t: t \geqslant 0 \right \}$ . Furthermore, in Corollary 3.5, we obtain a pointwise version with the space-rate which is uniform on compact sets. By [Reference Kaleta and Lőrinczi25, Corollary 2.2], there is a constant $c_3>0$ such that

$$ \begin{gather*} \phi_0(x) \geqslant c_6 \left(1 \wedge \frac{\nu(x)}{V(x)}\right), \quad x \in \mathbb{R}^d \end{gather*} $$

(this remains true for potentials without radial profiles). However, we always have

which leads to the following explicit estimate of the rate

(7.19)

The function on the right-hand side tends to infinity as $|x| \to \infty $ , but it is bounded on compact sets. Better estimates for can be found in [Reference Kaleta and Schilling26, Theorem 4.8]. Observe that the sufficient condition in (7.18) does not depend on $\nu $ .

(2) aGSD regime and uniform exponential quasi-ergodicity. By [Reference Kaleta and Lőrinczi25, Theorem 2.6],

(7.20)

Recall that the property on the right-hand side is called aGSD; see Definition 4.2.a). If we are in the aGSD regime – that is, if the growth of the potential is sufficiently large with respect to $|\log \nu (x)|$ – then the exponential quasi-ergodicity in Corollary 3.5 discussed above becomes uniform on the full space $\mathbb {R}^d$ . This is exactly what we saw in Corollary 4.4.a). The same conclusion follows from [Reference Knobloch and Partzsch31, Theorem 1, Corollary 2]. However, now we get much more. Combining Corollary 4.4 and (7.20), we get that the uniform quasi-ergodicity of the semigroup $\left \{U_t: t \geqslant 0 \right \}$ is equivalent to the condition

$$ \begin{gather*} \liminf_{|x| \to \infty} \frac{V(x)}{|\log\nu(x)|}>0. \end{gather*} $$

If $\nu (x)$ has polynomial decay at infinity (as is, for example, the case for the fractional Laplacian), then $|\log \nu (x)| \asymp \log |x|$ , for $|x|$ large enough, and the properties in (7.18) and (7.20) coincide. However, if $\nu $ decays at infinity faster than any polynomial, then (7.20) is more restrictive than (7.18). For example, if $\nu (x)$ is exponential at infinity (e.g., for relativistic operators), then $|\log \nu (x)| \asymp |x|$ , for large $|x|$ .

(3) pGSD and progressive uniform quasi-ergodicity. Finally, we characterize the quasi-ergodic behavior of the Schrödinger semigroups $\left \{U_t: t \geqslant 0 \right \}$ with confining potentials as above which do not satisfy (7.20), no matter how slow the potential may grow at infinity; this brings us out of the aGSD regime. Note that this problem was completely open. Since we want to be here as sharp as possible, we have to monitor precisely the threshold between the aGSD and the non-aGSD regime. For this reason, we follow [Reference Kaleta and Schilling26, Assumption (A4)] and impose an additional technical condition on the profile g of the potential V. More precisely, we require that g depends on the profile f of the Lévy density $\nu $ in a sufficiently regular way: we consider $R_0>0$ such that $f(R_0)<1$ and assume that $g(r) = h(|\log f(r)|)$ , $r \geqslant R_0$ , for an increasing function $h:[|\log f(R_0)|,\infty ) \to (0,\infty )$ such that $h(s)/s$ is monotone. Under this assumption, the class of Schrödinger semigroups $\left \{U_t: t \geqslant 0 \right \}$ with Lévy densities $\nu $ and potentials V is divided into two classes:

1. a) (aGSD regime): $\liminf _{|x| \to \infty } \frac {V(x)}{|\log \nu (x)|}>0$ , i.e. (7.20) holds;

2. b) (non-aGSD regime): $\lim _{|x| \to \infty } \frac {V(x)}{|\log \nu (x)|} =0$ ;

cf. [Reference Kaleta and Schilling26, Remark 5.1]. We already know from Part (2) that in the aGSD regime, the Schrödinger semigroup is uniformly exponentially quasi-ergodic, and, in fact, we have the equivalence of these two properties.

The rest of this section will be devoted to the analysis of the quasi-ergodic behavior in the non-aGSD regime. Of course, if (7.18) holds, then we know from Part (1) above that we have the exponential quasi-ergodicity with semi-explicit rate, which is uniform on compact sets. With [Reference Kaleta and Schilling26, Corollary 5.6 b)], we can now give even better estimates for the rate (7.19)

for some $c_4, c_5>0$ . Our theorems from Section 5 now yield much stronger results. In the paper [Reference Kaleta and Schilling26], we identify a new large-time regularity property of compact semigroups, which we call progressive intrinsic ultracontractivity (pIUC). This property holds for semigroups considered here. In [Reference Kaleta and Schilling26, Corollary 5.6 b)], we show that pIUC implies the following two-sided sharp estimates: there exist $\theta \in (0,1)$ , $t_2>0$ and an increasing function $\rho : [t_2, \infty ) \to (0,\infty )$ , with $\lim _{t \to \infty } \rho (t) = \infty $ , such that

(7.21)

(note that the comparison constants are uniform in x and t!). The function $\rho $ is the right-continuous generalized inverse function of $r \mapsto |\log f(r)|/g(r)$ ; see [Reference Kaleta and Schilling26, Lemma 5.2]. Note that (7.21) is the progressive ground state domination (pGSD) property; cf. Definition 4.2.b).

Therefore, we see that $\left \{U_t: t \geqslant 0 \right \}$ is pGSD with the exhausting family $\left \{K_t: t \geqslant t_2/\theta \right \}$ where $K_t = \overline {B}_{\rho (\theta t)}(0)$ , and we can apply Theorem 5.2 to show that the progressive uniform quasi-ergodicity on any space $L^p$ holds. We obtain that there exist $\gamma , t_0>0$ such that for every $a, b \in (0,1)$ such that $a+2b=1$ and $p \in [1,\infty ]$ , there is a constant $c_6>0$ such that

where

Because of Theorem 5.5, the measure m is the only quasi-stationary distribution of the Schrödinger semigroup $\left \{U_t: t \geqslant 0 \right \}$ – no matter how slowly V grows at infinity. We need to verify only the condition (5.5). To this end, we observe that by [Reference Kaleta and Schilling26, Corollary 5.6 b)], there are constants $c_7, c_8>0$ such that

for sufficiently large values of t. Since $V(x) \to \infty $ as $|x| \to \infty $ and $\rho (t) \to \infty $ as $t \to \infty $ , this implies (5.8), which is equivalent to (5.5).