Feller generators with measurable lower order terms

Abstract

We study perturbations of Feller generators by ‘lower order terms’ with measurable coefficients. We investigate which properties of the original semigroup—such as positivity, conservativeness and the Feller property—are passed on to the perturbed semigroup. We give several examples and discuss applications in the theory of martingale problems and stochastic differential equations with measurable coefficients.

1 Introduction

Given two operators A and B, it is a classical question to ask how (and, in particular, on which domain) the sum $A+B$ can be defined and which properties of A are preserved under the perturbation B, see e.g. [3, 26].

In this article, we consider this problem in the context of Feller semigroups and generators. We are interested in constructing Feller generators with measurable lower order terms, and this means that the perturbation does not take values in the space of continuous functions. In consequence, classical perturbation results from the theory of strongly continuous semigroups do not apply.

Let $(T(t))_{t \ge 0}$ be a Feller semigroup, that is, a sub Markovian, strongly continuous semigroup on the space ${\mathcal {C}}_\infty (\mathbb {R}^d)$ of continuous functions vanishing at infinity. If the domain of the (infinitesimal) generator A contains the test functions ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$, then the Courrège–von Waldenfels theorem, see e.g. [6, Thm. 2.21], shows that ${\mathcal {A}}:= A|_{{\mathcal {C}}_c^{\infty }(\mathbb {R}^d)}$ is a Lévy-type operator, i.e. an integro-differential operator of the form

$$\begin{aligned} \begin{aligned} {\mathcal {A}}f(x)&= b(x) \nabla f(x) + \frac{1}{2} {{\,\mathrm{\textrm{tr}}\,}}(Q(x) \nabla ^2 f(x)) \\&\quad + \int _{\mathbb {R}^d {\setminus } \{0\}} (f(x+y)-f(x)-\nabla f(x) \cdot \mathbbm {1}_{(0,1)}(|y|)) \, \nu (x,dy). \end{aligned}\nonumber \\ \end{aligned}$$

(1.1)

Here $(b(x), Q(x), \nu (x, \cdot ))_{x\in \mathbb {R}^d}$ is the (infinitesimal) characteristics consisting of the drift coefficients $b= (b_j): \mathbb {R}^d\rightarrow \mathbb {R}^d$, the diffusion coefficients $Q=(q_{ij}) : \mathbb {R}^d\rightarrow \mathbb {R}^{d\times d}$ and the jumping kernels $\nu : \mathbb {R}^d \rightarrow {\mathcal {M}}^+(\mathbb {R}^d)$. We note that any of these characteristics may vanish. In particular, it may happen that $Q\equiv 0$.

We are interested in the following questions: If a Lévy-type operator B is a lower order perturbation of A, then under which conditions is (a realization of) $A+B$ the generator of a semigroup, and which properties does the perturbed semigroup inherit from the original semigroup $(T(t))_{t \ge 0}$? Is the martingale problem for $A+B$ well-posed? Classical perturbation results can be used to tackle these questions if B maps D(A) into ${\mathcal {C}}_\infty (\mathbb {R}^d)$, which in particular implies that the infinitesimal characteristics of B depend continuously on x. In this article, we investigate perturbations B whose characteristics depend merely measurably on x. Allowing for discontinuous characteristics of B leads to a number of issues and subtleties. To give an example: If B maps D(A) into ${\mathcal {C}}_\infty (\mathbb {R}^d)$, then the sub-Markovianity of the perturbed semigroup can be verified using the positive maximum principle; this does not work any longer if B has discontinuous coefficients, i.e. if we need to work on the space ${\mathcal {B}}_b(\mathbb {R}^d)$ of bounded measurable functions rather than ${\mathcal {C}}_\infty (\mathbb {R}^d)$.

In fact, establishing the sub-Markov property of the perturbed semigroup turns out to be a rather delicate point. At the same time, being sub-Markovian is crucial for applications in the theory of stochastic processes, which we are interested in. To establish sub-Markovianity, we will use an approximation argument which is of independent interest, e.g. it can be used to prove continuous dependence of solutions of certain martingale problems on the coefficients (see Theorem 6.3). Let us mention that our questions can be formulated equivalently in the context of pseudo-differential operators. Namely, the Lévy-type operator ${\mathcal {A}}$ can be seen as a pseudo-differential operator

$$\begin{aligned} {\mathcal {A}}f(x) = - \int _{\mathbb {R}^d} q(x,\xi ) e^{ix \cdot \xi } {\hat{f}}(\xi ) \, d\xi , \qquad f \in {\mathcal {C}}_c^{\infty }(\mathbb {R}^d),\;x \in \mathbb {R}^d, \end{aligned}$$

(1.2)

where ${\hat{f}}(\xi )=(2\pi )^{-d} \int _{\mathbb {R}^d} f(x) e^{-ix \cdot \xi } \, dx$ is the Fourier transform of f, and

$$\begin{aligned} q(x,\xi ):= & {} -ib(x) \cdot \xi + \frac{1}{2} \xi \cdot Q(x) \xi \nonumber \\ {}{} & {} + \int _{\mathbb {R}^d {\setminus } \{0\}} (1-e^{iy \cdot \xi }+iy \cdot \xi \mathbbm {1}_{(0,1)}(|y|)) \, \nu (x,dy) \end{aligned}$$

(1.3)

is the so-called the symbol of ${\mathcal {A}}$. We study under which conditions the pseudo-differential operator with the symbol $q(x,\xi )+p(x,\xi )$ gives rise to a Feller semigroup if the symbol $p(x,\xi )$ of the perturbation B depends merely measurably on x.

If the symbol—or, equivalently, the characteristics—of a Lévy-type operator satisfies suitable smoothness conditions, then general results from symbolic calculus show that the closure of $({\mathcal {A}},{\mathcal {C}}_c^{\infty }(\mathbb {R}^d))$ is the generator of a Feller semigroup, see e.g. [17, 21]. Under the milder assumption that the coefficients are Hölder continuous, the situation is already much more complicated and there is an immense literature on the question whether the Lévy-type operator ${\mathcal {A}}$ gives rise to a Feller semigroup, see e.g. [6, 17, 20, 21, 33] for a survey. None of these results applies in our framework since we are dealing with discontinuous coefficients. For the particular case that the diffusion coefficient Q is uniformly (strictly) elliptic, there are general results in the literature which allow for discontinuous coefficients, see e.g. [20, Thm. 2.1.43] and also [56, 57] for processes on bounded domains. Moreover, the well-posedness of the $({\mathcal {A}},{\mathcal {C}}_c^{\infty }(\mathbb {R}^d))$-martingale problem is well studied in this case, cf. [55]. No such general results are available if the operator has a vanishing diffusion component $Q \equiv 0$. There are some perturbation results for martingale problems which allow for discontinuous coefficients, including bounded perturbations [13, Section 4.10] and perturbations of Lévy generators e.g. in [8, 23, 31, 45]. Note that already the existence of a solution to the martingale problem can be highly non-trivial if the coefficients are discontinuous: While there are general existence results for Lévy-type operators with continuous coefficients, cf. [17, Thm. 3.15], there are no such results for the discontinuous framework; see however [35] and the references therein. Let us point out that the existence of (unique) solutions to martingale problems associated with Lévy-type operators can be used to deduce the existence of (unique) weak solutions to Lévy-driven stochastic differential equations (SDEs)

$$\begin{aligned} dX_t = b(X_{t-}) \, dt + \sigma (X_{t-}) \, dL_t, \end{aligned}$$

see Sect. 7.4 for details. In particular, drift(-type) perturbations of Lévy processes (i.e. $\sigma \equiv 1$) have been studied quite intensively, [8, 27, 28, 37, 46, 58] to mention just a few classical and recent works.

We point out that our approach is basically operator theoretic and is based on a Miyadera–Voigt type perturbation result from [42], where an infinite-dimensional situation was considered. Here, we focus on the finite dimensional situation (so that the concept of a ‘Feller semigroup’ is available) and present results that are tailormade to be applied to integro-differential operators as considered above. However, we point out that we do not, in general, assume that the generator A has a specific form. Instead, we impose assumptions on the semigroup $(T(t))_{t\ge 0}$ generated by A. More precisely, we assume that this semigroup maps the space ${\mathcal {B}}_b(\mathbb {R}^d)$ of bounded and measurable functions to a space of Hölder continuous functions; in particular the semigroup $T(t))_{t\ge 0}$ enjoys the strong Feller property. We can now summarize our main results from Theorem 5.4 and Theorem 6.2. For unexplained terminology, we refer the reader to Sect. 2.

Theorem 1.1

Let $(T(t))_{t \ge 0}$ be a Feller semigroup with (strong) generator A such that ${\mathcal {C}}_c^\infty (\mathbb {R}^d)$ is a core for A. Assume that there are $\rho \in (0,2)$ and $\varphi \in L^1(0,1)$ such that

$$\begin{aligned} \Vert T(t) f\Vert _{{\mathcal {C}}_b^{\rho }} \le \varphi (t)\Vert f\Vert _{\infty }, \qquad t \in (0,1),\, f \in {\mathcal {B}}_b(\mathbb {R}^d), \end{aligned}$$

Let ${\hat{B}}$ be a Lévy-type operator

$$\begin{aligned} {\hat{B}}f(x)= & {} b(x) \nabla f(x) + \int _{\mathbb {R}^d {\setminus }\{0\}} (f(x+y)-f(x)\nonumber \\ {}{} & {} -\nabla f(x) \cdot y \mathbbm {1}_{(0,1)}(|y|)) \, \mu (x,dy), \end{aligned}$$

(1.4)

where the drift b and the jumping kernels $\mu (x,dy)$ depend measurably on $x \in \mathbb {R}^d$. Assume that

$$\begin{aligned} \sup _{x \in \mathbb {R}^d} \left( |b(x)| + \int _{\mathbb {R}^d} \min \{1,|y|^{\beta }\} \, \mu (x,dy)\right) <\infty \end{aligned}$$

for some $\beta \in [0,\rho )$ and that the compensated drift is zero in case that $\rho \le 1$. Then

• (a) The bp-closure of $A+{{\hat{B}}}$ is the full generator of a Markovian ${\mathcal {C}}_b$-Feller semigroup S, i.e. S leaves ${\mathcal {C}}_b(\mathbb {R}^d)$ invariant and $S(t)f \rightharpoonup f$ as $t \rightarrow 0$ for every $f \in {\mathcal {C}}_b(\mathbb {R}^d)$. Moreover, S enjoys the strong Feller property;

• (b) The $(A+{\hat{B}}, {\mathcal {C}}_c^\infty (\mathbb {R}^d))$-martingale problem is well-posed;

• (c) If additionally the tightness condition

  $$\begin{aligned} \lim _{R \rightarrow \infty } \sup _{x \in \mathbb {R}^d} \mu (x,\{y \in \mathbb {R}^d; |y| \ge R\})=0 \end{aligned}$$

  holds, then S is a ${\mathcal {C}}_\infty$-semigroup, i.e. S leaves ${\mathcal {C}}_\infty (\mathbb {R}^d)$ invariant and its restriction to that space is strongly continuous, and so S is a Feller semigroup.

The central assumption that $\Vert T(t)f\Vert _{{\mathcal {C}}_b^\rho } \le \varphi (t) \Vert f\Vert _\infty$ can be verified in many cases so that Theorem 1.1 applies to a wide class of Feller generators A. In Sect. 7 we present several classes of examples where it can be deduced from existing literature, that the above assumption is satisfied. This includes uniformly (strictly) elliptic differential operators (see Sect. 7.1), generators of certain Lévy processes (see Sect. 7.2) and operators of variable order (see Sect. 7.3). In Sect. 7.4, we show that the well-posedness of the martingale problem in (b) yields existence and uniqueness results for Lévy-driven SDEs.

Let us comment on the notion of the (infinitesimal) generator. In this article, we will be mostly working with the so-called full generator, see Sect. 2.2 below. The reason is that we will be dealing with Lévy-type operators with discontinuous coefficients. The strong generator, which is often considered in the literature, is too restrictive for our purpose; for example, in the situation of Theorem 1.1, the domain of the full generator of the perturbed semigroup contains ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ while this is not true, in general, for the strong generator. In particular, Theorem 1.1 allows us to construct (non-trivial) Feller semigroups whose strong generator has a small domain, in the sense that ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ is not contained in its domain.

This article is organized as follows. In Sect. 2, we recall some notions concerning kernel operators, semigroups and their generators. Here we also establish the connection between cores of Feller generators and bp-cores of the corresponding full generators; this is important for our applications to martingale problems. In Sect. 3 we recall a perturbation result from [42] and prove our first main result (Theorem 3.4). Section 4 concerns convergence results for perturbed semigroups, which is crucial for establishing sub-Markovianity in Theorem 1.1. Sections 2–4 concern abstract state spaces E rather than $\mathbb {R}^d$ and thus do not take the special structure of the operators ${\mathcal {A}}$ in (1.1) into account. We point out that these abstract results could also be of interest for different settings, e.g. if E is a domain in $\mathbb {R}^d$ or a Riemannian manifold (with or without boundary).

The remaining Sects. 5–7 apply the abstract results to Lévy-type operators as in (1.1). Section 5 concerns the actual perturbation result and Sect. 6 the corresponding martingale problem. The concluding Sect. 7 contains examples and discusses applications in the theory of stochastic differential equations.

Moreover, there are two appendices which contain results that are used in the proof of Theorem 5.4 and which we believe to be of independent interest.

2 Preliminaries

2.1 Kernel operators

Throughout, E is a locally compact Polish space. We endow E with its Borel $\sigma$-algebra ${\mathcal {B}}(E)$. The notation $C \Subset E$ means that C is a compact subset of E. The spaces of bounded and measurable resp. bounded and continuous functions on E are denoted by ${\mathcal {B}}_b(E)$ and ${\mathcal {C}}_b(E)$ respectively whereas ${\mathcal {C}}_\infty (E)$ refers to the space of continuous functions vanishing at $\infty$, i.e. of those continuous functions $f: E\rightarrow \mathbb {R}$ such that for every $\varepsilon >0$ we find a compact set $C \Subset E$ with $|f(x)|\le \varepsilon$ for all $x\in E{\setminus } C$. We denote by ${\mathcal {C}}_b^{\varrho }(\mathbb {R}^d)$ the classical Hölder spaces on $\mathbb {R}^d$, see e.g. [60, Def. 1, p. 4].

A kernel on E is a map $k: E\times {\mathcal {B}}(E) \rightarrow \mathbb {R}$ such that

• (i) the map $x\mapsto k(x,A)$ is measurable for every $A \in {\mathcal {B}}(E)$;

• (ii) $k(x, \cdot )$ is a (signed) measure for every $x \in E$;

• (iii) $\sup _{x\in E} |k|(x, E) <\infty$, where $|k|(x, \cdot )$ denotes the total variation of $k(x, \cdot )$.

If $k(x, \cdot )$ is a positive measure for every $x \in E$, then k is called a positive kernel; if every $k(x, \cdot )$ is a probability measure, k is a Markovian kernel.

With every kernel k on E we can associate a bounded linear operator K on ${\mathcal {B}}_b(E)$ by setting

$$\begin{aligned} (Kf)(x) \mathrel {\mathop :}=\int _E f(y) \, k(x, dy), \quad f \in {\mathcal {B}}_b(E),\; x \in E. \end{aligned}$$

(2.1)

We call an operator K of this form a kernel operator on ${\mathcal {B}}_b(E)$. It turns out that a bounded, linear operator on ${\mathcal {B}}_b(E)$ is a kernel operator if, and only if, K is continuous with respect to the weak topology $\sigma \mathrel {\mathop :}=\sigma ({\mathcal {B}}_b(E), {\mathcal {M}}_b (E))$ induced by the space ${\mathcal {M}}_b(E)$ of bounded signed measures on E (see e.g. [41, Prop. 3.5]). For sequences, $\sigma$-convergence is nothing else than bp-convergence (bp is short for bounded and pointwise), i.e. $f_n \rightarrow f$ with respect to $\sigma$ if, and only if, $\sup _{n \in \mathbb {N}} \Vert f_n\Vert _{\infty }<\infty$ and $f_n \rightarrow f$ pointwise. Indeed, by dominated convergence, bp-convergence implies $\sigma$-convergence; the converse follows from testing against Dirac measures and noting that a $\sigma$-convergent sequence is norm-bounded by the uniform boundedness principle. We will write ${\mathcal {L}}({\mathcal {B}}_b(E), \sigma )$ for the space of $\sigma$-continuous linear operators on E, i.e. kernel operators. Note that any such operator is automatically bounded. In what follows, we write $\rightharpoonup$ to indicate convergence with respect to $\sigma$ while we reserve $\rightarrow$ to indicate convergence with respect to the supremum norm. If K is a bounded operator defined via (2.1) on ${\mathcal {C}}_b(E)$ or ${\mathcal {C}}_\infty (E)$, then K is also called a kernel operator (on ${\mathcal {C}}_b(E)$ resp. ${\mathcal {C}}_\infty (E)$); it can be extended in a unique way to a kernel operator on ${\mathcal {B}}_b(E)$. As a consequence of Riesz’ representation theorem, every bounded operator on ${\mathcal {C}}_\infty (E)$ is a kernel operator and can thus be extended to a kernel operator on ${\mathcal {B}}_b(E)$, see e.g. [52, Cor. 21.12].

In applications, it is often of interest if a kernel operator on ${\mathcal {B}}_b(E)$ leaves one of the spaces ${\mathcal {C}}_b(E)$ or ${\mathcal {C}}_\infty (E)$ invariant. Of particular interest is the case where a kernel operator K maps ${\mathcal {B}}_b(E)$ into ${\mathcal {C}}_b(E)$. Such an operator is called strong Feller operator. For our perturbation results, it will be important to know under which conditions a kernel operator leaves the space

$$\begin{aligned} {\mathcal {B}}_\infty (E) \mathrel {\mathop :}=\big \{ f\in {\mathcal {B}}_b(E): \forall \varepsilon >0\, \exists \, C\Subset E \, \forall \, x\in E{\setminus } C: |f(x)| \le \varepsilon \big \}. \end{aligned}$$

invariant. We present two results on this topic.

Lemma 2.1

Let K be a kernel operator with kernel k. Then $K{\mathcal {B}}_\infty (E)\subset {\mathcal {B}}_\infty (E)$ if and only if $k(\cdot , C) \in {\mathcal {B}}_\infty (E)$ for every relatively compact set C.

Proof

As $K\mathbbm {1}_C = k(\cdot , C)$ and $\mathbbm {1}_C \in {\mathcal {B}}_\infty (E)$ for a compact set C, the stated condition is certainly necessary. To prove its sufficiency, consider a function $f\in {\mathcal {B}}_\infty (E)$ with compact support S. We find a sequence of simple functions ${\tilde{f}}_n$ that converges uniformly to f. If we set $f_n \mathrel {\mathop :}={\tilde{f}}_n \mathbbm {1}_S$, then also the sequence $f_n$ converges uniformly to f and, moreover, for $c\ne 0$ and every $n\in \mathbb {N}$ the set $\{f_n = c\}$ is relatively compact. Thus, $Kf_n\in {\mathcal {B}}_\infty (E)$. As $Kf_n \rightarrow Kf$ uniformly and since ${\mathcal {B}}_\infty (E)$ is closed with respect to the supremum norm, $Kf\in {\mathcal {B}}_\infty (E)$. The case of a general function $f\in {\mathcal {B}}_\infty (E)$ follows from this by approximating f uniformly with a sequence of measurable functions with compact support. $\square$

If the operator K is positive, then invariance of ${\mathcal {C}}_\infty (E)$ implies invariance of ${\mathcal {B}}_\infty (E)$.

Lemma 2.2

Let K be a positive kernel operator. If $K{\mathcal {C}}_\infty (E)\subset {\mathcal {C}}_\infty (E)$, then we have $K{\mathcal {B}}_\infty (E) \subset {\mathcal {B}}_\infty (E)$. If, in addition, K has the strong Feller property, then $K{\mathcal {B}}_\infty (E) \subset {\mathcal {C}}_\infty (E)$.

Proof

Let $0\le f\in {\mathcal {B}}_\infty (E)$. Given $\varepsilon >0$, pick a compact set C such that $|f(x)|\le \varepsilon /\Vert K\Vert$ for $x\in E{\setminus } C$. Then

$$\begin{aligned} 0\le f\le \Vert f\Vert _\infty \mathbbm {1}_C + \frac{\varepsilon }{\Vert K\Vert }\mathbbm {1}_{E\setminus C} \end{aligned}$$

and, consequently,

$$\begin{aligned} 0\le Kf \le \Vert f\Vert _\infty k(\cdot , C) + \varepsilon . \end{aligned}$$

Since $K{\mathcal {C}}_\infty (E)\subset K{\mathcal {C}}_\infty (E)$, [21, Lem. 3.2.15] yields $k(\cdot , C) \in {\mathcal {B}}_\infty (E)$ and thus $Kf\in {\mathcal {B}}_\infty (E)$ follows as $\varepsilon >0$ was arbitrary. The addendum follows from the identity ${\mathcal {C}}_\infty (E) = {\mathcal {B}}_\infty (E)\cap {\mathcal {C}}_b(E)$. $\square$

The following example shows that the assertion of Lemma 2.2 fails to hold if K is not a positive operator.

Example 2.3

Consider the kernel operator

$$\begin{aligned} (Kf)(x) := f(0)- \chi (x) f \left( \frac{1}{|x|} \right) , \qquad x \in \mathbb {R}, \end{aligned}$$

for a function $\chi \in {\mathcal {C}}_b(\mathbb {R})$ with $\chi _{\mathbb {R}\setminus (-2,2)} \equiv 1$ and $\chi |_{(-1,1)} \equiv 0$. Clearly, $Kf \in {\mathcal {C}}_\infty (\mathbb {R})$ for any continuous f and so, in particular, $K{\mathcal {C}}_\infty (\mathbb {R}) \subset {\mathcal {C}}_\infty (\mathbb {R})$. On the other hand, $K {\mathcal {B}}_\infty (\mathbb {R}) \subset {\mathcal {B}}_\infty (\mathbb {R})$ fails to hold as can be seen by considering e.g. $f := \mathbbm {1}_{[0,1] \cap {\mathbb {Q}}}$.

2.2 Semigroups, (pseudo)resolvents and generators

We now turn our attention to semigroups of kernel operators. A family $T= (T(t))_{t>0} \subset {\mathcal {L}}({\mathcal {B}}_b(E), \sigma )$ is called semigroup of kernel operators if

• (i) $T(t+s) = T(t)T(s)$ for all $s,t>0$.

• (ii) T is exponentially bounded, i.e. there are constants $\omega \in \mathbb {R}$, $M>0$ such that $\Vert T(t)\Vert \le Me^{\omega t}$. We say that T is of type $(M, \omega )$ to stress these constants.

• (iii) for every $f\in {\mathcal {B}}_b(E)$, the map $(t,x) \mapsto (T(t)f)(x)$ is jointly measurable.

It is not difficult to see that if T is a semigroup of kernel operators of type $(M,\omega )$ then for $f \in \mathcal {B}_{b}(E)$ and $\mu \in {\mathcal {M}}_b(E)$ the map $t\mapsto \langle T(t)f,\mu \rangle$ is measurable. Moreover, given $\uplambda \in \mathbb {C}$ with $\textrm{Re}\,\uplambda >0$ there is an operator $R(\uplambda ) \in {\mathcal {L}}({\mathcal {B}}_b(E), \sigma )$ such that

$$\begin{aligned} \langle R(\uplambda ) f, \mu \rangle = \int _0^\infty e^{-\uplambda t}\langle T(t)f, \mu \rangle \, dt \end{aligned}$$

(2.2)

for all $f\in {\mathcal {B}}_b(E)$ and $\mu \in {\mathcal {M}}_b(E)$.

This shows that, in the terms of [41, Def. 5.1], a semigroup of kernel operators is an integrable semigroup on the norming dual pair $({\mathcal {B}}_b(E), {\mathcal {M}}_b(E))$. It turns out that the family $(R(\uplambda ))_{\textrm{Re}\,\uplambda >\omega }$ is a pseudo-resolvent, i.e. it satisfies the resolvent identity $R(\uplambda )-R(\mu )=(\mu -\uplambda )R(\uplambda )R(\mu )$, cf. [41, Prop. 5.2]. For more details on pseudoresolvents we refer to [62] and [12, Sect. III.4.a].

In general, the family $(R(\uplambda ))_{\textrm{Re}\,\uplambda >\omega }$ does not consist of injective operators, and so it is not the resolvent of a (single-valued) operator. However, there is a multivalued operator ${{\hat{A}}}$ such that $R(\uplambda ) = (\uplambda - {{\hat{A}}})^{-1}$ for $\textrm{Re}\,\uplambda >\omega$ (see [16, Appendix A] for more information concerning multivalued operators). We call ${\hat{A}}$ the full generator of the semigroup T. The full generator can be characterized equivalently as follows:

$$\begin{aligned} (f,g) \in {\hat{A}} \iff \forall t>0\,:\, T(t) f-f = \int _0^t T_s g \, ds, \end{aligned}$$

(2.3)

cf. [41, Prop. 5.7]. In particular, our terminology is consistent with that used by Ethier–Kurtz [13, Sect. 1.5]. The domain $D({\hat{A}})$ of the multivalued operator ${\hat{A}}$ is defined by $D({\hat{A}}) := \{f; \exists g\,:\, (f,g) \in {\hat{A}}\}$.

We now introduce additional properties of semigroups. If T is of type (1, 0), then T is called a contraction semigroup of kernel operators. If T is a contraction semigroup of kernel operators and every operator T(t) is positive, we say that T is sub-Markovian. If additionally $T(t)\mathbbm {1}_{E} = \mathbbm {1}_E$ for all $t>0$, then T is called Markovian.

We will call T a ${\mathcal {C}}_\infty$-semigroup if $T(t){\mathcal {C}}_\infty (E)\subset {\mathcal {C}}_\infty (E)$ and the restriction of T to ${\mathcal {C}}_\infty (E)$ is strongly continuous, i.e. for every $f\in {\mathcal {C}}_\infty (E)$ we have $T(t)f \rightarrow f$ with respect to $\Vert \cdot \Vert _\infty$ as $t\rightarrow 0$. A ${\mathcal {C}}_\infty$-semigroup that is also sub-Markovian is called a Feller semigroup.

We will call T a ${\mathcal {C}}_b$-semigroup if $T(t){\mathcal {C}}_b(E)\subset {\mathcal {C}}_b(E)$ and if the restriction of T to ${\mathcal {C}}_b(E)$ is stochastically continuous, i.e. $T(t)f\rightharpoonup f$ as $t\rightarrow 0$ for $f\in {\mathcal {C}}_b(E)$. We note that for sub Markovian semigroups, this continuity condition can be equivalently characterized in terms of the associated kernels $p_t$ by asking that $p_t(x, B(x, \varepsilon )) \rightarrow 1$ as $t\rightarrow 0$ for every $x\in E$ and $\varepsilon >0$, see [21, Lem. 3.2.17]. A ${\mathcal {C}}_b$-semigroup that is also sub-Markovian is called ${\mathcal {C}}_b$-Feller semigroup.

Let us point out that the above-defined objects are compatible with the classical notions. To wit: if T is a ${\mathcal {C}}_\infty$-semigroup, then the operator $R(\uplambda )$ leaves the space ${\mathcal {C}}_\infty (E)$ invariant for $\textrm{Re}\,\uplambda >\omega$ and its restriction to that space is injective. Thus, the restriction of $(R(\uplambda ))_{\textrm{Re}\,\uplambda >\omega }$ is the resolvent of a (single valued) operator A. Moreover, A is the part of ${{\hat{A}}}$ in ${\mathcal {C}}_\infty (E)$, i.e.

$$\begin{aligned} D(A) = \{ u \in {\mathcal {C}}_\infty (E) : \exists f\in {\mathcal {C}}_\infty (E) \text{ s.t. } (u,f) \in {{\hat{A}}}\},\qquad Au = f. \end{aligned}$$

Identifying A with its graph, we have $A= {{\hat{A}}}\cap ({\mathcal {C}}_\infty (E)\times {\mathcal {C}}_\infty (E))$. By general semigroup theory, A coincides with the generator semigroup $T|_{{\mathcal {C}}_\infty (E)}$, defined as (norm-)derivative in 0.

Conversely, if we start with a strongly continuous semigroup on ${\mathcal {C}}_\infty (E)$, we may extend it to ${\mathcal {B}}_b(E)$ as any bounded linear operator on ${\mathcal {C}}_\infty (E)$ is a kernel operator. In fact, the extended semigroup is a semigroup of kernel operators in the sense defined above (this follows from [41, Lem. 6.1]). If the semigroup is sub-Markovian, then the extension is also stochastically continuous, see [19, Lem. 4.8.7].

We next discuss how to recover the full generator ${{\hat{A}}}$ from the operator A. To that end, we make use of operator cores. We recall that a subset $D\subset D(A)$ is called a core for A if for every $u\in D(A)$ we find a sequence $(u_n)\subset D$ with $u_n \rightarrow u$ and $Au_n \rightarrow Au$. To use cores in the context of multivalued operators, we employ the notion of bp-convergence defined in Sect. 2.1. A set $M \subset {\mathcal {B}}_b(E)$ is called bp-closed if with every bp-convergent sequence, it also contains its limit. The bp-closure of a set M is the smallest bp-closed subset of ${\mathcal {B}}_b(E)$ that contains M. We use these notions, mutatis mutandis, also in ${\mathcal {B}}_b(E)\times {\mathcal {B}}_b(E)$. We call a subset $M \subset {{\hat{A}}}$ a bp-core if the bp-closure of M equals ${{\hat{A}}}$.

Lemma 2.4

Let T be a ${\mathcal {C}}_\infty$-semigroup with full generator ${{\hat{A}}}$ and let A be the part of ${{\hat{A}}}$ in ${\mathcal {C}}_\infty (E)$. Let D be a core for A. Then $\{(u, Au) : u \in D\}$ is a bp-core for ${{\hat{A}}}$.

Proof

Let T be of type $(M, \omega )$ and fix $\uplambda >\omega$. As D is a core for A, the set $(\uplambda - A)D$ is $\Vert \cdot \Vert$-dense in ${\mathcal {C}}_\infty (E)$. It follows that the bp-closure of $(\uplambda - A)D$ contains ${\mathcal {C}}_\infty (E)$ and thus equals all of ${\mathcal {B}}_b(E)$ (see Proposition A.1).

Put $S\mathrel {\mathop :}=\{ (u, \uplambda u -Au) : u\in D\}$. We note that ${{\hat{A}}}$, and hence also $\uplambda - {{\hat{A}}}$, is $\sigma$-closed and thus, in particular, bp-closed. As $S\subset \uplambda - {{\hat{A}}}$ it follows that the bp-closure ${\bar{S}}$ is a subset of $\uplambda - {{\hat{A}}}$. Now define $C\mathrel {\mathop :}=\{ f\in {\mathcal {B}}_b(E) : \exists u \text{ s.t. } (u, f) \in {\bar{S}}\}$. Then C is bp-closed.

Indeed, Let $(f_n) \subset C$ be bp-convergent to f. Pick $u_n$ such that $(u_n, f_n ) \in {\bar{S}}$. As $R(\uplambda ) = (\uplambda - {{\hat{A}}})^{-1}$, we see that $(R(\uplambda )f_n, f_n) \in \uplambda - \hat{A}$. It follows that $u_n - R(\uplambda ) f_n \in \textrm{ker}(\uplambda -{{\hat{A}}}) = \{0\}$, i.e. $u_n = R(\uplambda ) f_n$. By the $\sigma$-continuity of $R(\uplambda )$, we have that $u_n \rightharpoonup u\mathrel {\mathop :}=R(\uplambda )f$. This implies that $(u,f) \in {\bar{S}}$ whence $f\in C$, proving that C is bp-closed as claimed.

As C contains $(\uplambda - A)D$, it follows that $C= {\mathcal {B}}_b(E)$. But this entails that ${\bar{S}}= \uplambda -{{\hat{A}}}$. We have proved that $\uplambda -{{\hat{A}}}$ is the bp-closure of S which is equivalent to the claim. $\square$

3 Perturbation of (strong) Feller semigroups

We begin by recalling a result from [42] concerning the perturbation of semigroups of kernel operators consisting of strong Feller operators. In this section (and also in the following section), we work in an abstract framework, so that neither ${{\hat{A}}}$ nor ${{\hat{B}}}$ are assumed to be integro-differential operators. Instead, we will use the following set of assumptions.

Hypothesis 3.1

Let T be a ${\mathcal {C}}_b$-semigroup of type $(M, \omega )$ that consists of strong Feller operators. We denote by ${{\hat{A}}}$ the full generator of the semigroup and write $(R(\uplambda ))_{\textrm{Re}\,\uplambda >\omega }$ for the Laplace transform of T. Moreover, let ${{\hat{B}}} : D({{\hat{B}}}) \rightarrow {\mathcal {B}}_b(E)$ be a single-valued linear operator with $D({{\hat{A}}}) \subset D({{\hat{B}}})$ which satisfies the following assumptions:

• (i) For $t>0$ the operator ${{\hat{B}}} T(t)$, initially defined on $D({{\hat{A}}})$, has an extension to an operator in ${\mathcal {L}}({\mathcal {B}}_b(E), \sigma )$ (which, by slight abuse of notation, we still denote by ${{\hat{B}}}T(t)$);

• (ii) ${{\hat{B}}}R(\uplambda ) \in {\mathcal {L}}({\mathcal {B}}_b(E),\sigma )$ for one/all $\uplambda >\omega$;

• (iii) the function $(t,x) \mapsto {{\hat{B}}} T(t)f(x)$ is measurable for $f\in {\mathcal {B}}_b(E)$;

• (iv) there is a function $\varphi$ which is integrable in a neighborhood of 0 such that $\Vert {{\hat{B}}} T(t)\Vert \le \varphi (t)$ for $t>0$.

We now recall the following result from [42], which is a variant of the classical Miyadera–Voigt perturbation theorem, see [12, Thm. III.3.14]. Note, however, that neither of the semigroups T and S is strongly continuous in general, so that [12, Thm. III.3.14] is not directly applicable.

Theorem 3.2

Assuming Hypothesis 3.1 the operator ${{\hat{A}}} + {{\hat{B}}}$ (defined on $D({{\hat{A}}}) \subset D({{\hat{B}}})$) is the full generator of a ${\mathcal {C}}_b$-semigroup S that consists of strong Feller operators. This semigroup satisfies for $t>0$ the Duhamel formula

$$\begin{aligned} S(t)f = T(t)f + \int _0^t S(t-s){{\hat{B}}} T(s)f\, ds. \end{aligned}$$

(3.1)

Moreover, we can develop S in its Dyson–Phillips series:

$$\begin{aligned} S(t) = \sum _{n=0}^\infty S_n(t)\quad \text{ where }\quad S_0(t) = T(t)\quad \text{ and }\quad S_{n+1}(t)f = \int _0^t S_n(t-s){{\hat{B}}} T(s)\, ds,\nonumber \\ \end{aligned}$$

(3.2)

which converges in the operator norm. Here, all integrals have to be understood in the weak ${\mathcal {M}}_b(E)$-sense as in (2.2).

Proof

See [42, Thm. 3.3]. $\square$

It is a rather obvious question whether the new semigroup S is a ${\mathcal {C}}_\infty$-semigroup if this is the case for T. Unfortunately, without further assumptions on ${{\hat{B}}}$ this may fail.

Example 3.3

For $E= \mathbb {R}$, consider the heat semigroup T, given by

$$\begin{aligned} (T(t)f)(x) \mathrel {\mathop :}=\frac{1}{\sqrt{2\pi t}} \int _\mathbb {R}\exp \left( -\frac{(x-y)^2}{4t} \right) f(y)\, dy\quad (x\in \mathbb {R}) \end{aligned}$$

for $f\in {\mathcal {B}}_b(\mathbb {R})$. It is well known that T is a ${\mathcal {C}}_\infty$-semigroup of strong Feller operators. Now consider the operator ${{\hat{B}}}: f\mapsto f(0) \cdot \mathbbm {1}_\mathbb {R}$. Then ${{\hat{B}}} \in {\mathcal {L}}({\mathcal {B}}_b(\mathbb {R}), \sigma )$ (whence it satisfies Hypothesis 3.1, see [42, Ex. 3.4]). It follows that ${{\hat{B}}} T(s)f = c\cdot \mathbbm {1}_\mathbb {R}$, for some constant $c=c(s,f)$. Consequently, $T(t-s){{\hat{B}}} T(s)f= c \mathbbm {1}_\mathbb {R}$ for all $t\ge s$. If $0<f \in {\mathcal {C}}_\infty (\mathbb {R})$ then $c_t(f)\mathrel {\mathop :}=\inf \{c(s,f) : 0\le s \le t\}>0$ for all $t>0$ and we can infer from the Dyson–Phillips expansion (3.2) that the perturbed semigroup S does not leave ${\mathcal {C}}_\infty (\mathbb {R})$ invariant. Indeed, fixing $t>0$, we see that $S(t)f \ge S_1(t)f \ge c_t(f)\mathbbm {1}_\mathbb {R}$, and the latter does not vanish at infinity.

Theorem 3.4

Assume in addition to Hypothesis 3.1 that T is a positive ${\mathcal {C}}_\infty$-semigroup.

• (a) If ${{\hat{D}}}$ is a bp-core for ${{\hat{A}}}$, then $\{ (u, f+ {{\hat{B}}} u) : (u,f) \in {{\hat{D}}}\}$ is a bp-core for ${{\hat{A}}} + {{\hat{B}}}$.

• (b) If ${{\hat{B}}}T(t) {\mathcal {B}}_\infty (E) \subset {\mathcal {B}}_\infty (E)$ for every $t>0$, then S is a ${\mathcal {C}}_\infty$-semigroup.

Proof

(a) Let ${{\hat{D}}}$ be a bp-core for ${{\hat{A}}}$ and denote by ${\bar{S}}$ the bp-closure of the set $\{(u, f+ {{\hat{B}}} u), (u, f) \in {{\hat{D}}}\}$. As ${{\hat{A}}} + {{\hat{B}}}$ is a full generator, it is $\sigma$-closed and thus in particular bp-closed, and it follows that ${\bar{S}}\subset {{\hat{A}}} +{{\hat{B}}}$. Note that this entails that for $(u,g) \in {\bar{S}}$ we have $u \in D({{\hat{A}}})$. We now define $W\mathrel {\mathop :}=\{(u, g- {{\hat{B}}} u) : (u, g) \in {\bar{S}}\}$. Clearly, ${{\hat{D}}} \subset W \subset {{\hat{A}}}$. We prove that W is bp-closed. To that end, let $(u_n, g_n-{{\hat{B}}} u_n)$ be a sequence in W (thus $(u_n, g_n) \in {\bar{S}}$) such that $u_n$ bp-converges to u and $f_n \mathrel {\mathop :}=g_n - {{\hat{B}}} u_n$ bp-converges to f. Then $(u_n, f_n) \in {\bar{A}}$ bp-converges to (u, f). By the bp-closedness of ${{\hat{A}}}$, we have $(u,f)\in {{\hat{A}}}$. Picking $\uplambda >\omega$, this is equivalent to $u= R(\uplambda ) (\uplambda u- f)$, and the same equality holds with u/f replaced by $u_n/f_n$. Using that ${{\hat{B}}} R(\uplambda )$ is $\sigma$-continuous by (ii) in Hypothesis 3.1, we infer

$$\begin{aligned} {{\hat{B}}} u_n = {{\hat{B}}} R(\uplambda )(\uplambda u_n - f_n) \rightharpoonup {{\hat{B}}} R(\uplambda ) (\uplambda u - f) = {{\hat{B}}} u. \end{aligned}$$

Setting $g\mathrel {\mathop :}=f+{{\hat{B}}} u$, the sequence $(u_n, g_n) \subset {\bar{S}}$ bp-converges to (u, g). As ${\bar{S}}$ is bp-closed, $(u,g) \in {\bar{S}}$ and thus $(u, f) \in W$. Since ${{\hat{D}}}$ is a bp-core for ${{\hat{A}}}$, it follows that $W= {{\hat{A}}}$. But this is equivalent to ${\bar{S}} = {{\hat{A}}} + {{\hat{B}}}$ and so $\{(u, f+ {{\hat{B}}} u), (u, f) \in {{\hat{D}}}\}$ is indeed a bp-core for ${{\hat{A}}} + {{\hat{B}}}$.

(b) We first prove by induction that the operators $S_n(t)$ appearing in the Dyson–Phillips series (3.2) map ${\mathcal {B}}_\infty (E)$ to ${\mathcal {C}}_\infty (E)$. In particular, they leave the space ${\mathcal {B}}_\infty (E)$ invariant.

For $n=0$ this follows from Lemma 2.2 and our assumption that T is a positive ${\mathcal {C}}_\infty$-semigroup that enjoys the strong Feller property.

Let us assume that, for some $n\in \mathbb {N}$, we have already proved that $S_n(t)$ maps ${\mathcal {B}}_\infty (E)$ to ${\mathcal {C}}_\infty (E)$ for every $t>0$. Fix $f\in {\mathcal {B}}_\infty (E)$ and $t>0$. By our assumption, we have ${{\hat{B}}}T(s)f \in {\mathcal {B}}_\infty (E)$ for every $s\in (0,t)$. By the induction hypothesis, $S_n(t-s)$ maps ${\mathcal {B}}_\infty (E)$ to ${\mathcal {C}}_\infty (E)$, whence $S_n(t-s){{\hat{B}}} T(s) f \in {\mathcal {C}}_\infty (E)$ for $s\in (0,t)$. Since ${\mathcal {C}}_\infty (E)$ is separable with dual space ${\mathcal {M}}_b (E)$, the Pettis measurability theorem implies that the function $s\mapsto S_n(t-s){{\hat{B}}}T(s)f$ is Bochner integrable which, in turn, implies that $S_{n+1}(t)f = \int _0^t S_n(t-s){{\hat{B}}} T(s)f\, ds \in {\mathcal {C}}_\infty (E)$.

Note that the series in (3.2) converges for small t even in the operator norm. From this it follows that for small t we have $S(t)f \in {\mathcal {C}}_\infty (E)$. As f was arbitrary, $S(t){\mathcal {C}}_\infty (E)\subset {\mathcal {C}}_\infty (E)$ for small t and hence, by the semigroup law, for all $t>0$. The strong continuity of the restriction of S to ${\mathcal {C}}_\infty (E)$ follows from that of T, the identity (3.1) and the observation that the integral in that formula converges to 0 in operator norm as $t\mapsto 0$.

$\square$

4 Resolvent convergence of the perturbed operators

In this section, we will consider a sequence ${\hat{B}}_n$ of perturbations that converges, in a certain sense, to the operator ${\hat{B}}$. We want to know under which assumptions and in which sense the pseudoresolvents $(\uplambda - ({{\hat{A}}} + {\hat{B}}_n))^{-1}$ converge to $(\uplambda - ({{\hat{A}}} + {{\hat{B}}}))^{-1}$ as $n\rightarrow \infty$. Besides being interesting in its own right, such a convergence result will also allow us to establish dissipativity and/or resolvent positivity for large classes of operators with measurable coefficients. The strategy to use approximation to prove positivity was already used in [14].

The key to prove our convergence result is the following Lemma.

Lemma 4.1

Assuming Hypothesis 3.1, we have $\Vert {{\hat{B}}} R(\uplambda )\Vert \rightarrow 0$ as $\uplambda \rightarrow \infty$. Moreover, if $\uplambda$ is so large that $\Vert {{\hat{B}}}R(\uplambda ) \Vert <1$, then

$$\begin{aligned} (\uplambda - ({{\hat{A}}} + {{\hat{B}}}))^{-1} = R(\uplambda ) \sum _{k=0}^\infty ({{\hat{B}}} R(\uplambda ))^k, \end{aligned}$$

where the latter series converges in operator norm.

Proof

This relation is a consequence of the Dyson–Phillips series expansion for the perturbed semigroup and standard in Miyadera–Voigt type perturbation theorems. A proof can be found in [42, Lem. 3.8]. $\square$

Proposition 4.2

Assume Hypothesis 3.1 and let $({{\hat{B}}}_n, D({{\hat{B}}}_n))$ be an operator such that Hypothesis 3.1 is also fulfilled with ${{\hat{B}}}$ replaced by ${{\hat{B}}}_n$ (but with the same semigroup T). Moreover assume that

• (i) $\sup _{n\in \mathbb {N}}\Vert {{\hat{B}}}_n R(\uplambda )\Vert \rightarrow 0$ as $\uplambda \rightarrow \infty$,

• (ii) for every $h\in {\mathcal {B}}_b(E)$ and large enough $\uplambda$ we have ${{\hat{B}}}_n R(\uplambda ) h \rightharpoonup {{\hat{B}}}R(\uplambda )h$, and

• (iii) whenever $(h_n)\subset {\mathcal {B}}_b(E)$ satisfies $h_n \rightharpoonup 0$, we have ${{\hat{B}}}_nR(\uplambda )h_n \rightharpoonup 0$ for large enough $\uplambda$.

In this case, for a large enough $\uplambda$ we have

$$\begin{aligned} (\uplambda - ({{\hat{A}}} + {{\hat{B}}}_n ))^{-1} f \rightharpoonup (\uplambda - ({{\hat{A}}} + {{\hat{B}}}))^{-1}f \end{aligned}$$

(4.1)

for all $f\in {\mathcal {B}}_b(E)$.

Proof

By assumption (i), we can pick $\uplambda _0$ sufficiently large that $\Vert {{\hat{B}}}_nR(\uplambda )\Vert \le 1/2$ for every $n\in \mathbb {N}$ and $\uplambda \ge \uplambda _0$. By Lemma 4.1, we have

$$\begin{aligned} (\uplambda - ({{\hat{A}}} + {{\hat{B}}}_n))^{-1} = R(\uplambda )\sum _{k=0}^\infty ({{\hat{B}}}_n R(\uplambda ))^k. \end{aligned}$$

(4.2)

Now let $f\in {\mathcal {B}}_b(E)$ and $\varepsilon >0$. We fix $N\in \mathbb {N}$ such that $\sum _{k=N+1}^\infty \Vert ({{\hat{B}}}_n R(\uplambda ))^kf\Vert \le \varepsilon$ for all $n\in \mathbb {N}$.

We prove that, for every $k\in \mathbb {N}$, we have $({{\hat{B}}}_n R(\uplambda ))^kf \rightharpoonup ({{\hat{B}}}R(\uplambda ))^kf$ as $n\rightarrow \infty$.

For $k=1$ this is exactly our assumption (ii). Assume that we know this convergence to be true for some $k\in \mathbb {N}$. We set $g_n \mathrel {\mathop :}=({{\hat{B}}}_n R(\uplambda ))^k f$ and $g\mathrel {\mathop :}=({{\hat{B}}} R(\uplambda ))^k f$. Then

$$\begin{aligned} ({{\hat{B}}}_n R(\uplambda ))^{k+1}f - ({{\hat{B}}} R(\uplambda ))^{k+1}f = {{\hat{B}}}_n R(\uplambda )(g_n-g) + ({{\hat{B}}}_nR(\uplambda )g - {{\hat{B}}}R(\uplambda )g) \rightharpoonup 0. \end{aligned}$$

Indeed, by the induction hypothesis $g_n-g\rightharpoonup 0$ so that the first term converges weakly to 0 by our assumption (iii), and the second term converges weakly to 0 by the case $k=1$.

Altogether, we find that $\sum _{k=0}^\infty ({{\hat{B}}}_n R(\uplambda ))^k f \rightharpoonup \sum _{k=0}^\infty ({{\hat{B}}} R(\uplambda ))^kf$. Given that the operator $R(\uplambda )$ is $\sigma$-continuous, the representation (4.2) and Lemma 4.1 yield (4.1) $\square$

If the operator $R(\uplambda )$ is positive (which is always the case in our main application) we obtain stronger convergence results.

Corollary 4.3

In the situation of Proposition 4.2, assume additionally that $R(\uplambda )\ge 0$ is positivity preserving for large enough $\uplambda$. Then the convergence in (4.1) is not only with respect to $\sigma$, but even uniform on compact subsets of E.

Proof

Fix $f\in {\mathcal {B}}_b(E)$ and put $g_n\mathrel {\mathop :}=\sum _{k=0}^\infty ({{\hat{B}}}_n R(\uplambda ))^kf$ and $g \mathrel {\mathop :}=\sum _{k=0}^\infty ({{\hat{B}}} R(\uplambda ))^kf$. Fixing $m\in \mathbb {N}$, we put $h_m \mathrel {\mathop :}=\sup _{n\ge m} |g_n-g|$. We have seen in the proof of Proposition 4.2 that $g_n \rightharpoonup g$ and, consequently, $h_m \downarrow 0$ pointwise.

As $R(\uplambda )$ is positive, we find for $n\ge m$

$$\begin{aligned} \big | (\uplambda - ({{\hat{A}}}+{{\hat{B}}}_n))^{-1}f - (\uplambda - ({{\hat{A}}} +{{\hat{B}}}))^{-1}f\big |&= |R(\uplambda )g_n - R(\uplambda )g|\\&\le R(\uplambda )|g_n-g|\\&\le R(\uplambda ) h_m. \end{aligned}$$

The latter converges to 0 as m, hence n, tends to $\infty$. However, as $R(\uplambda )$ is positivity preserving this is even a monotone convergence and thus the convergence is uniform on compact subsets of E by Dini’s theorem. $\square$

It is natural to ask whether the sequence of perturbed semigroups also converge. In general, weak convergence of the resolvents does not imply weak convergence of the corresponding semigroups, see [11] for a concrete example. In Sect. 6, we will use the theory of martingale problems to establish a convergence result for semigroups associated with Lévy-type operators.

We can also prove that certain features of the pseudoresolvents are stable under the convergence described in Proposition 4.2.

Corollary 4.4

In the situation of Proposition 4.2, the following holds true.

• (i) Let $\uplambda \in \mathbb {C}$ with $\textrm{Re}\,\uplambda >0$ be in the resolvent set of ${\hat{A}}+{\hat{B}}_n$ and assume that $\Vert (\uplambda - ({{\hat{A}}} + {{\hat{B}}}_n))^{-1}\Vert \le (\textrm{Re}\,\uplambda )^{-1}$ for all $n\in \mathbb {N}$. Then $\uplambda$ is in the resolvent set of ${{\hat{A}}} + {{\hat{B}}}$ and $\Vert (\uplambda - ({{\hat{A}}} + {{\hat{B}}}))^{-1}\Vert \le (\textrm{Re}\,\uplambda )^{-1}$. In this case, the semigroup S, generated by ${{\hat{A}}} + {{\hat{B}}}$, is contractive.

• (ii) Suppose that for $\uplambda \ge \uplambda _0$ we have $(\uplambda + ({{\hat{A}}} + {{\hat{B}}}_n))^{-1}\ge 0$ for all $n\in \mathbb {N}$. Then also $(\uplambda + ({{\hat{A}}} + {{\hat{B}}}))^{-1} \ge 0$ for $\uplambda \ge \uplambda _0$. In this case, the semigroup S generated by ${{\hat{A}}} + {{\hat{B}}}$ is positive.

Proof

We put $F_n(\uplambda ) \mathrel {\mathop :}=(\uplambda - ({{\hat{A}}} + {{\hat{B}}}_n))^{-1}$ and $F(\uplambda ) \mathrel {\mathop :}=(\uplambda - ({{\hat{A}}} + {{\hat{B}}}))^{-1}$ whenever these are defined.

(i) The sequence $F_n$ is an (operator-valued) holomorphic function that is, by assumption, locally bounded on $\{\textrm{Re}\,\uplambda > 0\}$. Fixing $f\in {\mathcal {B}}_b(E)$ and $\mu \in {\mathcal {M}}_b(E)$, the same is true for the scalar function $\varphi _n: \uplambda \mapsto \varphi _n(\uplambda ) \mathrel {\mathop :}=\langle F_n(\uplambda )f, \mu \rangle$. By Proposition 4.2, for real $\uplambda$ that are large enough, $\varphi _n(\uplambda )$ converges to $\varphi (\uplambda ) \mathrel {\mathop :}=\langle F(\uplambda )f, \mu \rangle$. It follows from Vitali’s theorem (see [2, Thm. 2.1]) that $\varphi$ has a holomorphic extension to the set $\{\textrm{Re}\,\uplambda >0\}$ and $\varphi _n$ converges locally uniformly to $\varphi$. As f and $\mu$ are arbitrary, it follows from a vector-valued analytic extension theorem [2, Thm. 3.5] that F has an analytic continuation to $\{\textrm{Re}\,\uplambda >0\}$ which proves that $\{\textrm{Re}\,\uplambda >0\}$ is contained in the resolvent set of ${\hat{A}}+{\hat{B}}$. Moreover

$$\begin{aligned} |\langle F(\uplambda ) f, \mu \rangle | = \lim _{n\rightarrow \infty } |\langle F_n(\uplambda ) f, \mu \rangle | \le \limsup _{n\rightarrow \infty } \Vert F_n(\uplambda )\Vert \Vert f\Vert \Vert \mu \Vert \le \frac{1}{\textrm{Re}\,\uplambda }\Vert f\Vert \Vert \mu \Vert \end{aligned}$$

for every $\textrm{Re}\,\uplambda >0$. This implies the estimate for the Laplace transform. As for the contractivity of the semigroup, we note that for $f\in {\mathcal {C}}_b(E)$ and ${\mathcal {M}}_b(E)$ the orbit $t\mapsto \langle S(t)f, \mu \rangle$ is continuous whence the (scalar) Post–Widder inversion formula (see [1, Thm. 1.7.7]) yields

$$\begin{aligned} \langle S(t)f, \mu \rangle = \lim _{n\rightarrow \infty } \Big \langle \Big (\frac{n}{t}\Big )^n \Big (\frac{n}{t} - {{\hat{A}}} - {{\hat{B}}}\Big )^{-n}f, \mu \Big \rangle . \end{aligned}$$

By the resolvent estimate, the absolute value of the right-hand side is at most $\Vert f\Vert \Vert \mu \Vert$ and it follows that S is contractive on ${\mathcal {C}}_b(E)$. But then it is also contractive on ${\mathcal {B}}_b(E)$.

(ii) For $0\le f\in {\mathcal {B}}_b(E)$, $0\le \mu \in {\mathcal {M}}(E)$ and $\uplambda \ge 0$, we have

$$\begin{aligned} \langle F(\uplambda )f, \mu \rangle = \lim _{n\rightarrow \infty } \langle F_n(\uplambda ) f, \mu \rangle \ge 0 \end{aligned}$$

as for $\uplambda \ge \uplambda _0$ we have $F_n(\uplambda ) \ge 0$ for all $n\in N$. Since f and $\mu$ were arbitrary, this proves $F(\uplambda )\ge 0$ for $\uplambda \ge \uplambda _0$. As above, we can infer positivity of the semigroup S from this by means of the Post–Widder inversion formula. $\square$

5 Perturbation of strong Feller semigroups by Lévy-type operators

In this section, we work in the Euclidean setting, $E=\mathbb {R}^d$, and consider perturbations of strong Feller semigroups by a class of integro-differential operators. To that end, we will make more concrete assumptions on our initial semigroup T which are tailor-made for this situation.

Hypothesis 5.1

Assume that $T=(T(t))_{t\ge 0}$ is a ${\mathcal {C}}_b$-semigroup of type $(M, \omega )$ with state space $E=\mathbb {R}^d$ that consists of strong Feller operators. Moreover, the following conditions hold for some $\rho >0$:

• (i) $T(t)f \in {\mathcal {C}}_b^\rho (\mathbb {R}^d)$ for all $t>0$ and $f\in {\mathcal {B}}_b(\mathbb {R}^d)$;

• (ii) There is a function $\varphi \in L^1(0,1)$ such that

  $$\begin{aligned} \Vert T(t)f\Vert _{{\mathcal {C}}_b^\rho } \le \varphi (t)\Vert f\Vert _\infty \end{aligned}$$

  for all $t \in (0,1)$ and $f\in {\mathcal {B}}_b(\mathbb {R}^d)$;

As before, we will denote the Laplace transform of our semigroup T by $(R(\uplambda ))_{\textrm{Re}\,\uplambda >\omega }$ and the full generator of T by ${{\hat{A}}}$. For future reference, we note the following consequence of Hypothesis 5.1.

Lemma 5.2

Assume Hypothesis 5.1. Then, for every $\uplambda \in \mathbb {C}$ with $\textrm{Re}\,\uplambda >\omega$, we have $R(\uplambda )f \in {\mathcal {C}}_b^{\rho }(\mathbb {R}^d)$ for all $f\in {\mathcal {B}}_b(\mathbb {R}^d)$ and

$$\begin{aligned} \Vert R(\uplambda )f\Vert _{{\mathcal {C}}_b^{\rho }} \le C(\uplambda ) \Vert f\Vert _\infty , \quad f\in {\mathcal {B}}_b(\mathbb {R}^d), \end{aligned}$$

for a constant $C(\uplambda )$ with $C(\uplambda ) \rightarrow 0$ as $\textrm{Re}\,\uplambda \rightarrow \infty$. In particular, $D({{\hat{A}}}) \subset {\mathcal {C}}_b^{\rho }(\mathbb {R}^d).$

Proof

For $r>0$ it follows from the exponential boundedness of the semigroup and Hypothesis 5.1(ii) that

$$\begin{aligned} \Vert T(1+r)f\Vert _{{\mathcal {C}}_b^\rho } \le \Vert T(1)\Vert _{{\mathcal {L}}({\mathcal {C}}_b^\rho , {\mathcal {B}}_b)} \Vert T(r)\Vert \, \Vert f\Vert _\infty \le M\varphi (1)e^{\omega r} \Vert f\Vert _\infty . \end{aligned}$$

Setting $\psi (t) = \varphi (t)$ for $t\in (0,1)$ and $\psi (t) = M\varphi (1) e^{\omega (t-1)}$ for $t \ge 1$, we get

$$\begin{aligned} \Vert T(t) f\Vert _{{\mathcal {C}}_b^{\rho }} \le \psi (t) \Vert f\Vert _{\infty }, \quad t >0, \; f \in {\mathcal {B}}_b(\mathbb {R}^d). \end{aligned}$$

(5.1)

Now fix $f \in {\mathcal {B}}_b(\mathbb {R}^d)$ and $x \in \mathbb {R}^d$. Since $\Vert T(t)\Vert \le M e^{\omega t}$ and $t \mapsto T(t) f(x)$ is Borel measurable, the integral $R(\uplambda ) f(x) = \int _{(0,\infty )} e^{-\uplambda t} T(t) f(x) \, dt$ exists as a Lebesgue integral for any $\uplambda$ with $\textrm{Re}\,\uplambda >\omega$. If $\rho \in (0,1)$, then it is immediate from (5.1) and the estimate

$$\begin{aligned} |R(\uplambda )f(x) - R(\uplambda )f(z)| \le \int _0^\infty e^{-\uplambda t} |T(t)f(x)-T(t)f(z)| \, dt \end{aligned}$$

that $\Vert R(\uplambda ) f\Vert _{{\mathcal {C}}_b^{\rho }} \le C(\uplambda ) \Vert f\Vert _{\infty }$, where $C(\uplambda ):= \int _0^\infty \psi (t) e^{-\uplambda t} \, dt$. If $\rho \ge 1$, then the differentiation lemma for parameter-depend integrals, see e.g. [52, Thm. 12.5] or [37, Prop. A.1], implies that $x \mapsto R(\uplambda ) f(x)$ is differentiable and

$$\begin{aligned} \frac{\partial }{\partial x_i} R(\uplambda ) f(x) = \int _0^\infty e^{-\uplambda t} \frac{\partial }{\partial x_i} T(t)f(x) \, dt \end{aligned}$$

for all $\textrm{Re}\,\uplambda >\omega$. Thus, by (5.1), $\Vert R(\uplambda ) f\Vert _{{\mathcal {C}}_b^{\rho }} \le C(\uplambda ) \Vert f\Vert _{\infty }$ with $C(\uplambda )$ as before. By dominated convergence, $C(\uplambda ) \rightarrow 0$ as $\textrm{Re}\,\uplambda \rightarrow \infty$. $\square$

We now introduce the integro-differential operator ${{\hat{B}}}$ that we will consider as a perturbation. We fix a function $\chi$ such that $\mathbbm {1}_{B(0,1)}\le \chi \le \mathbbm {1}_{B(0,2)}$ and put

$$\begin{aligned} {\hat{B}}f(x) = b(x) \cdot \nabla f(x) + \int _{\mathbb {R}^d {\setminus }\{0\}} (f(x+y)-f(x)-y \cdot \nabla f(x) \chi (y)) \, \mu (x,dy).\nonumber \\ \end{aligned}$$

(5.2)

We make the following standing assumption.

Hypothesis 5.3

The function $b: \mathbb {R}^d \rightarrow \mathbb {R}^d$ and the kernel $\mu : \mathbb {R}^d\times {\mathcal {B}}(\mathbb {R}^d{\setminus }\{0\}) \rightarrow [0, \infty ]$ satisfy

• (i) b is Borel measurable and bounded;

• (ii) $x \mapsto \int _{\mathbb {R}^d {\setminus } \{0\}} f(y) \, \mu (x,dy)$ is Borel measurable for every $f \in {\mathcal {C}}_c(\mathbb {R}^d {\setminus } \{0\})$.

• (iii) There is a constant $\beta \in (0,2)$ such that

  $$\begin{aligned} \Vert \mu \Vert _{\beta } := \sup _{x \in \mathbb {R}^d} \left( \int _{\mathbb {R}^d {\setminus } \{0\}} \min \{|y|^{\beta },1\} \, \mu (x,dy) \right) <\infty . \end{aligned}$$

  (5.3)

  Moreover, $\beta$ is assumed to be strictly smaller than the constant $\rho$ from Hypothesis 5.1.

• (iv) If $\rho \le 1$, then the compensated drift $b(\cdot ) - \int _{\mathbb {R}^d{\setminus } \{0\}} y\chi (y)\, \mu (\cdot , dy)$ is identically zero.

Occasionally we will additionally assume the following tightness assumption

$$\begin{aligned} \sup _{x\in \mathbb {R}^d} \mu (x, \{|y|>R\}) \rightarrow 0 \quad \text{ as }\quad R\rightarrow \infty . \end{aligned}$$

(5.4)

Let us comment briefly on these assumptions. Assumption (iii) implies that the above operator ${{\hat{B}}}$ is well-defined on ${\mathcal {C}}_b^\rho (\mathbb {R}^d)$. To see this, let us first assume that $\rho >1$. Then, clearly, the local part $b(\cdot ) \nabla f(\cdot )$ of ${{\hat{B}}}$ is well-defined on this space. As for the integral part, the elementary estimate

$$\begin{aligned} |f(x+y)-f(x)- y \cdot \nabla f(x) \chi (|y|)| \le 2\Vert f\Vert _{{\mathcal {C}}_b^{\rho }(\mathbb {R}^d)} \min \{1,|y|^{\rho }\} \end{aligned}$$

implies that $\Vert {\hat{B}}f\Vert _{\infty } \le 2\Vert \mu \Vert _{\beta } \Vert f\Vert _{{\mathcal {C}}_b^{\rho }}$. In the case $\rho \le 1$, the additional assumption (iv) entails that the operator ${{\hat{B}}}$ simplifies to

$$\begin{aligned} {{\hat{B}}} f(x) = \int _{\mathbb {R}^d{\setminus }\{0\}} (f(x+y) - f(x))\, \mu (x, dy) \end{aligned}$$

and we can argue similarly, making use of the Hölder continuity of f. The assumption $\beta <\rho$ yields that the operators ${{\hat{B}}} T(t)$ and ${{\hat{B}}} R(\uplambda )$ fulfill conditions (i)–(iv) of Hypothesis 3.1, whenever the semigroup T satisfies Hypothesis 5.1. The tightness condition (5.4) plays an important role in proving that the perturbed semigroup leaves the space ${\mathcal {C}}_\infty (\mathbb {R}^d)$ invariant. We note that (5.4) holds if the real part of the symbol

$$\begin{aligned} p(x,\xi ) := -ib(x) \cdot \xi + \int _{\mathbb {R}^d {\setminus } \{0\}} (1-e^{iy \cdot \xi } + iy \cdot \xi \chi (|y|)) \, \mu (x,dy) \end{aligned}$$

of the operator ${\hat{B}}$ is equicontinuous at $\xi =0$, i.e.

$$\begin{aligned} \lim _{|\xi | \rightarrow 0} \sup _{x \in \mathbb {R}^d} |\textrm{Re}\,p(x,\xi )|=0, \end{aligned}$$

(5.5)

see [50, (proof of) Thm. 4.4].

We can now formulate the main result of this section.

Theorem 5.4

Assume Hypotheses 5.1 and 5.3. Then the following hold true.

• (a) The operator ${{\hat{A}}} + {{\hat{B}}}$ is the full generator of a ${\mathcal {C}}_b$-semigroup $S= (S(t))_{t\ge 0}$ of strong Feller operators.

• (b) If T consists of sub-Markovian operators then so does S.

• (c) Assume additionally the tightness assumption (5.4), that T is a positive ${\mathcal {C}}_\infty$-semigroup and that the test functions ${\mathcal {C}}_c^\infty (\mathbb {R}^d)$ form a core for the generator A on ${\mathcal {C}}_\infty (\mathbb {R}^d)$. Then S is a ${\mathcal {C}}_\infty$-semigroup.

Remark 5.5

In the situation of Theorem 5.4(b), the perturbed semigroup S is Markovian if and only if the unperturbed semigroup T is Markovian. Indeed, we only need to check if the semigroup is conservative, i.e. if the semigroup leaves the function $\mathbbm {1}$ invariant. By Equation (2.3) this is the case if and only if the full generator maps $\mathbbm {1}$ to the zero function. As ${{\hat{B}}} \mathbbm {1}=0$, we see that $({{\hat{A}}} + {{\hat{B}}})\mathbbm {1}= 0$ is equivalent to ${{\hat{A}}} \mathbbm {1}=0$.

Example 5.6

Without the tightness assumption (5.4), it is not true in general that the perturbed semigroup S is a ${\mathcal {C}}_\infty$-semigroup even if this is the case for T. This can be seen for example by considering in dimension $d=1$ the kernel $\mu (x, \cdot ) \mathrel {\mathop :}=\delta _{-x}$, which obviously does not satisfy (5.4). Choose $b=0$ and let T be the heat semigroup. Then ${{\hat{B}}} f = f(0)\mathbbm {1}_\mathbb {R}- f$ and the perturbed semigroup S is (up to a rescaling by the factor $e^{-t}$) the semigroup from Example 3.3, which is not a ${\mathcal {C}}_\infty$-semigroup.

Corollary 5.7

Assume Hypotheses 5.1 and 5.3 and that T is a positive ${\mathcal {C}}_\infty$-semigroup. If ${\mathcal {C}}_c^\infty (\mathbb {R}^d)$ is a core for the generator A of $T|_{{\mathcal {C}}_\infty (\mathbb {R}^d)}$, then $\{ (f, Af +{{\hat{B}}} f) : f\in {\mathcal {C}}_c^\infty (\mathbb {R}^d)\}$ is a bp-core for the full generator ${{\hat{A}}} +{{\hat{B}}}$ of S.

Proof

If ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ is a core for the generator on ${\mathcal {C}}_\infty (\mathbb {R}^d)$, then, by Lemma 2.4, $\{(f,Af); f \in {\mathcal {C}}_c^{\infty }(\mathbb {R}^d)\}$ is a bp-core for ${\hat{A}}$. Thus, Theorem 3.4(a) shows that, $\{(f,Af+{\hat{B}}f); f \in {\mathcal {C}}_c^{\infty }(R^d)\}$ is a bp-core for the full generator ${\hat{A}}+{\hat{B}}$. $\square$

We now turn to the proof of Theorem 5.4.

Proof of parts (a) and (c) of Theorem 5.4

(a) Let us first prove prove that ${{\hat{B}}} f \in {\mathcal {B}}_b(\mathbb {R}^d)$ for $f\in {\mathcal {C}}_b^\rho (\mathbb {R}^d)$. This is obvious for the local part of ${{\hat{B}}} f$ so we focus on the integral part. From Hypothesis 5.3(ii), it follows that for any Borel subset S of $\mathbb {R}^d{\setminus }\{0\}$ the map $x\mapsto \int _{\mathbb {R}^d{\setminus }\{0\}} \mathbbm {1}_S(y)\, \mu (x, dy)$ is measurable. But then so is $x\mapsto \int _{\mathbb {R}^d{\setminus }\{0\}} \mathbbm {1}_{R\times S}(x,y)\, \mu (x, dy)$ for Borel sets $R\subset \mathbb {R}^d$ and $S\subset \mathbb {R}^d{\setminus }\{0\}$. An application of the functional monotone class theorem, see e.g. [52, Thm. 8.15], yields that $x \mapsto \int _{\mathbb {R}^d {\setminus } \{0\}} g(x,y) \, \mu (x,dy)$ is Borel measurable for any function g which is ${\mathcal {B}}(\mathbb {R}^d) \otimes {\mathcal {B}}(\mathbb {R}^d {\setminus } \{0\})$-measurable and satisfies $|g(x,y)| \le K \min \{1,|y|^{\rho }\}$ for some constant $K>0$. This readily gives that ${\hat{B}}f$ is Borel measurable. Note that the discussion before Theorem 5.4 shows that, in fact, ${{\hat{B}}}$ is a bounded linear operator from ${\mathcal {C}}_b^\rho (\mathbb {R}^d)$ to ${\mathcal {B}}_b(\mathbb {R}^d)$.

We now verify Hypothesis 3.1, then part (a) immediately follows from Theorem 3.2. As for Condition 3.1(i), the boundedness of ${{\hat{B}}} T(t)$ follows from Hypothesis 5.1(i) and the boundedness of ${{\hat{B}}}$ proved above. To prove continuity in the weak topology, let a bounded sequence $(f_n)_{n\in \mathbb {N}} \subset {\mathcal {B}}_b(\mathbb {R}^d)$ be given such that $f_n \rightarrow f$ pointwise. By Hypothesis 5.1(i), $T(t)f_n$ is bounded in ${\mathcal {C}}_b^\rho (\mathbb {R}^d)$. By the Arzelà–Ascoli theoren, passing to a subsequence, we may assume that $T(t)f_n$ converges locally uniformly to some continuous function. By the $\sigma$-continuity of T(t) the sequence $T(t)f_n$ converges pointwise to T(t)f, which implies that the only possible limit is T(t)f and it follows that $T(t)f_n \rightarrow T(t)f$ locally uniformly. Note that in the case where $\rho >1$ we also obtain that $\nabla T(t)f_n \rightarrow \nabla T(t)f$ locally uniformly. From this, it is immediate that the local part of ${{\hat{B}}}f_n$ converges pointwise to that of ${{\hat{B}}} f$. For the integral part this convergence follows from dominated convergence, noting that, as a consequence of the uniform boundedness in ${\mathcal {C}}_b^{\rho }(\mathbb {R}^d)$, we find an integrable majorant of the form $C\max \{1, |y|^\rho \}$.

The proof of Condition 3.1(ii) is similar, taking Lemma 5.2 into account. As for Condition 3.1(iii), we note that $(t,x) \mapsto T(t) f(x)$ is Borel measurable for every $f \in {\mathcal {B}}_b(\mathbb {R}^d)$ and using a reasoning similar to that in the first part of this proof, it follows that $(t,x) \mapsto {\hat{B}}T(t) f(x)$ is Borel measurable.

Condition 3.1(iv) is an immediate consequence of Hypothesis 5.1(ii) and the boundedness of ${{\hat{B}}}$.

(c) To prove this part, we use Theorem 3.4. We have to prove that ${{\hat{B}}} T(t) {\mathcal {B}}_\infty (\mathbb {R}^d)\subset {\mathcal {B}}_\infty (\mathbb {R}^d)$. We will only consider the case $\rho >1$; for $\rho \le 1$ the reasoning is a bit simpler because all terms involving the gradient vanish by Hypothesis 5.3(iv). Take $f\in {\mathcal {C}}_b^\rho (\mathbb {R}^d)\cap {\mathcal {C}}_c(\mathbb {R}^d)$ and choose $R>0$ such that the support of f is contained in the ball B(0, R). Taking into account that we now assume the tightness condition (5.4), we find for $x \in \mathbb {R}^d$ with $|x| \gg R$ sufficiently large that

$$\begin{aligned} |{\hat{B}} f(x)| = \left| \int f(x+y) \, \mu (x,dy) \right|&\le \Vert f\Vert _{\infty } \int _{\{|y| \ge r\}} \mu (x,dy) \xrightarrow []{r \rightarrow \infty } 0. \end{aligned}$$

Thus, ${{\hat{B}}} f \in {\mathcal {B}}_\infty (\mathbb {R}^d)$.

Now let $g\in {\mathcal {C}}_\infty (\mathbb {R}^d)$ be given. We denote the generator of $T|_{{\mathcal {C}}_\infty (\mathbb {R}^d)}$ by A. As ${\mathcal {C}}_c^\infty (\mathbb {R}^d)$ is a core, we find a sequence $(f_n)\subset {\mathcal {C}}_c^\infty (\mathbb {R}^d)$ such that $g_n \mathrel {\mathop :}=\uplambda f_n - A f_n \rightarrow g$, see [12, Ex. II.1.15]. It then follows that $f_n \rightarrow f=R(\uplambda )g \in D(A)$. By the above, ${{\hat{B}}} R(\uplambda )g_n = {{\hat{B}}} f_n \in {\mathcal {B}}_\infty (\mathbb {R}^d)$. As ${{\hat{B}}}R(\uplambda )$ is $\Vert \cdot \Vert _\infty$-continuous, it follows that ${{\hat{B}}} f = {{\hat{B}}} R(\uplambda )g = \lim _{n\rightarrow \infty } {{\hat{B}}}R(\uplambda ) g_n = \lim _{n\rightarrow \infty }{{\hat{B}}} f_n$ also belongs to ${\mathcal {B}}_\infty (\mathbb {R}^d)$. Consequently, ${{\hat{B}}} f \in {\mathcal {B}}_\infty (\mathbb {R}^d)$ whenever $f\in D(A)$.

As $T(t)D(A) \subset D(A)$ for every $t>0$, it follows that ${{\hat{B}}} T(t)f \in {\mathcal {B}}_\infty (\mathbb {R}^d)$ for $f\in D(A)$. But as D(A) is dense in ${\mathcal {C}}_\infty (\mathbb {R}^d)$ and ${{\hat{B}}} T(t)$ is $\Vert \cdot \Vert _{\infty }$-continuous, this remains valid for $f\in {\mathcal {C}}_\infty (\mathbb {R}^d)$. $\square$

Remark 5.8

The statement of Theorem 5.4(c) remains valid also without the assumption that ${\mathcal {C}}_c^\infty (\mathbb {R}^d)$ is a core for $T|_{{\mathcal {C}}_\infty (\mathbb {R}^d)}$, provided we assume that $b\in {\mathcal {B}}_\infty (\mathbb {R}^d; \mathbb {R}^d)$.

To see this, pick a sequence $\phi _n \in {\mathcal {C}}_c^\infty (\mathbb {R}^d)$ with $\mathbbm {1}_{B(0,n)} \le \phi _n \le \mathbbm {1}_{B(0,2n)}$ such that $\sup _n \Vert \phi _n\Vert _{{\mathcal {C}}_b^2(\mathbb {R}^d)}<\infty$. Given $f\in {\mathcal {C}}_b^\rho (\mathbb {R}^d) \cap {\mathcal {C}}_\infty (\mathbb {R}^d)$, we put $f_n \mathrel {\mathop :}=f\phi _n$. Then $f_n \in {\mathcal {C}}_b^\rho (\mathbb {R}^d)\cap {\mathcal {C}}_c(\mathbb {R}^d)$ and $f_n$ is a bounded sequence that converges to f uniformly on $\mathbb {R}^d$; moreover, $\nabla f_n$ is bounded and converges to $\nabla f$ locally uniformly in case that $\varrho >1$.

As $b\in {\mathcal {B}}_\infty (\mathbb {R}^d; \mathbb {R}^d)$ and the sequence $\nabla f_n$ is uniformly bounded, it is easy to see that $b\nabla f_n \rightarrow b\nabla f$ with respect to $\Vert \cdot \Vert _\infty$. Making use of the boundedness of $(f_n)$ in ${\mathcal {C}}_b^\rho (\mathbb {R}^d)$ and the tightness assumption (5.4) we can prove that also the nonlocal part converges with respect to $\Vert \cdot \Vert _\infty$. . In conclusion, ${{\hat{B}}} f_n \rightarrow {{\hat{B}}} f$ with respect to $\Vert \cdot \Vert _\infty$. As the proof of Theorem 5.4(c) shows that ${{\hat{B}}} f_n \in {\mathcal {B}}_\infty (\mathbb {R}^d)$, the same is true for its uniform limit ${{\hat{B}}} f$. This proves that ${{\hat{B}}} f\in {\mathcal {B}}_\infty (\mathbb {R}^d)$ for all $f\in {\mathcal {C}}_b^\rho (\mathbb {R}^d)\cap {\mathcal {C}}_\infty (\mathbb {R}^d)$ which is enough to ensure that ${{\hat{B}}}T(t)$ maps ${\mathcal {C}}_\infty (\mathbb {R}^d)$ to ${\mathcal {B}}_\infty (\mathbb {R}^d)$.

To prove part (b) of Theorem 5.4, we will employ the convergence result of Proposition 4.2. As this form of resolvent convergence is of independent interest, we formulate a separate lemma.

Lemma 5.9

Assume Hypothesis 5.1 and let sequences $(b_n)$ and $(\mu _n)$ be given that satisfy the assumptions in Hypothesis 5.3 for a common constant $\beta \in (0, \min \{2, \rho \})$. Denote the associated operators (see (5.2)) by ${{\hat{B}}}_n$. Moreover, we assume that b and $\nu$ are functions such that

• (i) $b_n(x) \rightarrow b(x)$ and $\mu _n(x,\cdot ) \rightarrow \mu (x,\cdot )$ vaguely for every $x \in \mathbb {R}^d$,

• (ii) $\sup _{n \in \mathbb {N}}\left( \Vert b_n\Vert _{\infty } + \Vert \mu _n\Vert _{\beta }\right) <\infty$ with $\Vert \cdot \Vert _{\beta }$ defined in (5.3).

• (iii) $\sup _{x\in K}\sup _{n\in \mathbb {N}}\mu _n (x, \{ |y|>R\}) \rightarrow 0$ as $R\rightarrow \infty$ for $K\Subset \mathbb {R}^d$.

Then, denoting the operator associated with b and $\mu$ by ${{\hat{B}}}$, we find

$$\begin{aligned} (\uplambda -({\hat{A}}+{\hat{B}}_n))^{-1} f \rightharpoonup (\uplambda -({\hat{A}}+{\hat{B}}))^{-1} f, \qquad f \in {\mathcal {B}}_b(\mathbb {R}^d), \end{aligned}$$

for large enough $\uplambda$.

Proof

We give the proof only for $\rho >1$. If $\rho \le 1$, then by Hypothesis 5.3(iv) all terms involving the gradient vanish in the computations below; apart from that, the reasoning is analogous.

Without loss of generality, we may assume $\beta \ge 1$ (otherwise consider ${\tilde{\beta }} := \max \{1,\beta \}$). We first note that also the functions b and $\nu$ satisfy Hypothesis 5.3 (see Lemma A.4), whence Theorem 5.4(a) yields that ${{\hat{A}}} + {{\hat{B}}}$ and, for every $n\in \mathbb {N}$, the operator ${{\hat{A}}} + {{\hat{B}}}_n$ is the full generator of a ${\mathcal {C}}_b$-semigroup. The discussion following Hypothesis 5.3 shows that the operator ${{\hat{B}}}_n$ defines a bounded linear operator from ${\mathcal {C}}_b^{\beta } (\mathbb {R}^d)$ to ${\mathcal {B}}_b(\mathbb {R}^d)$. Using our assumption (ii) above, we actually see that there is a constant $K>0$ such that

$$\begin{aligned} \Vert {\hat{B}}f\Vert _{\infty } + \sup _{n \in \mathbb {N}} \Vert {\hat{B}}_n f\Vert _{\infty } \le K \Vert f\Vert _{{\mathcal {C}}_b^{\beta }(\mathbb {R}^d)}, \qquad f \in {\mathcal {C}}_b^{\beta }(\mathbb {R}^d). \end{aligned}$$

(5.6)

We now check the assumptions of Proposition 4.2.

Assumption 4.2(i): By Lemma 5.2, we have

$$\begin{aligned} \Vert {{\hat{B}}}_n R(\uplambda )\Vert \le \Vert {{\hat{B}}}_n\Vert _{{\mathcal {L}}({\mathcal {C}}_b^\beta , {\mathcal {B}}_b)}\Vert R(\uplambda )\Vert _{{\mathcal {L}}({\mathcal {B}}_b, {\mathcal {C}}_b^\beta )}\le KC(\uplambda ) \rightarrow 0 \end{aligned}$$

as $\uplambda \rightarrow \infty$.

Assumption 4.2(ii): In view of Lemma 5.2, it suffices to prove that ${\hat{B}}_n f \rightharpoonup {\hat{B}}f$ for every $f \in {\mathcal {C}}_b^{\rho }(\mathbb {R}^d)$, so fix $f \in {\mathcal {C}}_b^{\rho }(\mathbb {R}^d)$. Because of (5.6), it actually suffices to show pointwise convergence. Since $b_n$ converges pointwise to b, it is clear that the local part of ${\hat{B}}_nf$ converges to the local part of ${\hat{B}}f$; therefore we assume in the following that the local part is zero. Write

$$\begin{aligned} {\hat{B}}_n f = I_n^{(r)} f + J_n^{(r,R)}f + K_n^{(R)} f, \end{aligned}$$

where

$$\begin{aligned} I_n^{(r)} f(x)&:= \int _{\{0<|y| \le r\}} (f(x+y)-f(x)-\nabla f(x) \cdot y) \, \mu _n(x,dy), \\ J_n^{(r,R)} f(x)&:= \int _{\{r<|y| <R\}} (f(x+y)-f(x)-\nabla f(x) \cdot y \chi (y)) \, \mu _n(x,dy), \\ K_n^{(R)} f(x)&:= \int _{\{|y| \ge R\}} (f(x+y)-f(x)) \, \mu _n(x,dy), \end{aligned}$$

for $r \in (0,1)$ and $R \in (1,\infty )$. We decompose the operator ${\hat{B}}$ in the same way, and we study the convergence for each of the terms separately. By an application of Taylor’s formula,

$$\begin{aligned} \sup _{n \in \mathbb {N}} |I_n^{(r)} f(x)| \le \Vert f\Vert _{{\mathcal {C}}_b^{\rho }(\mathbb {R}^d)} \sup _{n \in \mathbb {N}} \int _{\{|y| \le r\}} |y|^{\rho } \, \mu _n(x,dy). \end{aligned}$$

As $\rho >\beta$, it follows from (ii) that the right-hand side is finite and converges to 0 (uniformly in x) as $r \rightarrow 0$. As $\Vert \mu \Vert _{\beta }<\infty$, an analogous estimate holds for $I^{(r)} f$. Using

$$\begin{aligned} |K_n^{(R)} f(x)| \le 2 \Vert f\Vert _{\infty } \int _{\{|y| \ge R\}} \mu _n(x,dy) \end{aligned}$$

and the corresponding estimate for $K^{(R)} f$, assumption (iii) implies that $|K_n^{(R)} f(x)| + |K^{(R)} f(x)| \rightarrow 0$ uniformly in $n \in \mathbb {N}$ and x in compact subsets of $\mathbb {R}^d$ as $R \rightarrow \infty$. As f and $\nabla f$ are continuous, the vague convergence $\mu _n(x,\cdot ) \rightarrow \mu (x,\cdot )$ entails that $J_{n}^{r,R}f(x) \rightarrow J^{r,R} f(x)$ for any constants $r,R>0$ with $\mu (x,\{|y|=r\})=0$ and $\nu (x,\{|y|=R\})=0$. Since there are for each fixed x at most countably many radii with $\mu (x,\{|y|=r\})>0$, we can let $r \rightarrow 0$ and $R \rightarrow \infty$ along suitable sequences to deduce that ${\hat{B}}_n f(x) \rightarrow {\hat{B}}f(x)$.

Assumption 4.2(iii): Making use of Lemma 5.2 again, we see that it suffices to prove that ${\hat{B}}_n u_n \rightharpoonup 0$ for any sequence $(u_n)_{n \in \mathbb {N}}$ with $u_n \rightharpoonup 0$ and $\sup _{n\in \mathbb {N}} \Vert u_n\Vert _{{\mathcal {C}}_b^{\rho }}<\infty$.

Fix such a sequence. Then $\sup _{n \in \mathbb {N}} \Vert {\hat{B}}_nu_n\Vert _{\infty }<\infty$ is immediate from (5.6) and it only remains to prove pointwise convergence. Pick $\rho ' \in (\beta , \rho )$. By Taylor’s formula,

$$\begin{aligned} |{\hat{B}}_n u_n(x)|&\le \Vert b_n\Vert _{\infty } |\nabla u_n(x)| + \Vert u_n\Vert _{{\mathcal {C}}_b^{\rho '}(B_R)} \int _{\{|y| \le R\}} |y|^{\rho '} \, \mu _n(x,dy) \\&\quad + 2 \Vert u_n\Vert _{\infty } \int _{\{|y|>R\}} \, \mu _n(x,dy) \end{aligned}$$

for any $R>1$; here $B_R$ is the closed ball around x with radius R. Using (ii), the fact that $\rho >1$ and that $(b_n)_{n \in \mathbb {N}}$ is bounded, we find that given $\varepsilon >0$ there is some $R \gg 1$ such that

$$\begin{aligned} |{\hat{B}}_n u_n(x)| \le C \Vert u_n\Vert _{{\mathcal {C}}_b^{\rho '}(B_R)} + \varepsilon \end{aligned}$$

for a finite constant $C=C(R)$. In order to deduce that the left-hand side converges to 0, we use the subsequence principle. As $(u_n)$ is bounded in ${\mathcal {C}}_b^\rho$, by compact embedding, there is a subsequence $(u_{n''})_{n'' \in \mathbb {N}}$ which converges on the closed ball B[x, r] with respect to the $C^{\rho '}$-norm. Since we know that $u_n \rightarrow 0$ pointwise, the $C^{\rho }$-limit is also zero and so

$$\begin{aligned} \limsup _{n'' \rightarrow \infty } |{\hat{B}}_{n''} u_{n''}(x)| \le \varepsilon . \end{aligned}$$

Hence, $\limsup _{n \rightarrow \infty } |{\hat{B}}_n u_n(x)| \le \varepsilon$. As $\varepsilon >0$ is arbitrary, we conclude that ${\hat{B}}_n u_n(x) \rightarrow 0$ for all $x \in \mathbb {R}^d$.

Now Proposition 4.2 yields the claim. $\square$

We can now finish the proof of Theorem 5.4

Proof of Theorem 5.4(b)

Let us first consider kernels $\mu$ satisfying

$$\begin{aligned} \sup _{x\in K} \mu (x, \{|y|> R\}) \rightarrow 0\quad \text{ as }\quad R\rightarrow \infty \qquad \text{ for } \text{ all } K\Subset \mathbb {R}^d. \end{aligned}$$

(5.7)

Given b and $\mu$ as in Hypothesis 5.3, we write ${{\hat{B}}}_{b,\mu }$ for the operator defined via (5.2) with these particular coefficients and $S_{b,\mu } = (S_{b,\mu }(t))_{t\ge 0}$ for the semigroup generated by the operator ${{\hat{A}}} + {{\hat{B}}}_{b,\mu }$, which exists by part (a) of Theorem 5.4. Denote by F the family of pairs $(b,\mu )$ for which the assertion holds, i.e.

$$\begin{aligned} F= \{(b,\mu ) \, \, \text {satisfying}\,\, (5.7)\,\, \text {and}\,\, \text {Hypothesis}~5.3 \,:\, S_{b,\mu } \text {is sub-Markovian}\}. \end{aligned}$$

It follows from Lemma 5.9 and Corollary 4.4 that F is bp-closed. Therefore, it suffices to show that $(b,\mu ) \in F$ for any $b \in {\mathcal {C}}_c(\mathbb {R}^d;\mathbb {R}^d)$ and any $\mu \in {\mathcal {C}}_c\big (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus } \{0\})\big )$ with $\Vert \mu \Vert _{\beta }<\infty$ and (5.7). This is a consequence of the known fact that ${\mathcal {C}}_c(\mathbb {R}^d;\mathbb {R}^d)$ is bp-dense in ${\mathcal {B}}_b(\mathbb {R}^d;\mathbb {R}^d)$, and an analogous result for measures which we establish in Appendix A; see in particular Corollary A.6 (with $g(y)\mathrel {\mathop :}=\min \{|y|^{\beta },1\}$).

Fix $b \in {\mathcal {C}}_c(\mathbb {R}^d;\mathbb {R}^d)$ and $\mu \in {\mathcal {C}}_c\big (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus } \{0\})\big )$ with (5.7). Then, in particular, Theorem 5.4(c) and Remark 5.8 show that $S_{b, \mu }$ is a ${\mathcal {C}}_\infty$-semigroup; let us denote the generator of the restriction of $S_{b,\mu }$ to ${\mathcal {C}}_\infty (\mathbb {R}^d)$ by $L_{b, \mu }$.

It was seen in the proof of Theorem 5.4(c) that for $f\in {\mathcal {C}}_b^\rho (\mathbb {R}^d)\cap {\mathcal {C}}_\infty (\mathbb {R}^d)$ we have ${{\hat{B}}}_{b,\nu } f\in {\mathcal {B}}_\infty (\mathbb {R}^d)$. Moreover, $\Vert \mu \Vert _{\beta } < \infty$ and (5.7) give

$$\begin{aligned} \limsup _{R \rightarrow \infty } \sup _{x \in K} \int _{\{|y|>R\}} \, \mu (x,dy)=0 \qquad \lim _{r \rightarrow 0} \sup _{x \in K} \int _{\{|y| \le r\}} |y|^2 \, \mu (x,dy)=0, \end{aligned}$$

(5.8)

for all $K\Subset \mathbb {R}^d$. This entails the continuity of ${\hat{B}}_{b,\nu }f$, see Theorem B.1 (note that we can assume without loss of generality that $\chi$ is smooth by modifying the drift term accordingly). Hence, ${\hat{B}}_{b, \mu } f\in {\mathcal {C}}_\infty (\mathbb {R}^d)$.

Denoting by A the generator of the restriction of T to ${\mathcal {C}}_\infty (\mathbb {R}^d)$, we may consider $L\mathrel {\mathop :}=A+{{\hat{B}}}_{b, \mu }$, defined on D(A), as an operator on the space ${\mathcal {C}}_\infty (\mathbb {R}^d)$. We note that ${{\hat{B}}}_{b, \mu } R(\uplambda ){\mathcal {C}}_\infty (\mathbb {R}^d) \subset {\mathcal {C}}_\infty (\mathbb {R}^d)$. and from this and Lemma 4.1 it follows that, for large enough $\uplambda$, the operator $\uplambda - L$ is surjective.

As is well known (see e.g. [13, Thm. 4.2.2]) a strongly continuous semigroup on ${\mathcal {C}}_\infty (\mathbb {R}^d)$ is sub-Markovian if and only if its generator satisfies the positive maximum principle. By assumption, this is certainly true for A. However, if $f\in D(A)$ satisfies $f(x_0) = \max \{ f(x) : x\in \mathbb {R}^d\} \ge 0$, then $f(x_0 + y) - f(x_0) \ge 0$ for all $y\in \mathbb {R}^d$; if $\rho >1$, we also see that $\nabla f(x_0)= 0$. It follows from $D(A) \subseteq {\mathcal {C}}_b^{\rho }(\mathbb {R}^d)$, cf. Lemma 5.2, and the definition of the operator ${{\hat{B}}}_{b, \mu }$ and that ${{\hat{B}}}_{b, \mu } f(x_0) \le 0$. This shows that $L= A+{{\hat{B}}}_{b, \mu }$ satisfies the positive maximum principle.

By [13, Thm. 4.2.2], L generates a sub-Markovian semigroup on ${\mathcal {C}}_\infty (\mathbb {R}^d)$. However, $L_{b, \nu }$ is obviously an extension of L, so we must have $L=L_{b,\nu }$ and thus the restriction of S to ${\mathcal {C}}_\infty (\mathbb {R}^d)$ is sub-Markovian and so is then also S itself. This shows that $(b,\nu ) \in F$ and finishes the proof in this case.

To remove the tightness assumption (5.7), we consider the ‘small’ and the ‘large’ jumps created by $\mu$ separately. To that end, we put

$$\begin{aligned} \mu _s (x, dy) \mathrel {\mathop :}=\mathbbm {1}_{\{|y|\le 1\}}\mu (x, dy) \qquad \text{ and }\qquad \mu _l(x, dy) \mathrel {\mathop :}=\mathbbm {1}_{\{|y|> 1\}} \mu (x, dy). \end{aligned}$$

(5.9)

We split the operator ${{\hat{B}}} = {{\hat{B}}}_1 + {{\hat{B}}}_2$ where, in the notation above, ${{\hat{B}}}_1 = B_{b, \mu _s}$ and ${{\hat{B}}}_2 = {{\hat{B}}}_{0, \mu _l}$. As $\mu _s(x, \{|y|>1\}) \equiv 0$, we can apply the above to infer that ${{\hat{A}}} + {{\hat{B}}}_1$ is the full generator of a sub-Markovian semigroup. Noting that ${{\hat{B}}}_2$ is a bounded operator and taking the special structure of this operator into account, the results of [13, Sect. 4.10] (see in particular Exercise 3 on p. 261) yield that also ${{\hat{A}}} + {{\hat{B}}} = {{\hat{A}}} + {{\hat{B}}}_1 +{{\hat{B}}}_2$ is the full generator of a sub-Markovian semigroup. $\square$

6 Application: Martingale problems

We apply the results from the previous section to study perturbations of martingale problems associated with Lévy-type operators. We do not strive for full generality; we rather would like to illustrate how our results can be applied in this setting. We will make the following assumptions which ensure that T and the perturbed semigroup S are Feller semigroups, provided the tightness assumption (5.4) is satisfied. Note, however, that (5.4) is not assumed in this section.

Hypothesis 6.1

Assume that $T=(T(t))_{t \ge 0}$ is a semigroup of kernel operators such that

• (i) T satisfies Hypothesis 5.1 for some constant $\rho >0$,

• (ii) T is a Markovian ${\mathcal {C}}_\infty$-semigroup,

• (iii) the test functions ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ are a core for the generator A of $T|_{{\mathcal {C}}_\infty (\mathbb {R}^d)}$.

It follows from general theory (see, e.g., [48, Sect. 3.2]) that we can associate Markov process (actually a Feller process) with càdlàg paths with the semigroup T, i.e. there is a Markov process whose transition semigroup equals T.

There is another connection between stochastic processes and (Feller) semigroups via martingale problems. Let us briefly recall the relevant notions. By ${\mathcal {D}}([0,\infty ); \mathbb {R}^d)$ we denote the Skorohod space, i.e. the space of càdlàg functions $\omega : [0,\infty ) \rightarrow \mathbb {R}^d$. We equip ${\mathcal {D}}([0,\infty );\mathbb {R}^d)$ with the Skorohod topology, see [5, 13] for more information. Given a (possibly multi-valued) operator L with domain D(L), a set $D\subset D(L)$ and a measure $\mu$ on $\mathbb {R}^d$, a measure $\mathbb {P}^{\mu }$ on ${\mathcal {D}}([0,\infty ); \mathbb {R}^d)$ is called a solution to the $(L, D; \mu )$-martingale problem if (i) $\mathbb {P}^{\mu } (\omega _0 \in A) = \mu (A)$ for all Borel sets $A\subset \mathbb {R}^d$ and (ii) the process

$$\begin{aligned} M_t^{(f,g)}(\omega ) \mathrel {\mathop :}=f(\omega _t) - f(\omega _0) - \int _0^t g(\omega _s)\, ds, \quad w \in {\mathcal {D}}[0,\infty ), \end{aligned}$$

is a martingale under $\mathbb {P}^{\mu }$ with respect to the canonical filtration $(\sigma ( \omega _s : s\le t))_{t\ge 0}$ for any $(f,g) \in L$ with $f\in D$. In case that L is single valued on D, we have $g=Lf$.

We say that uniqueness in law holds for the (L, D)-martingale problem if any two solutions $\mathbb {P}_1$, $\mathbb {P}_2$ with the same initial distribution satisfy $\mathbb {P}_1=\mathbb {P}_2$. The (L, D)-martingale problem is well-posed if for every $x \in \mathbb {R}^d$ there exists a unique solution $\mathbb {P}^x$ to the $(L,D,\delta _x)$-martingale problem.

If $(X_t)$ is a Markov process associated with the semigroup T, then its distribution solves the $({{\hat{A}}}, D({{\hat{A}}}))$-martingale problem for the full generator ${{\hat{A}}}$ of T, see [13, Prop. 4.1.7]. Conversely, if the (L, D)-martingale problem is well-posed, then under each measure $\mathbb {P}^x$, the canonical process $(\omega _t)_{t\ge 0}$ is a Markov process [13, Thm. 4.4.2]; if (L, D) is, in a sense, ‘rich enough’ to determine a semigroup uniquely, then this semigroup is the transition semigroup of the process, see [13, Thm. 4.4.2].

Theorem 6.2

Let $(T(t))_{t \ge 0}$ be a semigroup satisfying Hypothesis 6.1 with full generator ${\hat{A}}$ and denote by A the generator of $T|_{{\mathcal {C}}_\infty (\mathbb {R}^d)}$. If ${\hat{B}}$ is a Lévy-type operator

$$\begin{aligned} {\hat{B}}f(x) = b(x) \nabla f(x) + \int _{\mathbb {R}^d {\setminus } \{0\}} (f(x+y)-f(x)-\nabla f(x) \cdot y \chi (y)) \, \mu (x,dy) \end{aligned}$$

with b(x) and $\nu (x,dy)$ satisfying Hypothesis 5.3, then the $(A+{{\hat{B}}}, {\mathcal {C}}_c^\infty (\mathbb {R}^d))$-martingale problem is well-posed.

Proof

By Theorem 5.4, there is a Markovian ${\mathcal {C}}_b$-semigroup $S=(S(t))_{t \ge 0}$ with full generator ${\hat{A}}+{\hat{B}}$. Corollary 5.7 shows that $\{ (f, Af +{{\hat{B}}}f) : f\in {\mathcal {C}}_c^\infty (\mathbb {R}^d)\}$ is a bp-core for ${{\hat{A}}} + {{\hat{B}}}$. Thus, by [13, Prop. 4.3.1], a measure $\mathbb {P}^{\mu }$ on ${\mathcal {D}}([0,\infty );\mathbb {R}^d)$ solves the $(A+{{\hat{B}}}, {\mathcal {C}}_c^\infty (\mathbb {R}^d))$-martingale problem if and only if it solves the $({{\hat{A}}} + {{\hat{B}}},D({\hat{A}}+{\hat{B}}))$-martingale problem. It follows from [13, Thm. 4.4.1], applied with $L= \overline{D({{\hat{A}}})}$, which includes ${\mathcal {C}}_\infty (\mathbb {R}^d)$ and is thus separating, that uniqueness holds for the $({{\hat{A}}} + {{\hat{B}}},D({\hat{A}}+{\hat{B}}))$-martingale problem and, hence, also for the $(A+{{\hat{B}}}, {\mathcal {C}}_c^\infty (\mathbb {R}^d))$-martingale problem.

It remains to establish the existence of solutions. First we consider the particular case that the tightness assumption (5.4) is additionally satisfied. In this case, Theorem 5.4 yields that S is actually a Feller semigroup and as is well known, see, e.g. [30, Theorem 4.10.3], this implies well-posedness for the martingale problem for the generator of $S|_{{\mathcal {C}}_\infty }$. In particular, there is a solution to the $(A+{{\hat{B}}}, {\mathcal {C}}_c^\infty (\mathbb {R}^d))$-martingale problem. To remove the tightness condition, we proceed as in the proof of Theorem 5.4(b), i.e. we write ${\hat{B}}={\hat{B}}_1+{\hat{B}}_2$ for a bounded operator ${\hat{B}}_2$ and a Lévy-type operator ${\hat{B}}_1$, whose jumping kernels are supported in the unit ball, cf. (5.9). The existence of a solution to the $(A+{{\hat{B}}}_1, {\mathcal {C}}_c^\infty (\mathbb {R}^d))$-martingale problem is clear from the first part. Since ${\hat{B}}_2$ is a bounded perturbation, existence of a solution to the martingale problem for ${\hat{A}}+{\hat{B}}={\hat{A}}+{\hat{B}}_1+{\hat{B}}_2$ follows from [13, Prop. 4.10.2]. $\square$

We can now also prove continuous dependence of the solutions of our martingale problem on the ‘coefficients’ b and $\mu$. To that end, we make the same assumptions as in Lemma 5.9. As a direct consequence, we obtain a convergence result for semigroups, which fits well to our earlier result on the resolvent convergence, cf. Proposition 4.2 and Corollary 4.3.

Theorem 6.3

Let $(T(t))_{t \ge 0}$ be a semigroup satisfying Hypothesis 6.1 for some constant $\rho >0$ and denote by ${\hat{A}}$ its full generator. Let $(b_n)_{n \in \mathbb {N}\cup \{\infty \} }$ and $(\mu _n)_{n \in \mathbb {N}\cup \{\infty \}}$ satisfy Hypothesis 5.3 for a common constant $\beta \in (0, \min \{2, \rho \})$. Denote the associated Lévy-type operators by ${{\hat{B}}}_n$ for $n\in \mathbb {N}\cup \{\infty \}$. Assume that

• (i) $b_n(x) \rightarrow b_\infty (x)$ and $\mu _n(x,\cdot ) \rightarrow \mu _\infty (x,\cdot )$ vaguely for every $x \in \mathbb {R}^d$,

• (ii) $\sup _{n \in \mathbb {N}}\left( \Vert b_n\Vert _{\infty } + \Vert \mu _n\Vert _{\beta }\right) <\infty$ with $\Vert \cdot \Vert _{\beta }$ defined in (5.3).

• (iii) $\sup _{x\in K}\sup _{n\in \mathbb {N}}\mu _n (x, \{ |y|>R\}) \rightarrow 0$ as $R\rightarrow \infty$ for $K\Subset \mathbb {R}^d$.

Then, if $\mathbb {P}_n^{x_n}$ solves the $(A+{{\hat{B}}}_n, {\mathcal {C}}_c^\infty (\mathbb {R}^d), \delta _{x_n})$-martingale problem and $\mathbb {P}_\infty ^x$ solves the $(A+{{\hat{B}}}_\infty , {\mathcal {C}}_c^\infty (\mathbb {R}^d), \delta _{x_n})$-martingale problem for some sequence $x_n \rightarrow x$, we have weak convergence $\mathbb {P}_n^{x_n} \rightarrow \mathbb {P}_\infty ^x$.

In particular, if $S_n(t)$ and $S_\infty (t)$ denote the perturbed semigroups with generator ${\hat{A}}+{\hat{B}}_n$ and ${\hat{A}}+{\hat{B}}_\infty$, respectively, then $S_n(t)f \rightarrow S_\infty (t)f$ pointwise for every $f \in {\mathcal {C}}_b(\mathbb {R}^d)$.

Proof

It was seen in the proof of Lemma 5.9 that under the assumptions above ${{\hat{B}}}_n u \rightharpoonup {{\hat{B}}}_\infty u$ for all $u\in D({{\hat{A}}})$. Thus, if $(u, f) \in {{\hat{A}}}$, we have $(u, f + {{\hat{B}}}_n u ) \in {{\hat{A}}} + {{\hat{B}}}_n$ and this converges in bp-sense to $(u, f + {{\hat{B}}}_\infty u)$. In particular, the sequence $g_n \mathrel {\mathop :}=f+{{\hat{B}}}_n u$ is uniformly bounded. By assumption,

$$\begin{aligned} M^{(u,g_n)}_t(\omega ) \mathrel {\mathop :}=u(\omega _t) - u(\omega _0) + \int _0^t g_n(\omega _s)\, ds \end{aligned}$$

is a $\mathbb {P}_n^{x_n}$-martingale. Combining [13, Thm. 3.9.4] and [13, Cor. 3.9.3], it follows that the sequence $(\mathbb {P}_n^{x_n})$ is tight. For an alternative argument yielding the tightness, see [35, Corollary 3.9].

We will invoke the subsequence principle, see e.g. [5, Thm. 2.6], to prove that $\mathbb {P}_n^{x_n} \rightarrow \mathbb {P}_{\infty }^x$ weakly.

To that end, observe that as a further consequence of Lemma 5.9, we have

$$\begin{aligned} R_n(\uplambda ) f:= (\uplambda -({\hat{A}}+{\hat{B}}_n))^{-1} f \rightharpoonup R_{\infty }(\uplambda )f:= (\uplambda -({\hat{A}}+{\hat{B}}))^{-1} f, \quad f \in {\mathcal {B}}_b(\mathbb {R}^d), \end{aligned}$$

for large enough $\uplambda$. In fact, Corollary 4.3 implies that the convergence is actually uniform on compact sets.

Take any subsequence of $(\mathbb {P}_n^{x_n})_{n \in \mathbb {N}}$. By tightness, there is a further subsequence converging to some measure, say, $\mathbb {Q}$. For simplicity of notation, we denote the convergent subsequence also by $\mathbb {P}_n^{x_n}$. For fixed $f \in {\mathcal {B}}_b(\mathbb {R}^d)$, $s \le t$, $0\le r_1< \ldots < r_k \le s$ and $h_j \in {\mathcal {C}}_b(\mathbb {R}^d)$, define

$$\begin{aligned} \Phi _n(\omega ) := \left[ R_n(\uplambda ) f(\omega _t) - R_n(\uplambda ) f(\omega _s) - \int _s^t (\uplambda R_n(\uplambda )f-f)(\omega _r) \, dr \right] \prod _{j=1}^k h_j(\omega _{r_j}), \end{aligned}$$

where $n \in \mathbb {N}\cup \{\infty \}$. Since $R_n(\uplambda )f \in D({\hat{A}}+{\hat{B}}_n)$ and $\mathbb {P}_n^{x_n}$ solves the martingale problem for ${\hat{A}}+{\hat{B}}_n$, we find that $\mathbb {E}_{\mathbb {P}_n^{x_n}}(\Phi _n):=\int \Phi _n \, d\mathbb {P}_n^{x_n}=0$ for all $n \in \mathbb {N}$. We claim that $\mathbb {E}_{\mathbb {Q}}(\Phi _{\infty })=0$. By the weak convergence, it holds that $\mathbb {E}_{\mathbb {P}_n^{x_n}}(\Phi _{\infty }) \rightarrow \mathbb {E}_{\mathbb {Q}}(\Phi _{\infty })$. If we can show that

$$\begin{aligned} \lim _{n \rightarrow \infty } |\mathbb {E}_{\mathbb {P}_n^{x_n}}(\Phi _n-\Phi _{\infty })| = 0, \end{aligned}$$

(6.1)

then it follows that $\mathbb {E}_{\mathbb {Q}}(\Phi _{\infty })=\lim _n \mathbb {E}_{\mathbb {P}_n^{x_n}}(\Phi _n)=0$. To prove (6.1), we note that, by the uniform boundedness of the resolvents, $M:=\sup _{n \in \mathbb {N}} \Vert \Phi _n\Vert _{\infty }+\Vert \Phi _{\infty }\Vert _{\infty }<\infty$. By tightness, given $\varepsilon >0$ we find $K \Subset {\mathcal {D}}([0,\infty ); \mathbb {R}^d)$ such that $\mathbb {P}_n^{x_n}(K^c)\le \varepsilon$ for all $n\in \mathbb {N}$. Note that $R:= \sup _{y \in K} \sup _{s \le t} |\omega _s|<\infty$. So, by the locally uniform convergence of the resolvents, we find some $N \in \mathbb {N}$ such that

$$\begin{aligned} |R_n(\uplambda ) f(\omega _r)-R_{\infty }(\uplambda ) f(\omega _r)| \le \varepsilon \end{aligned}$$

for all $n \ge N$, $r \in [0,t]$ and $y \in K$. Thus, for some constant $C>0$, we have

$$\begin{aligned} |\mathbb {E}_{\mathbb {P}_n^{x_n}}(\Phi _n-\Phi _{\infty })| \le C\varepsilon + M \varepsilon \end{aligned}$$

for all $n\ge N$. Hence,

$$\begin{aligned} \limsup _{n \rightarrow \infty } |\mathbb {E}_{\mathbb {P}_n^{x_n}}(\Phi _n-\Phi _{\infty })| \le (M +C)\varepsilon . \end{aligned}$$

As $\varepsilon >0$ was arbitrary, this finishes the proof of (6.1), and we get $\mathbb {E}_{\mathbb {Q}}(\Phi _{\infty })=0$. This implies that

$$\begin{aligned} M_t^{(f)}(\omega ) := R_{\infty } (\uplambda )f(\omega _t) - \int _0^t (\uplambda R_{\infty }(\uplambda )f-f)(\omega _r) \, dr, \quad \omega \in {\mathcal {D}}([0,\infty );\mathbb {R}^d), \end{aligned}$$

is a martingale with respect to $\mathbb {Q}$ for any $f \in {\mathcal {B}}_b(\mathbb {R}^d)$. Noting that

$$\begin{aligned} \{(R_{\infty }(\uplambda ) f, \uplambda R_{\infty }(\uplambda )f-f); f \in {\mathcal {B}}_b(\mathbb {R}^d)\} = {\hat{A}}+{\hat{B}} \end{aligned}$$

and $\mathbb {Q}(\omega _0 \in A) = \delta _x(A)$, this means that $\mathbb {Q}$ is a solution to the $({\hat{A}}+{\hat{B}},D({\hat{A}}),\delta _x)$-martingale problem and thus, by well-posedness, $\mathbb {Q}=\mathbb {P}_\infty ^x$. Hence, by the subsequence principle, $\mathbb {P}_n^{x_n} \rightarrow \mathbb {P}_\infty ^x$ weakly. $\square$

7 Examples

Our perturbation results for Feller semigroups and martingale problems require two sets of assumptions: one on the original semigroup T (cf. Hypothesis 5.1 resp. 6.1) and one on the perturbation ${\hat{B}}$ (cf. Hypothesis 5.3). The latter ensures that ${\hat{B}}$ is indeed a (lower order) perturbation of the generator of $(T(t))_{t \ge 0}$ and involves the parameter $\rho \in (0,2)$ from Hypothesis 5.1, which characterizes the regularizing properties of T. Given this parameter $\rho$, the conditions on ${\hat{B}}$ are typically easy to check, and so the main work is to verify the assumptions on T; in particular, to verify the regularity estimate

$$\begin{aligned} \Vert T(t) f\Vert _{{\mathcal {C}}_b^{\rho }} \le \varphi (t) \Vert f\Vert _{\infty }, \qquad t \in (0,1),\; f \in {\mathcal {B}}_b(\mathbb {R}^d), \end{aligned}$$

for some function $\varphi \in L^1(0,1)$. For a brief summary of some of our main results we refer to Theorem 1.1. In this section, we present examples of semigroups satisfying our assumptions and give some applications, e.g. in the theory of stochastic differential equations.

7.1 Differential operators

Let ${\mathcal {A}}$ be a second order differential operators on $\mathbb {R}^d$ of the form

$$\begin{aligned} {\mathcal {A}}f(x) \mathrel {\mathop :}=\textrm{tr}(Q(x)\nabla ^2 f(x)) \end{aligned}$$

(7.1)

for $Q(x)=(q_{ij}(x)) \in \mathbb {R}^{d\times d}$. Our next result gives conditions which ensure that the associated semigroup satisfies the assumptions of our main results.

Proposition 7.1

Assume that the coefficients $q_{ij}$ are bounded, Hölder continuous, symmetric (i.e. $q_{ij} = q_{ji}$) and uniformly (strictly) elliptic in the sense that there exists a constant $\eta >0$ such that

$$\begin{aligned} \sum _{i,j=1}^d q_{ij}(x)\xi _i\xi _j \ge \eta |\xi |^2, \quad x,\xi \in \mathbb {R}^d. \end{aligned}$$

Then the semigroup T associated with (7.1) satisfies Hypotheses 5.1 and 6.1 with arbitrary $\rho \in (0,2)$, and so Theorem 1.1 is applicable (with any $\rho \in (0,2)$).

Proof

First of all, we note that there is a Markovian Feller semigroup T associated with (7.1); this follows from [9, Thm. V.5.11], see also [13, Thm. 8.1.6]. Moreover, the second step of the proof of [9, Thm. V.5.11], with (0.40) replaced by (0.41) and (0.42) from the Appendix of [10], actually shows that

$$\begin{aligned} \Vert T(t) f\Vert _{{\mathcal {C}}_b^2(\mathbb {R}^d)} \le c t^{-1} \Vert f\Vert _{\infty }, \qquad f \in {\mathcal {B}}_b(\mathbb {R}^d). \end{aligned}$$

(7.2)

Since also $\Vert T(t) f\Vert _{\infty } \le \Vert f\Vert _{\infty }$ and the Hölder space ${\mathcal {C}}_b^{\gamma }(\mathbb {R}^d)$, $\gamma \in (0,2) {\setminus } \{1\}$, is a real interpolation space between ${\mathcal {C}}_b(\mathbb {R}^d)$ and ${\mathcal {C}}_b^2(\mathbb {R}^d)$, see [59, Thm. 2.7.2.1], an application of a classical interpolation theorem [59, Sec. 1.3.3] yields that $\Vert T(t) f\Vert _{{\mathcal {C}}_b^{\gamma }(\mathbb {R}^d)} \le M t^{-\rho /2} \Vert f\Vert _{\infty }$ for some constant $M=M(\rho )>0$ and any $f \in L^{\infty }$, $\rho \in [0,2] {\setminus } \{1\}$. Consequently, Hypothesis 5.1 is satisfied for any $\rho \in (0,2)$. Moreover, ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ is a core for the generator of $T|_{{\mathcal {C}}_\infty }(\mathbb {R}^d)$, cf. [13, Thm. 8.1.6], and so Hypothesis 6.1 holds. $\square$

Remark 7.2

Under the milder regularity assumption that the diffusion coefficients $q_{ij}$ are uniformly continuous, one can show with some more effort that the Hypothesis 5.1 is satisfied (for any $\rho \in (0,2)$), e.g. using results from [44]. It is, however, not clear whether ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ is still a core. In dimension $d=1$, one can use that cores are preserved under random time changes, cf. [32, Thm. 4.1], to deduce that ${\mathcal {C}}_c^{\infty }(\mathbb {R})$ is indeed a core if the diffusion coefficient is (uniformly) continuous. Thus, Proposition 7.1 and Corollary 7.3 below, hold in dimension $d=1$ also for uniformly continuous diffusion coefficients.

Combining Proposition 7.1 with Theorem 1.1, we get the following corollary.

Corollary 7.3

Let ${\mathcal {L}}$ be a Lévy-type operator of the form

$$\begin{aligned} \begin{aligned} {\mathcal {L}}f(x)&= b(x)\cdot \nabla f (x) + \textrm{tr}(Q(x) \nabla ^2f(x)) \\&\quad + \int _{\mathbb {R}^d{\setminus }\{0\}} (f(x+y) - f(x)-y \cdot \nabla f(x) \mathbbm {1}_{(0,1)}(|y|)) \, \mu (x, dy). \end{aligned}\nonumber \\ \end{aligned}$$

(7.3)

If the drift b is bounded and measurable, the diffusion coefficients are bounded, Hölder continuous, symmetric and uniformly (strictly) elliptic, the jumping kernels depend measurably on x and there is some $\beta \in (0,2)$ such that

$$\begin{aligned} \sup _{x \in \mathbb {R}^d} \left( \int _{\mathbb {R}^d {\setminus } \{0\}} \min \{|y|^{\beta },1\} \, \mu (x,dy) \right) <\infty , \end{aligned}$$

then there is a Markovian ${\mathcal {C}}_b$-Feller semigroup S whose full generator ${\hat{L}}$ is the bp-closure of $\{(f,{\mathcal {L}}f); f \in {\mathcal {C}}_c^{\infty }(\mathbb {R}^d)\}$. Moreover, the $({\mathcal {L}},{\mathcal {C}}_c^{\infty }(\mathbb {R}^d))$-martingale problem is well-posed. If additionally the family $(\mu (x,dy))_{x \in \mathbb {R}^d}$ is tight, then S is a ${\mathcal {C}}_\infty$-semigroup.

As already mentioned in the introduction, the fact that second order (or even higher order) differential operators with lower order coefficients that are merely measurable generate strongly continuous semigroups on ${\mathcal {C}}_\infty (\mathbb {R}^d)$ is known for a long time (see [54]). Here, we also allow for lower order integral terms that are merely measurable, providing an extension of Taira’s results on domains [56]. Also for the well-posedness of the martingale problem we require less regularity than e.g. in the classical results by Stroock [55], where continuous dependence of $\mu (x, \cdot )$ on x is assumed. We point out that recently, there are also first results for differential operators with merely measurable diffusion coefficients, see [61].

7.2 Lévy generators

Next we consider perturbations of operators ${\mathcal {A}}$ which are the generators of Lévy processes, that is, stochastic processes with càdlàg sample paths and independent and stationary increments. Recall that, by the Lévy–Khintchine formula, any Lévy process can be uniquely characterized (in distribution) by its characteristic exponent $\psi$ or by its Lévy triplet $(b,Q,\nu )$, cf. [49] for more information. An important example is the fractional Laplacian $-(-\Delta )^{\alpha /2}$, which is the generator of the isotropic $\alpha$-stable Lévy process. Its characteristic exponent is $\psi (\xi )=|\xi |^{\alpha }$, and the corresponding Lévy triplet is $(0,0,\nu )$ with

$$\begin{aligned} \nu (U) = c_{d,\alpha } \int _{U}\frac{1}{|y|^{d+\alpha }}\, dy \end{aligned}$$

(7.4)

for a normalizing constant $c_{d, \alpha }$.

In the following, we will consider general Lévy processes and assume that their characteristic exponent $\psi$ satisfies the Hartman–Wintner condition:

$$\begin{aligned} \lim _{|\xi | \rightarrow \infty } \frac{\textrm{Re}\,\psi (\xi )}{\log |\xi |} = \infty . \end{aligned}$$

(7.5)

This is a mild growth condition on $\textrm{Re}\,\psi$ which ensures the existence of a smooth transition density $p_t$ for each $t>0$ for the associated process, cf. [29] for a thorough discussion.

Proposition 7.4

Let $(X_t)_{t \ge 0}$ be a Lévy process with transition semigroup $(T(t))_{t \ge 0}$ and characteristic exponent $\psi$ satisfying the Hartman–Wintner condition (7.5). If there is a constant $\kappa >0$ such that the transition density $p_t$ of $X_t$ satisfies

$$\begin{aligned} \int _0^1\Big (\int _{\mathbb {R}^d} |\nabla p_t (x)|\, dx \Big )^\kappa dt < \infty . \end{aligned}$$

(7.6)

then Hypothesis 5.1 and 6.1 hold for any $\rho \le \kappa$, and so Theorem 1.1 is applicable (with $\rho \le \kappa$).

Proof

It is well known that $T(t) f(x) = \int f(x+y) p_t(y) \, dy$ is a ${\mathcal {C}}_\infty$-semigroup and because of the existence of the density, each T(t) is a strong Feller operator, see e.g. [19, pp. 438–39]. Moreover, ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ is a core for the generator of $T(t)|_{{\mathcal {C}}_\infty }$, see e.g. [6, Cor. 2.10]. Denote by ${\hat{A}}$ the corresponding full generator. The differentiation lemma for parameter-dependent integrals entails that T(t)f is differentiable for any $t>0$, $f \in L^{\infty }(\mathbb {R}^d)$, and

$$\begin{aligned} \Vert \nabla T(t) f\Vert _{\infty } \le \varphi (t) \Vert f\Vert _{\infty } \quad \text {with} \quad \varphi (t) := \max \left\{ 1,\int _{\mathbb {R}^d} |\nabla p_t(x)| \, dx\right\} . \end{aligned}$$

As $p_{2t}$ is the convolution of $p_t$ with itself, $\Vert \nabla ^2 T(t) f\Vert _{\infty } \le d^2 \varphi (t/2)^2 \Vert f\Vert _{\infty }$, cf. [37, Lem. 4.1]. Using that ${\mathcal {C}}_b^{\gamma }(\mathbb {R}^d)$, $\gamma \in (0,2) {\setminus } \{1\}$, is a real interpolation space between ${\mathcal {C}}_b(\mathbb {R}^d)$ and ${\mathcal {C}}_b^2(\mathbb {R}^d)$, see [59, Thm. 2.7.2.1], and applying an interpolation theorem [59, Sec. 1.3.3], we find that $\Vert T(t) f\Vert _{{\mathcal {C}}_b^{\gamma }(\mathbb {R}^d)} \le M \varphi (t/2)^{\gamma } \Vert f\Vert _{\infty }$ for some constant $M=M(\gamma )>0$ and any $f \in L^{\infty }$, $\gamma \in [0,2]$. $\square$

Gradient estimates of the form (7.6) for Lévy processes have been intensively studied in the last decade, e.g. [15, 24, 33, 39, 53] to mention but a few, and so Proposition 7.4 applies to a wide class of Lévy generators. Using the gradient estimates from [53, Ex. 1.5], we find that the assumptions of Proposition 7.4 are satisfied for Lévy measures which are dominated from below by a stable measure:

Example 7.5

Let $(T(t))_{t \ge 0}$ be the semigroup of a pure-jump Lévy process with Lévy measure $\nu$ satisfying

$$\begin{aligned} \nu (U) \ge \int _0^{r_0} \int _{{\mathbb {S}}} \mathbbm {1}_U(r \theta ) r^{-1-\alpha } \, dr \, \varrho (d\theta ), \quad U \in {\mathcal {B}}(\mathbb {R}^d {\setminus } \{0\}), \end{aligned}$$

for some constants $\alpha \in (0,2)$, $r_0>0$ and a measure $\varrho$ on the unit sphere ${\mathbb {S}}$ whose support is not contained in any proper linear subspace of $\mathbb {R}^d$. Then the assumptions of Proposition 7.4 are satisfied (with $\kappa <\alpha$).

Example 7.5 covers, in particular, stable operators, i.e. operators (1.1) with $b=0$, $Q=0$ and Lévy measures $\nu$ of the form $\nu (U) = \int _0^{\infty } \int _{{\mathbb {S}}} \mathbbm {1}_U(r \theta ) r^{-1-\alpha } \, dr \, \varrho (d\theta )$. Let us mention that Komatsu [31] was one of the first to study perturbations of stable operators by Lévy-type operators; however, his result requires quite strong assumptions on the regularity the density of $\nu$. A related result was obtained by Peng [23], who considers perturbations of the fractional Laplacian by time-dependent kernels $\mu (t,x,dy)$. Our approach has the advantage that it is not restricted to stable operators and works in a much more general setting, as demonstrated by Proposition 7.4

7.3 Operators of variable order

Our next application concerns perturbations of the fractional Laplacian of variable order $-(-\Delta )^{\alpha (\cdot )/2}$, which is defined on ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ by

$$\begin{aligned}{} & {} -(-\Delta )^{\alpha (\cdot )/2} f(x)\\ {}{} & {} \quad = c_{d,\alpha (x)} \int _{\{y \ne 0\}} (f(x+y)-f(x)-y \cdot \nabla f(x) \mathbbm {1}_{(0,1)}(|y|)) \frac{1}{|y|^{d+\alpha (x)}} \, dy \end{aligned}$$

for a certain normalizing constant $c_{d,\alpha (x)}$. This operator appears as generator of so-called processes of variable order, see e.g. [4, 17, 22, 30, 33] for the construction of such processes and more information about the probabilistic background. Denote by $(T(t))_{t \ge 0}$ the associated ${\mathcal {C}}_\infty$-Feller semigroup.

Proposition 7.6

If $\alpha : \mathbb {R}^d \rightarrow (0,2]$ be Hölder continuous with $\alpha _L := \inf _x \alpha (x)>0$, then $(T(t))_{t \ge 0}$ satisfies Hypothesis 5.1 and 6.1 with $\rho =\alpha _L$, and so Theorem 1.1 is applicable with $\rho =\alpha _L$.

Proof

Let $(T(t))_{t \ge 0}$ be the Markovian Feller semigroup associated with the operator $-(-\Delta )^{\alpha (\cdot )/2}$, see e.g. [4] or [33, Thm. 5.2] for the existence, and denote by ${\hat{A}}$ its full generator and by A its part in ${\mathcal {C}}_\infty (\mathbb {R}^d)$. It is known from the construction of the semigroup that ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ is a core and that $T(t) f(x) = \int f(y) p_t(x,y) \, dy$ for a bounded transition function $p_t$, cf. [33]. In consequence, by [6, Thm. 1.9, Thm. 1.14], each T(t) is a strong Feller operator. Moreover, by [36, Prop. 6.1],

$$\begin{aligned} \Vert T(t) f\Vert _{{\mathcal {C}}_b^{\beta }(\mathbb {R}^d)} \le c_{\beta } t^{-\beta /\alpha _L} \Vert f\Vert _{\infty }, \qquad f \in {\mathcal {B}}_b(\mathbb {R}^d),\;t \in (0,1), \end{aligned}$$

for any $\beta <\alpha _L$. In conclusion, Hypothesis 6.1 is satisfied. $\square$

Example 7.7

Consider

$$\begin{aligned} {\mathcal {L}}u = -(-\Delta )^{\alpha (\cdot )/2}u - (-\Delta )^{\beta (\cdot )/2} \end{aligned}$$

where $\alpha : \mathbb {R}^d\rightarrow (0,2]$ is Hölder continuous with $\alpha _L \mathrel {\mathop :}=\inf _x \alpha (x) > 0$ and $\beta : \mathbb {R}^d\rightarrow (0,2)$ is a measurable function with $\sup _x \beta (x) < \alpha _L$. Then, by Proposition 7.6 and Theorem 1.1, the bp-closure of $\{(f,{\mathcal {L}}f); f \in {\mathcal {C}}_c^{\infty }(\mathbb {R}^d)\}$ is the generator of a Markovian ${\mathcal {C}}_b$-Feller semigroup S and the $({\mathcal {L}},{\mathcal {C}}_c^{\infty }(\mathbb {R}^d))$-martingale problem is well-posed. If additionally $\inf _x \beta (x)>0$, then the tightness condition (5.4) is satisfied, and S is a ${\mathcal {C}}_\infty$-Feller semigroup.

Let us mention that Theorem 6.3 also yields the continuous dependence of these solutions on the parameter $\beta$. More precisely, if $\beta _n$, $n \in \mathbb {N}$ are measurable functions with $\beta _n(x) \rightarrow \beta (x)$ for all $x \in \mathbb {R}^d$ and $0<\inf _n \inf _x \beta (x) \le \sup _n \sup _x \beta _n(x)<\alpha _L$, then $\mathbb {P}^{(\beta _n,x)} \rightarrow \mathbb {P}^{(\beta ,x)}$ weakly for every $x \in \mathbb {R}^d$. In particular, the associated semigroups satisfy $S^{(\beta _n)}(t)f(x) \rightarrow S^{(\beta )}(t)f(x)$ for any $t>0$, $x\in \mathbb {R}^d$ and $f \in {\mathcal {C}}_b(\mathbb {R}^d)$.

Proposition 7.6 can be extended to a wider class of non-local operators, that is, we can replace the fractional Laplacian of variable order $-(-\Delta )^{\alpha (\cdot )/2}$ by other operators of variable order, see [33, Sect. 5.1] for some further examples.

7.4 Non-local operators and Lévy-driven SDEs

In this section, we show how our perturbation results can be applied in the theory of stochastic differential equations (SDEs). The key is a result by Kurtz [43], which states that the existence of solutions to Lévy-driven SDEs can be equivalently characterized by the existence of solutions to martingale problems associated with certain Lévy-type operators (1.1). For a comprehensive study of Lévy-driven SDEs, e.g. classical existence and uniqueness results, we refer to Ikeda–Watanabe [18] and Protter [47]. Our first result is about SDEs with additive Lévy noise.

Proposition 7.8

Let $(L_t)_{t \ge 0}$ be a Lévy process with characteristic exponent $\psi$ satisfying the Hartman–Wintner condition (7.5) and $b \in {\mathcal {B}}_b(\mathbb {R}^d; \mathbb {R}^d)$. If the transition density $p_t$ of $(L_t)_{t \ge 0}$ satisfies $\int _0^1 \int _{\mathbb {R}^d} |\nabla p_t(x)| \ dx \, dt < \infty$, then the SDE

$$\begin{aligned} dX_t = b(X_t) \, dt + dL_t, \qquad X_0=x, \end{aligned}$$

(7.7)

has a unique weak solution. Moreover, the solution does not explode in finite time and gives rise to a Markovian Feller semigroup with symbol

$$\begin{aligned} q(x,\xi ) = -i b(x) \cdot \xi + \psi (\xi ), \qquad x,\xi \in \mathbb {R}^d. \end{aligned}$$

Proof

This is immediate from Proposition 7.4 (with $\kappa =1$) and the results of [43]. $\square$

Proposition 7.8 applies, in particular, to any Lévy process with characteristic exponent $\psi$ satisfying $\textrm{Re}\,\psi (\xi ) \ge c |\xi |^{\alpha }$, $|\xi | \gg 1$, for some $\alpha >1$; indeed, (7.5) is then trivial and the integrability condition on $\nabla p_t$ follows from [53, Thm. 3.2]. Let us mention that Tanaka et. al [58, pp. 82, 83] obtained the existence of a unique solution under the milder assumption that $\frac{1}{\textrm{Re}\,\psi (\xi )} = o\left( |\xi |^{-1}\right)$ as $|\xi | \rightarrow \infty$. Furthermore, there is ongoing research about the optimal assumptions on the drift b to ensure the existence of a unique weak solution to (7.7) for a given Lévy process; e.g. [8] considers isotropic $\alpha$-stable drivers, $\alpha >1$, and (generally unbounded) drifts b from some Kato class.

Proposition 7.9

Let $(L_t)_{t \ge 0}$ be a one-dimensional isotropic $\alpha$-stable Lévy process for $\alpha \in (1,2)$, i.e. $(L_t)_{t \ge 0}$ is a Lévy process with characteristic exponent $\psi (\xi )=|\xi |^{\alpha }$. Let $\sigma : \mathbb {R}\rightarrow \mathbb {R}$ be a Hölder continuous function with $0<\inf _x \sigma (x) \le \sup _x \sigma (x)<\infty$. For every $b \in {\mathcal {B}}_b(\mathbb {R})$, there is a unique weak solution to the SDE

$$\begin{aligned} dX_t = b(X_{t-}) \, dt + \sigma (X_{t-}) \, dL_t, \qquad X_0 = x. \end{aligned}$$

(7.8)

The solution does not explode in finite time and gives rise to a Markovian ${\mathcal {C}}_\infty$-Feller semigroup with symbol

$$\begin{aligned} q(x,\xi ) = -ib(x) \cdot \xi + |\sigma (x)|^{\alpha } |\xi |^{\alpha }. \end{aligned}$$

Proof

Since $\sigma$ is Hölder continuous, the SDE $dY_t = \sigma (Y_{t-}) \, dL_t$ has a unique weak solution which gives rise to a Feller process, see e.g. [40, Thm. 2.1] or [33, Thm. 5.23]. Denote by $(T(t))_{t \ge 0}$ the associated semigroup and by ${\hat{A}}$ the full generator. It is known that ${\mathcal {C}}_c^{\infty }(\mathbb {R})$ is a core for the generator of $T|_{{\mathcal {C}}_\infty (\mathbb {R})}$, cf. [33, 40]. The strong Feller property of T(t) follows from [6, Thm. 1.14] and the fact that $Y_t$ has a bounded transition density $p_t$, see e.g. [33]. Moreover, [36, (proof of) Prop. 4.5] yields the regularity estimate

$$\begin{aligned} \Vert T(t) f\Vert _{{\mathcal {C}}_b^{\beta }(\mathbb {R})} \le c_{\beta } t^{-\beta /\alpha } \Vert f\Vert _{\infty }, \qquad f \in {\mathcal {B}}_b(\mathbb {R}),\;t>0, \end{aligned}$$

(7.9)

for $\beta <\alpha$. Choosing $\beta \in (1,\alpha )$ and setting ${\hat{B}}f(x)=b(x) \nabla f(x)$, we see that the assumptions of Theorem 5.4 are satisfied, and so there is a Markovian ${\mathcal {C}}_\infty$-semigroup with full generator ${\hat{A}}+{\hat{B}}$. Using the well-posedness of the $(A+{\hat{B}},{\mathcal {C}}_c^{\infty }(\mathbb {R}))$-martingale problem, which holds by Theorem 6.2, it follows from [43] that the SDE (7.8) has a unique weak solution. $\square$

Remark 7.10

Proposition 7.9 can be extended in several ways:

• (i) We can consider a wider class of driving Lévy processes $(L_t)_{t \ge 0}$. There are two key ingredients which we need. Firstly, the existence of a unique weak solution to the SDE $dY_t = \sigma (Y_{t-}) \, dL_t$; there is extensive literature on this topic, see e.g. [7, 33, 38, 40, 63] and the references therein. Secondly, a regularity estimate $\Vert T(t) f\Vert _{{\mathcal {C}}_b^{\beta }(\mathbb {R}^d)} \le \varphi (t) \Vert f\Vert _{\infty }$, for the corresponding semigroup, where $\varphi \in L^1(0,1)$ and $\beta >1$. Although there are quite some works on regularizing properties of semigroups associated with SDEs, many of them study only $\beta$-Hölder regularity for $\beta \le 1$, and this is not enough for our purposes.

• (ii) Using localization techniques for martingale problems, one can relax the growth assumptions on b and $\sigma$; see [34, Thm. 4.2] and [17, Thm. 5.3] for useful results in this direction.

Proposition 7.11

Let $(L_t)_{t \ge 0}$ be a one-dimensional isotropic $\alpha$-stable Lévy process for $\alpha \in (0,2)$. Let $(M_t)_{t \ge 0}$ be a one-dimensional Lévy process independent of $(L_t)_{t \ge 0}$ with Lévy triplet $(b,0,\nu )$ such that $\int _{\{|y| \le 1\}} |y|^{\beta } \, \nu (dy) < \infty$ for some $\beta \in [0,\alpha )$. If $\alpha \le 1$ also assume that the compensated drift $b-\int _{\{|y|<1\}} y \, \nu (dy)$ is zero. Let $\sigma : \mathbb {R}\rightarrow (0,\infty )$ be a Hölder continuous mapping with $0<\inf _x \sigma (x) \le \sup _x \sigma (x)<\infty$. For every $\kappa \in {\mathcal {B}}_b(\mathbb {R})$, the SDE

$$\begin{aligned} dX_{t-} = \kappa (X_{t-}) \, dM_t + \sigma (X_{t-}) \, dL_t, \qquad X_0=x \end{aligned}$$

has a unique weak solution. It does not explode in finite time and it gives rise to a Markovian Feller semigroup with the symbol

$$\begin{aligned} q(x,\xi ) = \psi (\kappa (x) \xi ) + |\sigma (x)|^{\alpha } |\xi |^{\alpha }, \end{aligned}$$

where $\psi$ denotes the characteristic exponent of $(M_t)_{t \ge 0}$.

Note that Proposition 7.11 can be extended to other driving Lévy processes $(L_t)_{t \ge 0}$, cf. Remark 7.10.

Proof of Proposition 7.11

Because of the integrability condition on the Lévy measure $\nu$, an application of Taylor’s formula shows that the operator

$$\begin{aligned} {\hat{B}}f(x)&\mathrel {\mathop :}=b \cdot \nabla f(x) \mathbbm {1}_{\{\alpha>1\}}\\&\quad +\int _{\{y \ne 0\}} (f(x+\kappa (x) y)-f(x)- f'(x) \kappa (x) y \mathbbm {1}_{(0,1)}(|y|) \mathbbm {1}_{\{\alpha >1\}}) \, \nu (dy) \end{aligned}$$

satisfies $\Vert {\hat{B}}f\Vert _{\infty } \le C \Vert f\Vert _{{\mathcal {C}}_b^{\beta }(\mathbb {R})}$ for some constant $C>0$. Moreover, Hypothesis 5.3 holds. Note that the kernels $\mu (x,dy)$ associated with ${\hat{B}}$ satisfy the tightness condition (5.4) because

$$\begin{aligned} \mu (x,\{|y|\ge R\})&= \nu (\{y \in \mathbb {R}{\setminus } \{0\}; |\kappa (x) y| \ge R\})\\&\le \nu (\{y \in \mathbb {R}{\setminus } \{0\}; |y| \ge R/\Vert \kappa \Vert _{\infty }\}) \xrightarrow []{R \rightarrow \infty } 0. \end{aligned}$$

Now we can proceed exactly as in the proof of Proposition 7.9; the only difference is that we consider the just-defined operator ${\hat{B}}$ rather than the drift operator. $\square$

Example 7.12

Let $\alpha \in (0,2]$ and $\beta \in (0,\alpha )$. Let $(L_t)_{t \ge 0}$ be an isotropic $\alpha$-stable Lévy process and $(M_t)_{t \ge 0}$ an isotropic $\beta$-stable Lévy process, which is independent of $(L_t)_{t \ge 0}$. Let $\sigma$ be a Hölder continuous function with $0<\inf _x \sigma (x) \le \sup _x \sigma (x)<\infty$. Then, by Proposition 7.11 and Theorem 6.3:

• (i) For every $\kappa \in {\mathcal {B}}_b(\mathbb {R}^d)$, the SDE

  $$\begin{aligned} dX_t = \kappa (X_{t-}) \, dM_t + \sigma (X_{t-}) \, dL_t, \qquad X_0 = x, \end{aligned}$$

  (7.10)

  has a unique weak solution. It gives rise to a Markovian Feller semigroup.

• (ii) If $\kappa _n \in {\mathcal {B}}_b(\mathbb {R})$, $n \in \mathbb {N}$, is a sequence of measurable functions such that $\kappa _n {\mathop {\rightarrow }\limits ^{\textrm{bp}}}\kappa$, then the solutions $X^{(\kappa _n)}$ to (7.10) (with $\kappa$ replaced by $\kappa _n$) converge weakly to $X=X^{(\kappa )}$ in the Skorohod space ${\mathcal {D}}[0,\infty )$.

Appendices

Appendix A. bp-convergence of measure-valued functions

We recall that if E is a metric space, then a set $M \subset {\mathcal {B}}_b(E)$ is called bp-closed if whenever $(f_n) \subset M$ is a sequence with $f_n {\mathop {\rightarrow }\limits ^{\textrm{bp}}}f$, then also $f\in M$. The bp-closure of $M\subset {\mathcal {B}}_b(E)$ is the smallest bp-closed subset of ${\mathcal {B}}_b(E)$ that contains M. If the bp-closure of M is ${\mathcal {B}}_b(E)$ we say that M is bp-dense in ${\mathcal {B}}_b(E)$.

We recall the following general result from [13, Prop. 3.4.2].

Proposition A.1

In a metric space E, the set ${\mathcal {C}}_b(E)$ is bp-dense in ${\mathcal {B}}_b(E)$. If E is additionally locally compact and $\sigma$-compact, then also ${\mathcal {C}}_c(E)$ is bp-dense in ${\mathcal {B}}_b(E)$.

In this appendix, we aim to extend this result to functions that take values in the cone ${\mathcal {M}}^+(E)$ of positive, locally finite measures on E, where E is a locally compact, separable metric space. To that end, we endow ${\mathcal {M}}^+(E)$ with the vague topology, induced by the space ${\mathcal {C}}_c(E)$ of continuous functions with compact support. As is well known, see [25, Thm. A.2.3(i)], ${\mathcal {M}}^+(E)$ is a Polish space in the vague topology, i.e. the topology is induced by a complete metric and the corresponding metric space is separable.

As a matter of fact, we will not be interested in arbitrary E, but only in $E=\mathbb {R}^d{ \setminus } \{0\}$. Fix a continuous function $g: \mathbb {R}^d {\setminus }\{0\} \rightarrow [0,\infty )$. Given a measure $\nu \in {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\})$, we put

$$\begin{aligned} |\nu |_{g} \mathrel {\mathop :}=\int _{\mathbb {R}^d{\setminus }\{0\}} g(y) \, \nu (dy). \end{aligned}$$

We denote by $K^g (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$ the set of measurable mappings $\mu : \mathbb {R}^d\rightarrow {\mathcal {M}}^+(\mathbb {R}^d{\setminus } \{0\})$ such that

$$\begin{aligned} \Vert \mu \Vert _g \mathrel {\mathop :}=\sup _{x\in \mathbb {R}^d}|\mu (x)|_g < \infty \quad \text{ and } \quad \lim _{R\rightarrow \infty } \sup _{x\in K} \mu (x, \{ |y|> R\}) = 0\big \} \, \, \text {for all }K \Subset \mathbb {R}^d. \end{aligned}$$

We note that, as a consequence of [25, Thm. A.2.3(iv)] the map $\mu : \mathbb {R}^d \rightarrow {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\})$ is measurable if and only if $x\mapsto \int _{\mathbb {R}^d} f(y) \mu (x, dy)$ is measurable for every $f\in {\mathcal {C}}_c(\mathbb {R}^d{\setminus }\{0\})$ if and only if $x\mapsto \mu (x, A)$ is measurable for every relatively compact set A.

Definition A.2

We say that a sequence $(\mu _n) \subset K^g (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$ converges boundedly and pointwise to $\mu : \mathbb {R}^d \rightarrow {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\})$, if

• (i) $\sup _n\Vert \mu _n\Vert _g <\infty$,

• (ii) $\mu _n(x,\cdot ) \rightarrow \nu (x,\cdot )$ vaguely for every $x\in \mathbb {R}^d$ and

• (iii) for every $K\Subset \mathbb {R}^d$ we have $\lim _{R\rightarrow \infty } \sup _{n\in \mathbb {N}} \sup _{x\in K} \mu _n (x, \{|y|> R\}) =0$.

We write $\mu _n {\mathop {\rightarrow }\limits ^{bp}}\nu$ to indicate bounded and pointwise convergence.

Remark A.3

We note that if $g(x) = \varphi (|x|)$ for some function $\varphi : (0,\infty ) \rightarrow (0,\infty )$ with $\varphi (t) \uparrow \infty$ as $t\rightarrow \infty$, then condition (i) entails condition (iii) in Definition A.2.

We note that we did not assume in the above definition that the limit $\mu$ belongs to $K^g (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$. As it turns out, this holds true automatically.

Lemma A.4

Let $(\mu _n) \subset K^g (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$. If $\mu _n {\mathop {\rightarrow }\limits ^{\textrm{bp}}}\mu$, then also $\mu$ belongs to $K^g (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$.

Proof

We first note that $\mu$ is measurable as pointwise limit of measurable functions. We now pick $C>0$ such that $\Vert \mu _n\Vert _g \le C$ for all $n\in \mathbb {N}$ and fix a sequence $(f_k)\in {\mathcal {C}}_c(\mathbb {R}^d{\setminus }\{0\})$ such that $0\le f_k(y) \uparrow g(y)$. As $\mu _n(x) \rightarrow \mu (x)$ vaguely, we obtain for fixed $k\in \mathbb {N}$ and arbitrary $x\in \mathbb {R}^d$ that

$$\begin{aligned} \int _{\mathbb {R}^d} f_k(y) \mu (x, dy)&= \lim _{n\rightarrow \infty } \int _{\mathbb {R}^d} f_k(y) \mu _n(x, dy) \le \limsup _{n\rightarrow \infty } \int _{\mathbb {R}^d} g(y)\, \mu _n(x, dy) \le C. \end{aligned}$$

Taking the limit $k\rightarrow \infty$, monotone convergence implies

$$\begin{aligned} \int _{\mathbb {R}^d} g(y)\, \mu (x, dy) \le C \end{aligned}$$

for every $x\in \mathbb {R}^d$, proving that $\Vert \mu \Vert _g \le C<\infty$. To prove that for a compact set K, we have $\lim _{R\rightarrow \infty }\sup _{x\in K}\mu (x, \{|y|>R\})=0$, we pick, given $\varepsilon >0$, a radius $R>0$ such that $\mu _n(x, \{|y|>R\}) \le \varepsilon$ for all $x\in K$ and $n\in \mathbb {N}$. Approximating $\mathbbm {1}_{\{|y|>R\}}$ by an increasing sequence of functions in ${\mathcal {C}}_c(\mathbb {R}^d)$, we can repeat the above argument to prove that $\mu (x, \{|y| >R\}) \le \varepsilon$ for all $x\in K$. $\square$

In what follows, we use the notions of bp-closedness, bp-closure and bp-denseness with the obvious meaning also for subsets of $K^g(\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$. We can now prove the main result of this appendix, a version of Proposition A.1 for measure-valued functions.

Theorem A.5

The set ${\mathcal {C}}_c(\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\})) \cap K^g(\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$ is bp-dense in $K^g(\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$.

Proof

We write $K^g_C$ for the set which is claimed to be bp-dense and denote by ${\mathcal {F}}$ its bp-closure. We proceed in several steps.

Step 1: We prove that if $\mu (x, A) = \varphi (x) \cdot \nu (A)$ for some $\varphi \in {\mathcal {B}}_b(\mathbb {R}^d)$ and $\nu \in {\mathcal {M}}^+(\mathbb {R}^d{\setminus } \{0\})$ with $|\nu |_g <\infty$, then $\mu \in {\mathcal {F}}$.

Indeed, if $\varphi \in {\mathcal {C}}_c(\mathbb {R}^d)$, then a function $\mu$ of the above form belongs to $K^g_C$, and hence to ${\mathcal {F}}$. However, if $\varphi _n {\mathop {\rightarrow }\limits ^{\textrm{bp}}}\varphi$ it is easy to see that $\varphi _n(\cdot )\nu {\mathop {\rightarrow }\limits ^{\textrm{bp}}}\varphi (\cdot )\nu$; thus the latter belongs ${\mathcal {F}}$ if this is true for the approximating sequence. Invoking Proposition A.1, it follows that indeed for every $\varphi \in {\mathcal {B}}_b(\mathbb {R}^d)$ the above function $\mu$ belongs to ${\mathcal {F}}$.

Step 2: We prove that ${\mathcal {F}}$ is a convex cone.

Fix $\mu \in {\mathcal {F}}$ and put

$$\begin{aligned} {\mathcal {F}}_\mu \mathrel {\mathop :}=\big \{ {{\tilde{\mu }}} \in {\mathcal {F}} : s\mu + t{{\tilde{\mu }}} \in {\mathcal {F}} \text{ for } \text{ all } s, t \in (0,\infty )\}. \end{aligned}$$

A moment’s thought shows that ${\mathcal {F}}_\mu$ is bp-closed. If $\mu \in K^g_C$, then clearly every element of $K^g_C$ belongs to ${\mathcal {F}}_\mu$ as $K^g_C$ is a cone. Thus ${\mathcal {F}}\subset {\mathcal {F}}_\mu$ and hence ${\mathcal {F}}={\mathcal {F}}_\mu$. Now let $\mu \in {\mathcal {F}}$ be arbitrary. Then we have $K^g_C \subset {\mathcal {F}}_\mu$ by what was just proved. Using that $K^g_C$ is bp-dense in ${\mathcal {F}}$ once again, we find ${\mathcal {F}}_\mu = {\mathcal {F}}$ for every $\mu \in {\mathcal {F}}$ which proves that ${\mathcal {F}}$ is indeed a cone.

Step 3: It follows from Step 1 and Step 2 that every function of the form

$$\begin{aligned} \mu (x,\cdot ) = \sum _{k=1}^n \mathbbm {1}_{A_k}(x)\nu _k(\cdot ), \end{aligned}$$

where $A_1, \ldots , A_n \in {\mathcal {B}}(\mathbb {R}^d)$ and $\nu _1, \ldots , \nu _n \in {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\})$ with $|\nu _k|_g <\infty$ for all $k=1, \ldots , n$, belongs to ${\mathcal {F}}$. Given any sequence $(A_k) \subset {\mathcal {B}}(\mathbb {R}^d)$ of pairwise disjoint sets and a tight sequence $\nu _k \in {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\})$ with $\sup _n|\nu _n|_g <\infty$ also the function

$$\begin{aligned} \mu (x, \cdot ) \mathrel {\mathop :}=\sum _{k=1}^\infty \mathbbm {1}_{A_k}(x)\nu _k(\cdot ) \end{aligned}$$

belongs to ${\mathcal {F}}$; indeed, the partial sums are all in ${\mathcal {F}}$ and so their bp-limit $\mu$ belongs to ${\mathcal {F}}$ by the closedness of ${\mathcal {F}}$.

Step 4 Fix $\mu \in K^g(\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$ with $\lim _{R\rightarrow \infty } \sup _{x\in \mathbb {R}^d} \mu (x, \{|y|>R\}) = 0$. We put

$$\begin{aligned} S_\mu&\mathrel {\mathop :}=\Bigg \{ \nu \in {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}) : |\nu |_g \le \Vert \mu \Vert _g \text{ and } \\&\qquad \nu (\{|y|> R\}) \le \sup _{x\in \mathbb {R}^d} \mu (x, \{|y|> R\}) \text{ for } \text{ all } R> 0\Bigg \}. \end{aligned}$$

Arguing similar as in the proof of Lemma A.4, we see that $S_\mu$ is a closed subset of ${\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\})$. Consequently, $S_\mu$ is separable, whence we find a dense sequence $(\nu _n) \subset S_\mu$ that is dense in $S_\mu$.

Let us fix a metric $\rho$ that generates the topology on $S_\mu$. We write $B(\nu , \varepsilon )$ for the open ball in $S_\mu$ of radius $\varepsilon$ and with center $\nu \in S_\mu$. We can now recursively define sets $B_k^{(n)}$ by $B_1^{(n)} = B(\nu _1, n^{-1})$ and

$$\begin{aligned} B_{k+1}^{(n)} \mathrel {\mathop :}=B(\nu _{k+1}, n^{-1})\setminus (B_1^{(n)}\cup \cdots \cup B_k^{(n)}). \end{aligned}$$

Then for every $n\in \mathbb {N}$ the sets $(B_k^{(n)})_{k\in \mathbb {N}}$ are pairwise disjoint and their union equals $S_\mu$. Let us set

$$\begin{aligned} \mu _n (x, \cdot ) \mathrel {\mathop :}=\sum _{k=1}^\infty \mathbbm {1}_{\mu ^{-1}(B_j^{(n)})}(x)\nu _j(\cdot ). \end{aligned}$$

By Step 3, the function $\mu _n$ belongs to ${\mathcal {F}}$. However, we have $\Vert \mu _n \Vert _g \le \Vert \mu \Vert _g$ for every $n\in \mathbb {N}$ as every $\nu _j$ belongs to $S_\mu$ and for $x\in \mathbb {R}^d$ we have $\rho (\mu _n(x), \mu (x)) \le n^{-1}$ so that $\mu _n(x) \rightarrow \mu (x)$. By the definition of the set $S_\mu$ also condition (iii) in Definition A.2 is satisfied so that $\mu _n {\mathop {\rightarrow }\limits ^{\textrm{bp}}}\mu$. It follows that $\mu \in {\mathcal {F}}$.

Step 5 Given $\mu \in K^g(\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d{\setminus }\{0\}))$, we set $\mu _n \mathrel {\mathop :}=\mathbbm {1}_{B(0,n)}\mu$. By Step 4, we have $\mu _n \in {\mathcal {F}}$, as $\sup _{x\in \mathbb {R}^d}\mu _n(x, \{|y|>R\})\rightarrow 0$ as $R\rightarrow \infty$. On the other hand, we have $\mu _n {\mathop {\rightarrow }\limits ^{\textrm{bp}}}\mu$, and we conclude that $\mu \in {\mathcal {F}}$. $\square$

We now combine Proposition A.1 and Theorem A.5 into a single result which will be useful in the main part of this article.

Corollary A.6

Assume that F is a subset of ${\mathcal {B}}_b(\mathbb {R}^d; \mathbb {R}^d)\times K^g(\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d\setminus \{0\}))$ such that

• (i) ${\mathcal {C}}_c(\mathbb {R}^d;\mathbb {R}^d)\times \Big [{\mathcal {C}}_c\big (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d\setminus \{0\})\big ) \cap K^g\big (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d\setminus \{0\})\big )\big ]\subset F$

• (ii) whenever $((f_n, \mu _n))_{n\in \mathbb {N}} \subset F$ and $f_n {\mathop {\rightarrow }\limits ^{\textrm{bp}}} f$ and $\mu _n {\mathop {\rightarrow }\limits ^{\textrm{bp}}} \mu$, then also $(f, \mu )\in F$.

Then $F= {\mathcal {B}}_b(\mathbb {R}^d)\times K^g(\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d\setminus \{0\}))$.

Proof

Fixing $f\in {\mathcal {C}}_c(\mathbb {R}^d; \mathbb {R}^d)$ and considering sequences of the form $(f, \mu _n)_{n\in \mathbb {N}}$ bp-converging to $(f,\mu )$, Theorem A.5 shows ${\mathcal {C}}_c(\mathbb {R}^d)\times K^g (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d\setminus \{0\})) \subset F$. Let us now fix $\mu \in K^g (\mathbb {R}^d, {\mathcal {M}}^+(\mathbb {R}^d\setminus \{0\}))$ and consider sequences $(f_n, \mu )$ bp-converging to $(f, \mu )$. Then Proposition A.1 yields the claim. $\square$

Appendix B. Continuity of Lévy-type operators in terms of the symbol and the characteristics

In the proof of one of our perturbation results, Theorem 5.4, we used that the integro-differential operator

$$\begin{aligned} {\hat{B}}f(x) = b(x) \cdot \nabla f(x) + \int _{\mathbb {R}^d \setminus \{0\}} (f(x+y)-f(x)-y \cdot \nabla f(x) \chi (y)) \, \nu (x,dy)\nonumber \\ \end{aligned}$$

(B.1)

maps (sufficiently) smooth functions to continuous functions. On ${\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$, the operator can be considered as a pseudo-differential operator with symbol $q(x,\xi )$, cf. (1.2), and so one can equivalently asked for conditions on the symbol ensuring that ${\hat{B}}f \in {\mathcal {C}}(\mathbb {R}^d)$ for $f \in {\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$. The following result answers this question; it is somewhat more refined than what we need for our proof but we believe the result to be of independent interest.

Theorem B.1

Let ${\hat{B}}$ be as in (B.1) and denote by

$$\begin{aligned} q(x,\xi ) = -i b(x) \cdot \xi + \int _{\mathbb {R}^d \setminus \{0\}} (1-e^{iy \cdot \xi }+iy \cdot \xi \chi (y)) \, \nu (x,dy) \end{aligned}$$

(B.2)

the associated symbol; here $\chi \in {\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$ is a continuous cut-off function with $\mathbbm {1}_{B(0,1)} \le \chi \le \mathbbm {1}_{B(0,2)}$. If q is locally bounded, then the following statements are equivalent.

• (i) ${\hat{B}}f$ is continuous for every $f \in {\mathcal {C}}_c^{\infty }(\mathbb {R}^d)$.

• (ii) $x \mapsto q(x,\xi )$ is continuous for all $\xi \in \mathbb {R}^d$.

• (iii) Each of the following conditions is satisfied.

  • (a) $x \mapsto b(x)$ is continuous.

  • (b) $x \mapsto \nu (x,\cdot )$ is vaguely continuous on $(\mathbb {R}^d \backslash \{0\},{\mathcal {B}}(\mathbb {R}^d \backslash \{0\}))$.

  • (c) The family $(\nu (x,\cdot ))_{x \in K}$, is tight for any compact set $K \Subset \mathbb {R}^d$, i.e. $\lim _{R \rightarrow \infty } \sup _{x \in K} \nu (x, \{|y| \ge R\}) = 0$.

  • (d) $\lim _{r \rightarrow 0} \sup _{x \in K} \int _{\{|y| \le r\}} |y|^2 \, \nu (x,dy)=0$ for any compact set $K \Subset \mathbb {R}^d$.

If one (hence, all) of the conditions is satisfied and ${\hat{B}}: {\mathcal {C}}_b^{\beta }(\mathbb {R}^d) \rightarrow {\mathcal {B}}_b(\mathbb {R}^d)$ is a bounded operator for some $\beta >0$, then ${\hat{B}}f$ is continuous for all $f \in {\mathcal {C}}_b^{\beta }(\mathbb {R}^d) \cap {\mathcal {C}}_\infty (\mathbb {R}^d)$.

Remark B.2

• (a) The implications (ii) $\iff$ (i) $\implies$ (iii) remain valid for any Borel measurable cut-off function $\chi$ such that $\mathbbm {1}_{B(0,1)} \le \chi \le \mathbbm {1}_{B(0,2)}$.

• (b) A symbol q of the form (B.2) is locally bounded if, and only if, for any compact set $K \subseteq \mathbb {R}^d$ there exists a constant $c>0$ such that $|q(x,\xi )| \le c(1+|\xi |^2)$ for all $x \in K$, $\xi \in \mathbb {R}^d$. By [51, Lem. 2.1, Rem. 2.2], this is equivalent to

  $$\begin{aligned} \forall K \Subset \mathbb {R}^d \, \, :\, \sup _{x \in K} |b(x)| + \sup _{x \in K} \int _{\mathbb {R}^d \setminus \{0\}} \min \{|y|^2, 1\} \, \nu (x,dy) <\infty . \end{aligned}$$

  (B.3)

Proof of Theorem B.1

The implication (i) $\implies$ (ii) follows from [50, Thm. 4.4]. Moreover, if $x \mapsto q(x,\xi )$ is continuous, then we find from the local boundedness of q and the dominated convergence theorem that

$$\begin{aligned} x \mapsto {\hat{B}}f(x) = - \int _{\mathbb {R}} q(x,\xi ) {\hat{f}}(\xi ) e^{ix \xi } \, d\xi \end{aligned}$$

is continuous for all $f \in {\mathcal {C}}_c^{\infty }(\mathbb {R})$, and this proves (ii) $\implies$ (i). It remains to show the equivalence (ii) $\iff$ (iii). First of all, we note that the conditions (iii)(c) and (iii)(d), resp., are equivalent to saying that for any convergent sequence $x_n \rightarrow x$ it holds that

$$\begin{aligned} \lim _{R \rightarrow \infty } \sup _{n \in \mathbb {N}} \nu (x_n,\{\{|y| \ge R\}) = 0 \quad \text {and} \quad \lim _{r \rightarrow 0} \sup _{n \in \mathbb {N}} \int _{\{|y| \le r\}} |y|^2 \, \nu (x_n,dy) = 0, \nonumber \\ \end{aligned}$$

(B.4)

resp. Fix $x \in \mathbb {R}^d$ and any sequence $x_n \rightarrow x$. The functions $\xi \mapsto q(x_n,\xi )$ and $\xi \mapsto q(x,\xi )$ are continuous negative definite functions, and therefore $\phi _n := e^{-q(x_n,\cdot )}$ resp. $\phi :=e^{- q(x,\cdot )}$ are characteristic functions of infinitely divisible distributions, say, $\mu _n$ resp. $\mu$. Clearly, $q(x_n,\xi ) \rightarrow q(x,\xi )$ for each $x \in \mathbb {R}^d$, if and only if, $\phi _n(\xi ) \rightarrow \phi (\xi )$ for all $\xi \in \mathbb {R}^d$, and by Lévy’s continuity theorem this is equivalent to weak convergence $\mu _n \rightarrow \mu$. The weak convergence $\mu _n \rightarrow \mu$, in turn, is equivalent to $b(x_n) \rightarrow b(x)$, $\nu (x_n,\cdot ) \rightarrow \nu (x,\cdot )$ vaguely and (B.4), cf. [49, Thm. 8.7]. This finishes the proof. $\square$

Corollary B.3

Let ${\hat{B}}$ be as in (B.1) for a continuous cut-off function $\chi$ with $\mathbbm {1}_{B(0,1)} \le \chi \le \mathbbm {1}_{B(0,2)}$, and denote by $q(x,\xi )$ the symbol of ${\hat{B}}$, cf. (B.2). If there is for every compact set $K \subseteq \mathbb {R}^d$ some constant $\alpha \in (0,2)$ such that

$$\begin{aligned} \sup _{x \in K} |b(x)| + \sup _{x \in K} \left( \int _{\{|y| \le 1\}} |y|^{\alpha } \, \nu (x,dy) \right) <\infty , \end{aligned}$$

(B.5)

and $(\nu (x,\cdot ))_{x \in K}$ is tight, then the following statements are equivalent:

• (i) ${\hat{B}}f$ is continuous for every $f \in C_c^{\infty }(\mathbb {R}^d)$,

• (ii) $x \mapsto q(x,\xi )$ is continuous for all $\xi \in \mathbb {R}^d$,

• (iii) $x \mapsto b(x)$ is continuous and $x \mapsto \nu (x,\cdot )$ is vaguely continuous.

Proof

Because of (B.5), the symbol q is locally bounded, cf. Remark B.2, and moreover (iii)(c),(d) are clearly satisfied. Thus, the assertion is immediate from Theorem B.1. $\square$