The fractional Laplacian with reflections

Motivated by the notion of isotropic $\alpha$-stable L\'evy processes confined, by reflections, to a bounded open Lipschitz set $D\subset \mathbb{R}^d$, we study some related analytical objects. Thus, we construct the corresponding transition semigroup, identify its generator and prove exponential speed of convergence of the semigroup to a unique stationary distribution for large time.

1. Introduction 1.1.Setting, goals, and results.Consider the isotropic α-stable Lévy process (Y t , t ≥ 0) in R d .The intensity of jumps for this process is given by ν(x, dy) = c d,α |x − y| −d−α dy, which is the integro-differential kernel of the fractional Laplacian on R d (for details, see below).Given an open set D ⊂ R d , our interest lies in a Markov process (X t , t ≥ 0) that coincides with Y as long as Y remains within D. However, at the time τ D of the first exit of Y from D, we intend to perform a reflection: instead of allowing X to leave D by going to the point z := Y τ D ∈ D c , our process X should be restarted immediately (at time τ D ) at a point y ∈ D, without venturing into D c .The point y is chosen (randomly) according to a probability measure µ(z, dy), which depends on z, otherwise independently of prior events.Based on this heuristic description, we expect the intensity of jumps of (X t , t ≥ 0) to be the following integral kernel on D, (1.1) γ(x, dy) := ν(x, dy) + D c ν(x, dz)µ(z, dy).
Actually, the outcome of this work is not a Markov process, but a conservative Markovian semigroup (K(t), t > 0) having γ as the kernel of the generator.We also prove that (K(t), t > 0) has a unique stationary density and is exponentially asymptotically stable.Regarding the construction and analysis of the process X, we will cover them in [17].Now, let us state our assumptions on D and µ: (i) For every A ∈ B(D), the map z → µ(z, A) is Borel measurable, and for every z ∈ D c , µ(z, •) is a Borel probability measure on D. (ii) There exists a compact set H ⊂ D such that ϑ := inf z∈D c µ(z, H) > 0.
Song, and Vondraček [40] for the halfline using the Lamperti transform.See also Rivero [56] and Fitzsimmons [32].Kim, Song, and Vondraček [41] explore halfspaces using Dirichlet forms; see also Chen and Song [23].Recent results for the entire real line can be found in Pantí, Pardo, and Rivero [53] and Iksanov and Pilipenko [37]; see also Chaumont, Pantí, and Rivero [21].Furthermore, Bobrowski [6] studies resurrections or concatenations on metric graphs via exit laws and resolvents.We should add that a general probabilistic approach to concatenation or piecing-out of Markov processes was proposed by Ikeda, Nagasawa, and Watanabe [36], see also Meyer [52].Sharpe [62] provides an introduction and further references to the method.For a recent presentation and references to concatenation of right processes, we refer to Werner [66].
1.3.Approach, perspectives, and content.We construct the Markovian transition semigroup corresponding to the kernel γ by the method of nonlocal Schrödinger perturbations of integral kernels by Bogdan and Sydor [19].In the special case of transition kernels, the method may handle analytic aspects of concatenation of Markov processes.Namely, given a sub-Markovian kernel, a suitable repulsion kernel defines a larger transition kernel, possibly Markovian, which corresponds to a specific partial differential equation with (nonlocal) boundary conditions.For instance, see Corollary 5.6.
In the present setting we consider a class of repulsion kernels µ for the Dirichlet heat kernel of the fractional Laplacian.On the one hand, the restriction to the fractional Laplacian is dictated merely by the technical convenience and general interest in this operator, so generalizations are quite obvious.On the other hand, the papers [9,3,28,65,40,41] make do without the lower bound for the repulsion kernel that we assume in Hypothesis 1.1(ii).Therefore the above references and the present paper should be considered as different ramifications of the problem of constructing operators, semigroups and Markov processes with specific boundary conditions.As we already mentioned, this area of research is motivated by the Neumann-type boundary-value problems and the problem of piecing-out or concatenation of Markov processes.Its objective, beyond the construction, are the questions of the large-time and boundary behavior of the resulting semigroup and process, as well as applications to nonlocal differential equations with those boundary conditions.
The paper is organized as follows.Section 2 provides essential preliminaries, including an introduction to the fractional Laplacian and related potential theory.It also introduces the Dirichlet heat kernel (p D (t), t > 0) of the set D. In Section 3, we proceed to construct the kernel (k(t), t > 0) that defines the semigroup (K(t)).We also prove that D k(t, x, y)dy = 1 for all x ∈ D and t > 0. Section 4 delves into the study of the resolvent of (K(t)).In Section 5, we characterize the generator of the semigroup and discuss the associated boundary conditions.Additionally, we provide a solution to a typical boundary value problem.In Section 6 we prove the existence of a unique invariant measure (density) and the exponential convergence of the semigroup to the stationary measure for large time.The reader may also find a few examples illustrating our results.Example 3.7 and Remark 3.8 (restarting at a fixed point x 0 ∈ D) show that, in general, the semigroup (K(t)) is neither symmetric nor does it act on L 2 .Examples 4.6, 4.7, and 4.8 focus on the boundary conditions induced by µ.In Remark 6.3, we propose directions for future research and generalizations.
As we mentioned, the construction of the semigroup (K(t)) is purely analytic, based on nonlocal Schrödinger perturbation, or Duhamel formula applied to p D .The reader interested in the existence and properties of a Markov process resulting from (K(t)) is referred to the forthcoming paper [17] by the authors.

Preliminaries
We often use := for definitions, e.g., N 0 := {0, 1, 2 . ..} and N := {1, 2, . ..}. 1 A denotes the indicator function of A and 1 is the indicator of the full space.We let d ∈ N and consider the Euclidean space R d .All the sets, functions, measures and kernels considered in the paper are Borel.If not stated otherwise, functions take values in the extended real line.For x ∈ R d and r ∈ (0, ∞) we denote by B(x, r) = {y ∈ R d : |y − x| < r} the ball with radius r and center at x.We require integrals to be nonnegative or absolutely convergent.The constant c d,α is chosen in such a way that According to Fourier inversion and the Lévy-Khinchine formula, there is a convolution semigroup of smooth probability densities (p(t), t > 0) such that (2.1) The above scaling implies in particular that (2.3) It is well known that p(1, x) ≈ (1 + |x|) −d−α for x ∈ R d , see, e.g., [14,remarks after Theorem 21] or Kwaśnicki [48, (2.11)].Here ≈ indicates that the ratio of both sides is bounded from above and below by a (strictly positive) constant.We call such comparisons sharp.Thus, Setting p(t, x, y) := p(t, y − x), x, y ∈ R d , t > 0, we introduce a (translationinvariant) transition density.On the space D([0, ∞)) of càdlàg functions (paths) ω : [0, ∞) → R d , we define the canonical process Y t (ω) = ω t , t ≥ 0, and define Markovian measures P x , x ∈ R d , as follows.For (starting points) x ∈ R d , (times) 0 < t 1 < t 2 < . . .< t n and (windows) A 1 , A 2 , . . ., A n ⊂ R d we let By the Kolmogorov extension theorem, these finite-dimensional distributions uniquely determine P x , the law of the Markov process (Y t ) starting from x.We let E x denote the corresponding expectation.As it turns out, Y is the isotropic α-stable process, a specific symmetric Lévy process in R d with the Lévy triplet (0, ν, 0), see, e.g., Sato [57,Section 11].To analyze Y we use the standard complete right-continuous filtration (F t , t ≥ 0), see Protter [54,Theorem I.31].In passing we also recall that every Lévy process is Feller, see [54] or Böttcher, Schilling and Wang [20], meaning that the operator semigroup (2.5) This is the fractional Laplacian (it is also common to use the notation −(−∆) α/2 for this operator).The limit exists, e.g., for u ∈ C ∞ c (R d ), the smooth functions with compact support.The operator ∆ α/2 extends to the infinitesimal generator of the Feller semigroup defined by (2.5) on C 0 (R d ).For a discussion of the many equivalent definitions of ∆ α/2 we refer to [48].

Exit times. The time of the first exit
By translation invariance and scaling of p(t, x, y), the law of {x + Y t , t ≥ 0} under P 0 is the same as the law of {Y t , t ≥ 0} under P x for x ∈ R d , and, under P 0 , the law of {cY t ≥ 0} equals that of {Y c α t ≥ 0}.Hence, τ x+U has the same law under P 0 as τ U under P x and, under P 0 , cτ U has the same law as τ c α U .
Recall that D is a nonempty bounded open subset of R d which is Lipschitz (in the sense of [11, p. 156]).In particular, D is regular, meaning that (2.7) This follows from the radial symmetry of p t (x), Zaremba's exterior cone property of D and Blumenthal's 0-1 law, as in [25,Section 4.4].We also refer to [5,VII.3,IV] for analytic definitions and treatment of regularity and to connections to the theory of stochastic processes.It is well known that the regularity of D is equivalent to solvability of the Dirichlet problem with arbitrary continuous data; see the above references.This condition will be important in Section 2.4.
The boundedness of D assures that the process leaves D in finite time: (2.9) The random variable τ D leads to important analytic objects, especially It is also well known that p D (t, x, y) is jointly continuous and positive for all (t, x, y) ∈ (0, ∞) × D × D. In view of (2.9), the joint distribution of (τ D , Y τ D − , Y τ D ) under P x is given by the following Ikeda-Watanabe formula By (2.12), the function is the density of the harmonic measure of D: By the same reason, (2.15) The survival probability P x (τ D > t) can be expressed in two ways as follows: (2.17) Indeed, the first equation follows from the Ikeda-Watanabe formula (2.12), and the second from (2.10).Combining this with (2.15) yields, for all t > 0, x ∈ D, We also have For future use, we record the following fact on the survival probability.
For clarity we note that some of the arguments in [22,Theorem 2.4] refer to Chung and Zhao [26], who deal with the Brownian motion, but the arguments apply more generally.Alternatively, we may use the sharp explicit bounds for P x (τ B(0,r) > t) given in Bogdan, Grzywny, Ryznar [13,Lemma 6].
2.3.The killed semigroup.In this section, we consider (P D (t), t > 0), the semigroup of the process killed upon leaving D. Thus, (2.20)For our development in Section 5, we need to characterize the C b -generator of the C b -Feller semigroup P D ; see A.4 for the definition.The operator is a (typically strict) extension of the generator of the Feller semigroup on C 0 (D).Whereas the latter is defined as the derivative at t = 0 of the orbits with respect to the norm • ∞ , the C b -generator is defined via the Laplace transform of the semigroup.To this end we will use a general Theorem A.5, providing several equivalent characterizations of C b -generators, and the following simple lemma.
The final ingredient to characterize the C b -generator A D of P D in terms of the (unkilled) semigroup P = P R d is the Dynkin operator D, defined as for any function u ∈ C b (R d ) and x ∈ R d , for which the limit exists.We also denote by ũ the extension of a function u Proof.That u ∈ D(A D ) and A D u = f is equivalent to (i) is an immediate consequence of the equivalence of (i) and (iii) in Theorem A.5.We shall prove (iii) ⇒ (ii) ⇔ (i) ⇒ (iii).In the proof of (i) ⇔ (ii), we follow the ideas of [2, Theorem 2.3] which, however, is concerned with the space C 0 (D) instead of C b (D).
(i) ⇒ (ii).By Lemma 2.2 we have P D (t)f ∈ C 0 (D) for all t > 0. It follows that the Laplace transform of P D takes values in C 0 (D) which, in turn, implies that D(A D ) ⊂ C 0 (D).To prove (2.25), we compare the difference quotients in (2.24) and (2.25).Arguing as in the proof of [2, Theorem 2.3], we see that Given ε > 0, we may pick R so large that D ⊂ B(0, R) and R −α ≤ ε.If s → 0 then P (s)ũ → ũ uniformly on R d , in particular on B(0, R).We may thus pick t 0 > 0 so that |I 1 | ≤ εP x (τ D ≤ t) for all t ≤ t 0 .As for I 2 , we infer from Lemma 2.3 that By the choice of R, we see that . By [20, Proposition 2.27(d) and Theorem 5.1] or Lévy inequality and [15, Remark 1], we have P x (τ B(x,r) ≤ t) ≤ M t for some M ∈ (0, ∞) and all t ≤ t 0 .Therefore for all such t, Since ε > 0 was arbitrary, the difference quotient in (2.25) has the same limit as that in (2.24).
is a martingale with respect to P x .Now consider the stopping time τ B(x,r) , where r > 0 is so small that B(x, r) ⊂ D. Then τ B(x,r) ∧ τ D = τ B(x,r) .Noting that u and f are bounded functions, it follows from optional stopping that Then, from the discussion of scaling in Subsection 2.1, we get (iii) ⇒ (ii).This follows from [48, Lemma 3.4].
2.4.The Dirichlet problem.We are interested in the following operators where x ∈ D, λ ≥ 0, and g is a nonnegative or integrable function on D c .From the Ikeda-Watanabe formula it is immediate that If g is also continuous at ∂D, then H D (λ)g extends continuously to D and the extension equals g on ∂D.
Proof.The result is well known, but the following argument is of some interest.
We have (2.15).Therefore, as a consequence of the continuity (with respect to x ∈ D) of the integrand in (2.27) with g ≡ 1 and λ = 0, by Vitali's convergence theorem (see, e.g., [60,Chapter 22]), the integrand is uniformly integrable for x in every compact subset of D. By majorization, the integrand in (2.27) for general g and λ is also uniformly integrable, so Now assume that g is continuous at ∂D and let x 0 ∈ ∂D and x ∈ D. Taking (2.15) into account again, we find For arbitrary δ > 0 we get Recall that D is regular, whence lim sup x→x 0 P x (τ D > δ) = 0, see [24, (9)] or [26,Proposition 1.19].So, lim sup Since δ > 0 was arbitrary, I 1 (x) → 0 as x → x 0 .We also have Proof.As in the proof of Proposition 2.4, we make use of the Dynkin operator D, defined by (2.23).Fix x ∈ D and let r > 0 be so small that B(x, r) ⊂ D. To obtain a function defined on the whole of R d , we extend h by setting h(x) = g(x) for x ∈ D c .By Lemma 2.5, this yields a continuous function on D. By the strong Markov property, On the other hand, from the discussion in Subsection 2.1, because h is continuous by Lemma 2.5 and bounded, so we can use the dominated convergence theorem.Adding (2.29) and (2.30), we get Dh(x) = λh(x) and the claim follows from [48, Lemma 3.3 and 3.4], including h ∈ D(A D ).

The transition kernel with reflections
In this section we define the transition kernel k(t) aforementioned in the Introduction.To this end we define the kernel ϕ : (0, ∞) × D × B(D) by Let us give an informal interpretation of ϕ.We consider x ∈ D as the value of Y 0 = X 0 , i.e., the starting point of both the processes; t as the value of τ D , the first exit time of Y from D. By the Ikeda-Watanabe formula, If the process X exists as described in the Introduction, then ϕ(t, x, dw)dt is bound to be the joint distribution of (τ D , X τ D ).Of course, we mention X and Y only to develop intuition -in this paper we merely construct a specific transition density k(t), using the analytic data: p D (t, x, y), ν(x, y) and µ(z, •), but we do not analyze the process X, for which see [17].
We note some simple properties of the kernel ϕ.Proof.Part (a) follows from Tonelli's theorem, the fact that every µ(z, •) is a probability measure, and (2.15).The proof of (b) is similar, using (2.18) instead of (2.15).
We shall use the operator S, defined as follows.For then we slightly abuse notation by also defining We shall apply S to f (t, x, y) = p D (t, x, y).Given the interpretation of ϕ, tentatively introduces a single reflection from D c before time t.To accommodate more reflections, we iterate S and define for t > 0, x, y ∈ D. Here, as usual, S 0 := I, the identity.Given a set A ⊂ D and a function f Applying this to (3.3) and using Tonelli's theorem again, yields Being a series of positive functions, k is well defined, with values in [0, ∞].We also have the following Duhamel (or perturbation) formula: We shall gradually prove that k is a transition probability density.We first establish the Chapman-Kolmogorov equations.Proof.The equality may be obtained as in Bogdan, Hansen and Jakubowski [16, Lemma 2].In fact, it is a special case of Bogdan and Sydor [19, Lemma 3], with the transition kernel k(s, x, t, A) there equal to p D (t − s, x, A) and the perturbing kernel J(u, z, du 1 dz 1 ) given by V (z, dz 1 )1 (u,∞) (u 1 )du 1 , where du 1 is the Lebesgue measure on R and V (x, A) := D c ν(x, dz)µ(z, A).
We next prove that k is sub-Markovian.Recall that by Lemma 3.1(b), Lemma 3.3.For all t > 0 and x ∈ D we have k(t, x, D) ≤ 1.
Since p D (t, x, D) ≤ 1, it follows by induction that By (3.4) and letting n → ∞, we verify the claim.
By iterating (3.6) we obtain the identity Making use of the lower bound in Hypothesis 1.1, we can actually establish that k is a Markovian kernel.
Proof.In view of Lemma 3.3, it suffices to prove that k(t, x, D) ≥ 1 for all t > 0 and x ∈ D. We fix an arbitrary T ∈ (0, ∞) and proceed in two steps.
Step 1: We prove a positive lower bound for p D (t, x, D)+Sp D (t, x, D) uniform for t ∈ (0, T ) and x ∈ D. Let H and ϑ be as in Hypothesis 1.1(ii).By Lemma 2.1 and (2.17), we find η > 0 such that p D (s, w, D) ≥ η for s ∈ (0, T ) and w ∈ H. Then for t ∈ (0, T ) and x ∈ D, by (2.18) and (2.19) we get We conclude that all the above integrals are comparable.This will be quite useful later on, but for now we only deduce that for t ∈ (0, T ) and x ∈ D, Step 2: We prove that k(t, x, D) = 1 for t > 0 and x ∈ D. Indeed, let Clearly 0 ≤ ℓ ≤ 1. Iterating (3.5), for t ∈ (0, T ) and x ∈ D we obtain We will focus on the semigroup K = (K(t), t > 0) associated to the transition kernels (k(t), t > 0).More precisely, given f ∈ B b (D) we put As a consequence of Theorem 3. Example 3.7.An important special case is when µ(z, A) does not depend on z.
For instance, let µ(z, •) = δ x 0 , the Dirac measure at a (fixed) point x 0 ∈ D. By (3.1), ϕ(t, x, dw) = (P D (t)κ D ) (x)δ x 0 (dw), where κ D (v) := ν(v, D c ) for v ∈ D. So, for f ≥ 0, t > 0, and x, y ∈ D, where * denotes the convolution on R. In particular, for t > 0, x, y ∈ D, but it also follows that Remark 3.8.The kernel k from Example 3.7 is quite singular.Indeed, we have k(t, x, y) ≥ Sp D (t, x, y) and the inner (space) integral in (3.12) is where c := inf{ν(v, D c ) : v ∈ D} > 0, s ≤ s 0 < ∞, and we used (2.17).It follows that for t ≤ s 0 and x in any given compact subset of D, where C > 0 by Lemma 2.1.Then, if y is close to x 0 , p D (s, x 0 , y) ≈ p(s, x 0 , y), so by [16, p. 249], x, y)f (y)dy ≡ ∞.So the semigroup (K(t)) does not even act on L 2 (D), in particular it is not associated to a Dirichlet form.Moreover, the kernels k(t, x, y) are not symmetric, otherwise symmetry, the equality K(t)1 D = 1 D , and Schur's test would make K(t) a contraction on L 2 (D) for t > 0, a contradiction.We will also see in Remark 5.2 that (K(t)) is not a Feller semigroup.This motivates our approach by resolvent kernels in the next section.

The Laplace transform of the semigroup
We now study the Laplace transform R(λ) of (K(t)), defined by and relate it to the Laplace transform R D (λ) of (P D (t)).To this end we introduce the operator Φ(λ), if it has a (necessarily unique) continuous extension to D. We then identify f and its extension to D.   Proof.It follows from Lemma 4.1, that Φ(λ) 2 defines a strong Feller operator on D. As is well known, see [55, §1.3], its square, i.e., Φ(λ) 4 , is an ultra-Feller operator, i.e., it maps bounded subsets of B b (D) to equicontinuous subsets of C(D).In particular, Φ(λ) 4 is a compact operator.By a variant of the Fredholm alternative, see [43,Theorem 15.4],I − Φ(λ) is invertible if and only if it is injective.The latter was proved in Lemma 4.2.So, 1 belongs to the resolvent set of Φ(λ).By the Krein-Rutman Theorem, see [58, Proposition V.4.1], the spectral radius r(Φ(λ)) belongs to the spectrum of Φ(λ).Since Φ(λ) ≤ 1 and 1 belongs to the resolvent set of Φ(λ), we must have r(Φ(λ)) < 1, which is equivalent to the claim.
We can now relate the resolvents R(λ) and R D (λ).Summarizing, we obtain the Laplace transform (in λ) of Sh by applying Φ(λ) to the Laplace transform of h.By this observation and induction, (3.4) yields We now come to the main result of this section, in which we characterize the closure of the range of R(λ).Given a function f ∈ C b (D), we let and we define the space C µ (D) by By Hypothesis 1.1(iii), the map Then µ satisfies Hypothesis 1.1.Note that µ(z, •) = δ 0 when |z| = 1, so the space C (D) is the same as in Example 4.6, but the extension f µ is different: Note that we may rephrase the second statement of  Now fix x 0 ∈ ∂D.Putting u = R(λ)f , we find Here the second equality uses Lemma 4.4, the third Lemma 4.1 and the fact that R D (λ)f ∈ C 0 (D).The fourth equality uses dominated convergence and the last Lemma 4.4 again.This shows that u ∈ C µ (D).As the latter is closed and contains the range of R(λ), it also contains the closure of the range.
To prove the converse, we only need to show that the range of R(λ) is dense in C µ (D).To that end, let f ∈ C µ (D) and g := f − Φ(λ)f .By Lemma 4.1, g ∈ C 0 (D).Since the semigroup of the killed process is strongly continuous on C 0 (D), the domain of its generator is dense in C 0 (D).We thus find a bounded sequence as n → ∞.Given ε > 0, we may, as a consequence of Lemma 4.3, pick N so large, that k≥N Φ(λ) k ≤ ε.Taking Lemma 4.4 into account, we get As ε > 0 was arbitrary, we see that R(λ)u n → f .

The generator of the semigroup
Proposition 5.1.
Proof.As is well known (see, e.g.[45, Remark 2.5]), K(t)f → f in the norm • ∞ if f belongs to the domain of the generator of K, i.e., the range of R(λ).By Theorem 4.9, the latter is a dense subset of C µ (D) so a standard approximation argument shows that the same is true for every f ∈ C µ (D).
To see the converse, let X := {f ∈ B b (D) : K(t)f → f as t → 0}.Then X is a closed subspace of B b (D) that is invariant under the semigroup K.Moreover, the restriction of (K(t)) to X is strongly continuous.By the Hille-Yosida theorem, the generator of the restriction to X, which is nothing more than the part of the full generator in X, is dense in X.But then X must be contained in the closure of the range of R(λ), i.e., C µ (D).Remark 5.2.It follows from Proposition 5.1 that the semigroup K is not, in general, a Feller semigroup, since typically C 0 (D) ⊂ C µ (D), whence the orbits of functions in C 0 (D) are not • ∞ -continuous.
We can now prove the first main result of this section.Let us consider the second term on the right hand side of (5.1).We have where the last equality uses Lemma 3.1(b) and Equation (2.17).By [24,Lemma 2], the latter converges to 0 as t → 0, uniformly on compact subsets of D.
Let now f ∈ C b (D).We have seen that the integral in (5.1) converges locally uniformly to 0 as t → 0. By Lemma 2.2, P D (t)f converges locally uniformly to f .Thus, (5.1) yields that K(t)f → f locally uniformly as t → 0.
Let us now prove the strong Feller property.To that end, fix t > 0, x ∈ D and f ∈ B b (D).Note that for s ∈ (0, t) we have K(t)f = K(s)K(t − s)f .We set g s := K(t − s)f .By (5.1) (with f replaced by g s and t replaced by s) and (5.2), The latter converges locally uniformly to 0 as s → 0, so P D (s)g s converges locally uniformly to K(t)f .Since P D has the strong Feller property (Lemma 2.2), the functions P D (s)g s are continuous.But then so is K(t)f .We can now characterize the C b -generator A of the semigroup K.In the following theorem, we use the notation u µ introduced in (4.2).
To prove the implication (i) ⇒ (iii), we note that, by Proposition 2.4 and (the proof of) Lemma 2.6, the function R D (λ)(λu−f )+Φ(λ)u belongs to the domain of the Dynkin operator and The implication (iii) ⇒ (ii) follows once again from [48, Lemma 3.4], whereas (iv) is merely a reformulation of (iii).

Asymptotic behavior of the semigroup
In order to establish the existence of an invariant measure, we employ the lower-bound technique of Lasota [49,Theorem 6.1].Here is the first step.
Note that x → p D (s, x, v)ν(v, z) is strictly positive and continuous for almost all triplets (s, v, z).Fatou's lemma implies that the function x → P x (τ D < t/2) is lower semicontinous.Because of the regularity (2.7), the function tends to 1 as x approaches the boundary (see [26,Theorem 1.23]), so at some point of D it attains its minimum, say C > 0. Thus k(t, x, y) ≥ cCϑ =: δ for all x ∈ D, y ∈ H.
(b) By the Chapman-Kolmogorov equations in Lemma 3.  We next consider the adjoint of the operators K(t), t > 0. For a finite measure κ on D we put Then K(t) * κ is again a measure on D and, by Tonelli's theorem and Theorem 3.4, K(t) * κ(D) = κ(D).Moreover, K(t) * κ is absolutely continuous with respect to Lebesgue measure, so we may think of K(t) * as operating on L 1 (D).With this in mind, the operators K(t) * are Markov operators in the sense of Lasota and Mackey [50], Lasota and York [51] and Komorowski [42].We call a probability measure κ a stationary distribution if K(t) * κ = κ for all t > 0.Then, again, K(t) * κ is absolutely continuous with respect to the Lebesgue measure so that any stationary distribution must have a density, called stationary density.
Proof.By Lemma 6.1(b), lim inf T →∞ 1 T T 0 K(t, x, H)dt > 0 for x ∈ D. This is the lower bound mentioned at the beginning of the section.By Da Prato and Zabczyk [27, Remark 3.1.3],we get a stationary distribution.Since the proof in not given in [27,Remark 3.1.3]we refer the reader to Lasota and York [51, Theorem 3.1], Lasota [49,Theorem 6.1] or Komorowski [42,Theorem 3.1] for the proofs in the discrete-time case.
In view of Lemma 6.1(c), [44,Theorem 1.3] (see also [35]), applied to P = K(1) yields the existence of constants c > 0 and e −ω = γ > 0 such that K(n) * ν − κ TV ≤ cγ n for all n ∈ N and probability measures ν on D. An arbitrary number t > 0 can be written as t = n+r, where n ∈ N 0 and r ∈ [0, 1).Then, (i) The compactness of D is crucial in this section; Lemma 6.1 and Theorem 6.2 easily fail for D = R d , since then K(t) = P (t).On the other hand, we conjecture that they hold if the boundedness of D is replaced by that of x → E x τ D , e.g., if D is an (unbounded) right circular cylinder.(ii) The repulsion kernel µ enters the construction of k(t) only through the return kernel V (x, A) := D c ν(x, z)µ(z, A)dz.We expect results for kernels V on D satisfying V (•, D) ≤ ν(•, D c ) describing general reflections.(iii) It is interesting to apply our methods to study semigroups of resurrections or recurrent extensions of Markov processes with scaling, see [32,56,53,37,40].In the same vein, it is important to study the semigroup corresponding to the quadratic form and the Neumann problem in [28]; see also [12,65] for motivation.In these settings, Hypothesis 1.1(ii) fails.
Appendix A. C b -Feller semigroups By Remark 5.2, the semigroup (K(t), t > 0) is, in general, not a Feller semigroup, so in this paper we use a different semigroup concept, namely the notion of C b -Feller semigroup.This can be seen as a special case of the theory of "semigroups on norming dual pairs", introduced in [45,46].As this is not a standard notion, we introduce this concept in this appendix and reformulate the relevant results from [45,46] in our special case.Throughout, we let E ⊂ be an open subset of R d , or, more generally, a Polish space.A kernel on E is a map k : E × B(E) → C such that (i) the map x → k(x, A) is measurable for all A ∈ B(E) (ii) the map A → k(x, A) defines a measure on E for every x ∈ E and (iii) we have sup x |k|(x, E) < ∞, where |k|(x, •) refers to the total variation of k(x, •).
A bounded linear operator T on C b (E) is called a kernel operator, if there exists a kernel k such that As it turns out, being a kernel operator can be characterized by an additional continuity condition with respect to the weak topology σ := σ(C b (E), M (E)) induced by the space of bounded (complex/signed) measures.We note that for a sequence of functions (f n ) ⊂ C b (E), the convergence with respect to σ is nothing else than bp-convergence (bp is short for bounded, pointwise), which means sup n f n ∞ < ∞ and f n → f pointwise.Indeed, that bp-convergence implies σ-convergence follows from the dominated convergence theorem whereas the converse implication follows easily using the uniform boundedness principle.
Proof.The equivalence of (i) and (ii) is proved in [46,Proposition 3.5] and the implication (ii) ⇒ (iii) is trivial in view of the above comment.To see (iii) ⇒ (i), let ϕ(f ) = (T f )(x).By (iii), it follows that ϕ(f n ) → 0 whenever f n bpconverges to 0. Now [7,Theorem 7.10.1]implies that ϕ(f ) = E f dν x for some Baire (hence Borel, as E is Polish) measure ν x .The measurable dependence of ν x on x can now be proved in a standard way, see the proof the implication (i) ⇒ (ii) in [46,Proposition 3.5].
In what follows, the space of bounded, σ-continuous operators (equivalently: kernel operators) on C b (E) is denoted by L (C b (E), σ).Note that any operator T ∈ L (C b (E), σ) can uniquely be extended to a bounded linear operator on all of B b (E), by merely plugging f ∈ B b (E) into the right hand side of (A. 1).In what follows we do not distinguish between T and its extension to B b (E).
We are now ready to define what a C b -semigroup is.To simplify the exposition, we restrict ourselves to sub-Markovian semigroups, as all the semigroups appearing in this article have this property.Obviously, a kernel operator T with kernel k is (sub-)Markovian if and only if the kernel k is (sub-)Markovian, i.e. k(x, •) is a (sub-)probability measure for every x ∈ E.
with the following properties: (i) T (t) is a sub-Markovian kernel operator for every t > 0; (ii) T (t + s) = T (t)T (s) for all t, s > 0; (iii) for f ∈ C b (E) we have T (t)f → f as t → 0, uniformly on compact subsets of E.
In case that E is locally compact, it follows along the lines of [59, Lemma 3.1], which is concerned with the case E = R d , that a Feller semigroup on C 0 (E) can be extended to a C b -Feller semigroup on C b (E).We should point out, however, that a C b -Feller semigroup in the above sense does not necessarily leave the space C 0 (E) invariant.In that respect, our definition of C b -Feller semigroup slightly differs from that in [38,Definition 4.8.6]where a C b -Feller semigroup is assumed to be Feller.
Recalling the connection between bp-convergence and σ-convergence, we see that the requirement (iii) in the above definition in particular implies that T t f → f as t → 0 with respect to σ and thus, by the semigroup law and the σ-continuity of the operators T (t), that T (t)f → T (s)f as t ↓ s for every f ∈ C b (E), i.e. the orbits t → T (t)f are right-continuous with respect to σ.In particular, the orbits have enough measurability to define the Laplace transform of a C b -Feller semigroup by setting As is well known, if (R(λ), λ > 0) consists of injective operators and satisfies the resolvent identity, then there exists a unique operator A (= λ − R(λ) −1 ) such that R(λ) = (λ − A) −1 .
The above gives an "integral" definition of the (C b -)generator by means of the Laplace transform of the semigroup.Often, a differential definition of the generator is preferred and we show next that several differential definitions are in fact equivalent to the above.In one of them, we make use of the so-called strict topology β 0 on C b (E).This topology is defined as follows: Let F 0 (E) denote the set of functions ϕ : E → R that vanish at infinity, i.e. for every ε > 0 there exists a compact set H ⊂ E with |ϕ(x)| ≤ ε for all x ∈ E \ H. Then the strict topology β 0 is the locally convex topology generated by the seminorms {p ϕ : ϕ ∈ F 0 }, where p ϕ (f ) = ϕf ∞ .This topology is consistent with the duality (C b (E), M (E)), i.e., the dual space (C b (E), β 0 ) ′ is M (E), see [39,Theorem 7.6.3].In fact, it is the Mackey topology of the dual pair (C b (E), M (E)), i.e. the finest locally convex topology on C b (E) that yields M (E) as a dual space, see [61,Theorem 4.5 and 5.8].This implies that a kernel operator is automatically also β 0 -continuous.By [39,Theorem 2.10.4],β 0 coincides on • ∞ -bounded subsets on C b (E) with the topology of uniform convergence on compact subsets of E. Thus, condition (iii) in Definition A.2 can be reformulated by saying T (t)f → f with respect to β 0 as t → 0 for every f ∈ C b (E).Taking the β 0 -continuity of the operators T (t) into account, it follows that for every f ∈ C b (E) the orbit t → T (t)f is β 0 right-continuous.(iii) sup{t −1 T (t)u − u ∞ : t ∈ (0, 1)} < ∞ and t −1 (T (t)u(x) − u(x)) → f (x) as t → 0 for all x ∈ E. (iv) t −1 (T (t)u − u) → f with respect to σ as t → 0. (v) t −1 (T (t)u − u) → f with respect to β 0 as t → 0. (vi) sup{t −1 T (t)u − u ∞ : t ∈ (0, 1)} < ∞ and t −1 (T (t)u − u) → f as t → 0 uniformly on compact subsets of E.
(iii) ⇒ (iv) follows from the dominated convergence theorem, whereas (iv) ⇒ (i) is a consequence of [45, Theorem 2.10], applied with τ M = σ, which corresponds to choosing M as the finite subsets of Y = M (E).
As β 0 is the Mackey topology of the pair (C b (E), M (E)), we have β = τ M where M denotes the collection of all absolutely convex subsets of Y = M (E) which are σ(M (E), C b (E))-compact.Thus the equivalence (i) ⇔ (v) also follows from [45,Theorem 2.10], this time applied with τ M = β 0 .The remaining equivalence (v) ⇔ (vi) follows from the fact that β 0 coincides with the topology of uniform convergence on compact subset of E on • ∞ -bounded subsets of C b (E) and the already established implications (v) ⇒ (i) ⇒ (iii).
If (T (t), t > 0) is a C b -Feller semigroup then, by the β 0 -continuity of the operators T (t) and (iii) in Definition A.2, for every f ∈ C b (E) the orbit t → T (t)f is right-continuous with respect to β 0 .It is a natural question, whether each orbit is actually β 0 -continuous, but, to the best of our knowledge, it is still open.However, if (T (t), t > 0) additionally enjoys the strong Feller property, i.e.T (t)B b (E) ⊂ C b (E) for all t > 0, then this is indeed the case.

Hypothesis 1 . 1 .
Let D = ∅ be an open bounded Lipschitz subset of R d and let µ : D c × B(D) → [0, 1] have the following properties:
the space of continuous functions f : R d → R satisfying f (x) → 0 as |x| → ∞, invariant and is strongly continuous on that space.As in the Introduction, we write ν(x, y) = ν(y − x) = c d,α |y − x| −d−α and ν(x, dy) = ν(x, y)dy and for u : R d → R and x ∈ R d we define

(
ii) ⇒ (i).Since (λ − A D ) −1 exists for λ > 0 and maps D(A D ) onto C b (D), this can be proved as in [2, Theorem 2.3].(i) ⇒ (iii).Adding an isolated point † as a cemetery state to D, we can consider the stopped process (Y t∧τ D ) t≥0 as a Markov process with state space E = D ∪ { †}.Here Y t∧τ D = † for t ≥ τ D .The transition semigroup of this process is P D , if we extend a function g in C b (D) to E by setting g( †) = 0.It follows from the implication (iii) ⇒ (ii) in Theorem A.5, that for u, f as in (i) the pair (u, f ) belongs to the full generator in the sense of [29, Equation (5.5) of Chapter 1].By [29, Proposition 4.1.7],the process because f is continuous and bounded and we can use the dominated convergence theorem.This shows that Du(x) = f (x), for all x ∈ D. At this point[48,  Lemma 3.3] yields (iii).

Lemma 4 . 2 .
If f ∈ B b (D), λ > 0 and Φ(λ)f = f , then f = 0. Proof.By Lemma 4.1, f = Φ(λ)f ∈ C(D).Assume that sup D f > 0. Note that sup D c µf ≤ sup D f .Using (4.1), for every x ∈ D we get Φ(λ)f (x) ≤ E x e −λτ D sup D f < sup D f,because P x -a.s.we have τ D > 0, by the right-continuity of the trajectories of the process Y .In particular, the maximum of f is attained at ∂D.Let H and ϑ be as in Hypothesis 1.1(ii).Then sup H f = (1 − δ) sup D f for some δ > 0. As Φ(λ)f = f , Lemma 4.1 yields for z ∈ ∂D,

Theorem 4 . 9 .
For λ > 0, the closure of the range of R(λ) equals C µ (D).Proof.Let us first prove that the range of R(λ) is contained in C µ (D).To that end, fix f ∈ B b (D).As P D (t)f ∈ C 0 (D) we have R D (λ)f ∈ C 0 (D) ⊂ C(D) for any f ∈ B b (D).Using Lemma 4.1 and induction, we have Φ(λ) k R D (λ)f ∈ C µ (D) ⊂ C(D) for all k ≥ 1 and Lemma 4.3 and Lemma 4.4 imply that R(λ)f ∈ C(D).

Remark 5 . 5 .Corollary 5 . 6 .
The condition u ∈ C µ (D) that appears in the statements (ii) -(iv) of Theorem 5.4 can be seen as a boundary condition that a function u ∈ C b (D) necessarily satisfies if it belongs to D(A ).The condition is equivalent to lim D∋x→z u(x) = D u(y)µ(z, dy), z ∈ ∂D.Let γ be given by (1.1) and fix λ > 0. Then for every f ∈ C b (D) there exists a unique function u ∈ C µ (D) satisfying λu(x) − lim ε→0 + {|y−x|>ε}∩D (u(y) − u(x))γ(x, dy) = f (x), x ∈ D. Proof.Since A is the generator of a C b -Feller semigroup, (0, ∞) belongs to the resolvent set of A .Thus, for every λ > 0 and f ∈ C b (D) the equation λu − A u = f has a unique solution u:= R(λ) ∈ D(A ).Now the claim follows from the characterization of A in Theorem 5.4(iv).

Remark 6 . 3 .
Here are some observations and open problems.

Lemma A. 1 .
Let T ∈ L (C b (E)) be a bounded linear operator.The following are equivalent:

Theorem A. 5 .
Let (T (t), t > 0) be a C b -Feller semigroup with C b -generator A. Then for u, f ∈ C b (E), the following assertions are equivalent.(i) u ∈ D(A) and Au = f .(ii) For every t > 0 and x ∈ E, we have T (t)u(x) − u(x) = t 0 T (s)f (x) ds.
see, e.g., Bogdan and Byczkowski [10, Subsection 2.3].We consider Y τ D , the position at the exit time, and Y τ D − := lim s↑τ D Y s , the position just before the exit.Thanks to the Lipschitz geometry of D, [26,D (t, x, y)dy, x ∈ D.Proof.Due to[24,page 68], the semigroup P D is a Feller semigroup and enjoys the strong Feller property, because these properties hold true for the unkilled semigroup P and D is regular.At this point [59, Lemma 3.1] implies that P D is a C b -semigroup.Note that[59]is actually concerned with Feller semigroups on R d , but the proof of[59, Lemma 3.1]applies to open subsets of R d , too.The last statement of Lemma 2.2 can be proved as in[26, Theorem 2.7].
4, K(t) is a Markovian operator on B b (D).
[12,rk 3.6.Many results of this section do not need the full strength of Hypothesis 1.1.Namely, Lemma 3.1, 3.2 and 3.3 do not require parts (ii) and (iii) of Hypothesis 1.1 and Theorem 3.4 and Corollary 3.5 do not use part (iii).The Lipschitz condition on D can be replaced throughout the paper by the regularity (2.7) and "not hitting the boundary" (2.8); this follows from our proofs.For instance, an open set D ⊂ R d with the complement D c satisfying the so-called volume density condition[12]is regular by the arguments following (2.7).If its boundary has zero Lebesgue measure, then (2.8) holds too; see[12, Corollary  A.2].The boundedness of D may be dropped in Lemma 3.1, 3.2, 3.3, Theorem 3.4, and Corollary 3.5, but it is crucial, e.g., in Section 6; see Remark 6.3.