Abstract
Motivated by the notion of isotropic \(\alpha \)-stable Lévy 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.
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Acknowledgements
We thank Adam Bobrowski, Jan van Casteren, Damian Fafuła, Piotr Garbaczewski, Wolfhard Hansen, Tadeusz Kulczycki, Tomasz Klimsiak, Tomasz Komorowski, Andrey Pilipenko, Victor Rivero, Tomasz Szarek, Paweł Sztonyk and Zoran Vondraček for discussions, comments and references. We thank the referee for insightful comments, including those on Hypothesis 1.1(iii), which strongly influenced the presentation and content of the paper.
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Appendix
Appendix
1.1 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 \(E\subset \) be an open subset of \({\mathbb {R}^d}\), or, more generally, a Polish space. A kernel on E is a map \(k: E\times \mathscr {B}(E)\rightarrow \mathbb {C}\) such that (i) the map \(x \mapsto k(x,A)\) is measurable for all \(A\in \mathscr {B}(E)\) (ii) the map \(A\mapsto k(x,A)\) defines a measure on E for every \(x\in E\) and (iii) we have \(\sup _x |k|(x, E) <\infty \), where \(|k|(x,\cdot )\) refers to the total variation of \(k(x,\cdot )\).
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 \(\sigma \mathrel {\mathop :}=\sigma (C_b(E), \mathscr {M}(E))\) induced by the space of bounded (complex/signed) measures. We note that for a sequence of functions \((f_n) \subset C_b(E)\), the convergence with respect to \(\sigma \) is nothing else than bp-convergence (bp is short for bounded, pointwise), which means \(\sup _n \Vert f_n\Vert _\infty <\infty \) and \(f_n\rightarrow f\) pointwise. Indeed, that bp-convergence implies \(\sigma \)-convergence follows from the dominated convergence theorem whereas the converse implication follows easily using the uniform boundedness principle.
Lemma A.1
Let \(T\in \mathscr {L}(C_b(E))\) be a bounded linear operator. The following are equivalent:
-
(i)
T is a kernel operator;
-
(ii)
T is \(\sigma \)-continuous;
-
(iii)
If \((f_n) \subset C_b(E)\) bp-converges to \(f\in C_b(E)\), then \(Tf_n\) bp-converges to Tf.
Proof
The equivalence of (i) and (ii) is proved in [46, Proposition 3.5] and the implication (ii) \(\Rightarrow \) (iii) is trivial in view of the above comment. To see (iii) \(\Rightarrow \) (i), let \(\varphi (f) = (Tf)(x)\). By (iii), it follows that \(\varphi (f_n) \rightarrow 0\) whenever \(f_n\) bp-converges to 0. Now [7, Theorem 7.10.1] implies that \(\varphi (f) = \int _E f\, d\nu _x\) for some Baire (hence Borel, as E is Polish) measure \(\nu _x\). The measurable dependence of \(\nu _x\) on x can now be proved in a standard way, see the proof the implication (i) \(\Rightarrow \) (ii) in [46, Proposition 3.5].\(\square \)
In what follows, the space of bounded, \(\sigma \)-continuous operators (equivalently: kernel operators) on \(C_b(E)\) is denoted by \(\mathscr {L}(C_b(E), \sigma )\). Note that any operator \(T\in \mathscr {L}(C_b(E), \sigma )\) can uniquely be extended to a bounded linear operator on all of \(B_b(E)\), by merely plugging \(f\in B_b(E)\) into the right hand side of Eq. 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. \(k(x, \cdot )\) is a (sub-)probability measure for every \(x\in E\).
Definition A.2
A \(C_b\)-Feller semigroup is a family \((T(t), t>0) \subset \mathscr {L}(C_b(E), \sigma )\) 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 \(f\in C_b(E)\) we have \(T(t)f \rightarrow f\) as \(t\rightarrow 0\), 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 \(E={\mathbb {R}^d}\), 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 \(\sigma \)-convergence, we see that the requirement (iii) in the above definition in particular implies that \(T_tf \rightarrow f\) as \(t\rightarrow 0\) with respect to \(\sigma \) and thus, by the semigroup law and the \(\sigma \)-continuity of the operators T(t), that \(T(t)f \rightarrow T(s)f\) as \(t\downarrow s\) for every \(f\in C_b(E)\), i.e. the orbits \(t\mapsto T(t)f\) are right-continuous with respect to \(\sigma \). In particular, the orbits have enough measurability to define the Laplace transform of a \(C_b\)-Feller semigroup by setting
for any \(f\in C_b(E)\), \(\nu \in \mathscr {M}(E)\) and \(\lambda >0\).
Lemma A.3
Let \((T(t), t>0)\) be a \(C_b\)-Feller semigroup. Then, for every \(\lambda >0\), Equation A.2 defines an operator \(R(\lambda ) \in \mathscr {L}(C_b(E), \sigma )\). Moreover, the family \((R(\lambda ), \lambda >0)\) consists of injective operators and satisfies the resolvent identity
for all \(\lambda _1, \lambda _2>0\).
Proof
By [46, Theorem 6.2] any \(C_b\)-Feller semigroup is integrable in the sense of [46, Definition 5.1]. Now the resolvent identity for the operators \(R_\lambda \) follows from [46, Proposition 5.2]. That the operators \(R(\lambda )\) are injective is a consequence of [45, Theorem 2.10].\(\square \)
As is well known, if \((R(\lambda ), \lambda >0)\) consists of injective operators and satisfies the resolvent identity, then there exists a unique operator A (\(= \lambda - R(\lambda )^{-1}\)) such that \(R(\lambda ) = (\lambda - A)^{-1}\).
Definition A.4
Let \((T(t), t>0)\) be a \(C_b\)-Feller semigroup. The \(C_b\)-generator of \((T(t), t>0)\) is the unique operator A such that \(R(\lambda ) = (\lambda - A)^{-1}\) for all \(\lambda >0\), where the operators \(R(\lambda )\) are given by Eq. A.2, and its domain is \(D(A) \mathrel {\mathop :}=\textrm{rg}R(\lambda )\), which is independent of \(\lambda >0\).
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 \(\beta _0\) on \(C_b(E)\). This topology is defined as follows: Let \(\mathscr {F}_0(E)\) denote the set of functions \(\varphi : E\rightarrow \mathbb {R}\) that vanish at infinity, i.e. for every \(\varepsilon >0\) there exists a compact set \(H\subset E\) with \(|\varphi (x)|\le \varepsilon \) for all \(x\in E\setminus H\). Then the strict topology \(\beta _0\) is the locally convex topology generated by the seminorms \(\{p_\varphi : \varphi \in \mathscr {F}_0\}\), where \(p_\varphi (f) = \Vert \varphi f\Vert _\infty \). This topology is consistent with the duality \((C_b(E), \mathscr {M}(E))\), i.e., the dual space \((C_b(E), \beta _0)'\) is \(\mathscr {M}(E)\), see [39, Theorem 7.6.3]. In fact, it is the Mackey topology of the dual pair \((C_b(E), \mathscr {M}(E))\), i.e. the finest locally convex topology on \(C_b(E)\) that yields \(\mathscr {M}(E)\) as a dual space, see [61, Theorem 4.5 and 5.8]. This implies that a kernel operator is automatically also \(\beta _0\)-continuous. By [39, Theorem 2.10.4], \(\beta _0\) coincides on \(\Vert \cdot \Vert _\infty \)-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 \(T(t)f\rightarrow f\) with respect to \(\beta _0\) as \(t\rightarrow 0\) for every \(f\in C_b(E)\). Taking the \(\beta _0\)-continuity of the operators T(t) into account, it follows that for every \(f\in C_b(E)\) the orbit \(t\mapsto T(t)f\) is \(\beta _0\) right-continuous.
Theorem A.5
Let \((T(t), t> 0)\) be a \(C_b\)-Feller semigroup with \(C_b\)-generator A. Then for \(u,f\in C_b(E)\), the following assertions are equivalent.
-
(i)
\(u\in D(A)\) and \(Au=f\).
-
(ii)
For every \(t>0\) and \(x\in E\), we have \(T(t)u(x) - u(x) = \int _0^t T(s)f(x)\,\textrm{d}s\).
-
(iii)
\(\sup \{ t^{-1} \Vert T(t)u-u\Vert _\infty : t\in (0,1)\} <\infty \) and \(t^{-1}(T(t)u(x) - u(x)) \rightarrow f(x)\) as \(t\rightarrow 0\) for all \(x\in E\).
-
(iv)
\(t^{-1}(T(t)u - u) \rightarrow f\) with respect to \(\sigma \) as \(t\rightarrow 0\).
-
(v)
\(t^{-1}(T(t)u-u) \rightarrow f\) with respect to \(\beta _0\) as \(t\rightarrow 0\).
-
(vi)
\(\sup \{t^{-1}\Vert T(t)u-u\Vert _\infty : t\in (0,1)\} <\infty \) and \(t^{-1}(T(t)u-u) \rightarrow f\) as \(t\rightarrow 0\) uniformly on compact subsets of E.
Proof
(i) \(\Rightarrow \) (ii). By [46, Proposition 5.7](i), \(\langle T(t)u - u, \nu \rangle = \int _0^t \langle T(s)f, \nu \rangle \,\textrm{d}s\) for all \(t>0\) and \(\nu \in \mathscr {M}(E)\). Picking \(\nu = \delta _x\), we get (ii).
(ii) \(\Rightarrow \) (iii). We have \(t^{-1}(T(t)u(x) - u(x)) = t^{-1}\int _0^t T(s)f(x)\,\textrm{d}s \rightarrow f(x)\) as \(t\rightarrow 0\), by the continuity of \(s\mapsto T(s)f(x)\) in 0. Moreover,
for all \(t>0\).
(iii) \(\Rightarrow \) (iv) follows from the dominated convergence theorem, whereas (iv) \(\Rightarrow \) (i) is a consequence of [45, Theorem 2.10], applied with \(\tau _{\mathfrak {M}} = \sigma \), which corresponds to choosing \(\mathfrak {M}\) as the finite subsets of \(Y= \mathscr {M}(E)\).
As \(\beta _0\) is the Mackey topology of the pair \((C_b(E),\mathscr {M}(E))\), we have \(\beta = \tau _{\mathfrak {M}}\) where \(\mathfrak {M}\) denotes the collection of all absolutely convex subsets of \(Y= \mathscr {M}(E)\) which are \(\sigma (\mathscr {M}(E), C_b(E))\)-compact. Thus the equivalence (i) \(\Leftrightarrow \) (v) also follows from [45, Theorem 2.10], this time applied with \(\tau _{\mathfrak {M}}= \beta _0\). The remaining equivalence (v) \(\Leftrightarrow \) (vi) follows from the fact that \(\beta _0\) coincides with the topology of uniform convergence on compact subset of E on \(\Vert \cdot \Vert _\infty \)-bounded subsets of \(C_b(E)\) and the already established implications (v) \(\Rightarrow \) (i) \(\Rightarrow \) (iii). \(\square \)
If \((T(t), t> 0)\) is a \(C_b\)-Feller semigroup then, by the \(\beta _0\)-continuity of the operators T(t) and (iii) in Definition A.2, for every \(f\in C_b(E)\) the orbit \(t\mapsto T(t)f\) is right-continuous with respect to \(\beta _0\). It is a natural question, whether each orbit is actually \(\beta _0\)-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. \(T(t)B_b(E) \subset C_b(E)\) for all \(t>0\), then this is indeed the case.
Lemma A.6
Let \((T(t), t> 0)\) be a \(C_b\)-Feller semigroup that enjoys the strong Feller property. Then \((T(t), t> 0)\) has the following additional properties. Here, in parts (b) and (c) we set \(T(0)=I\).
-
(a)
For every \(f\in B_b(E)\), the map \((0,\infty )\times E \ni (t,x) \mapsto T(t)f(x)\) is continuous.
-
(b)
For every \(f\in C_b(E)\), the map \([0,\infty ) \ni t \mapsto T(t)\) is \(\beta _0\)-continuous.
-
(c)
For every \(f \in C_b(E)\) and \(t_0\in [0,\infty )\), we have \(T_tf \rightarrow T_{t_0}f\) as \(t\rightarrow t_0\) uniformly on compact subsets of E.
Proof
(a) follows from [5, Proposition V.2.10]. See also [47, Theorem 3.7], which shows that the continuity assumption in [5] can be weakened to a measurability assumption. It follows from (a) and (iii) in Definition A.2, that for \(f\in C_b(E)\) the map \([0,\infty )\times E \ni (t,x) \mapsto T_tf(x)\) is continuous. Now (b) and (c) follow from [45, Theorem 4.4].\(\square \)
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Bogdan, K., Kunze, M. The Fractional Laplacian with Reflections. Potential Anal 61, 317–345 (2024). https://doi.org/10.1007/s11118-023-10111-7
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DOI: https://doi.org/10.1007/s11118-023-10111-7