Analytic Properties of Markov Semigroup Generated by Stochastic Differential Equations Driven by Lévy Processes

We consider the stochastic differential equation (SDE) of the form dXx(t)=σ(X(t−))dL(t)Xx(0)=x,x∈ℝd,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{rcl} dX^ x(t) &=& \sigma(X(t-)) dL(t) \\ X^ x(0)&=&x,\quad x\in{\mathbb{R}}^ d, \end{array}\right. \end{array} $$\end{document} where σ:ℝd→ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma :{\mathbb {R}}^ d\to {\mathbb {R}}^ d$\end{document} is globally Lipschitz continuous and L={L(t):t≥0} is a Lévy process. Under this condition on σ it is well known that the above problem has a unique solution X. Let (Pt)t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathcal {P}_{t})_{t\ge 0}$\end{document} be the Markovian semigroup associated to X defined by Ptf(x):=Ef(Xx(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left ({\mathcal {P}}_{t} f\right ) (x) := \mathbb {E} \left [ f(X^ x(t))\right ]$\end{document}, t≥0, x∈ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\in {\mathbb {R}}^{d}$\end{document}, f∈ℬb(ℝd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\in \mathcal {B}_{b}({\mathbb {R}}^{d})$\end{document}. Let B be a pseudo–differential operator characterized by its symbol q. Fix ρ∈ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho \in \mathbb {R}$\end{document}. In this article we investigate under which conditions on σ, L and q there exist two constants γ>0 and C>0 such that BPtuH2ρ≤Ct−γuH2ρ,∀u∈H2ρ(ℝd),t>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| B {\mathcal{P}}_{t} u \right|_{H^{\rho}_{2}} \le C \, t^{-\gamma} \,\left| u \right|_{H^{\rho}_{2}}, \quad \forall u \in {H^{\rho}_{2}}(\mathbb{R}^{d} ),\, t>0. $$\end{document}


Introduction
The Blumenthal-Getoor index was first introduced in [6] in order to analyze the Hölder continuity of the sample paths, the r-variation, r ∈ (0, 2] and the Hausdorff-dimension of the paths of Lévy processes. Straightforward calculations give that the Blumenthal-Getoor index of an r-stable process is r. Lévy processes with Blumenthal-Getoor index less than 1 (resp. greater than 1) have paths of finite variation (resp. infinite variation). The Brownian motion has finite 2-variation. By using Hoh's symbol, Schilling introduced in [24], see also [4], a generalized Blumenthal-Getoor index which enabled him to characterize the Hölder continuity of the samples paths of a stochastic process. In [25], Schilling and Schnurr described the long term behavior in terms of this generalized index, for more details see also [4]. In [11] Glau gives a classification of Lévy processes via their symbols. To be more precise, Glau defines the Sobolev index of a Lévy process by a certain growth condition of its symbol.
In the present paper we investigate analytic properties of the Markovian semigroup generated by a SDE driven by a Lévy process. The main result in this article is Theorem 2.1 which is important, for instance, in nonlinear filtering with Lévy noise where one has to analyze the Zakai equation with jumps (see [10]). The leading operator of the Zakai equation is a pseudo-differential operator defined by the Hoh symbol of the driving noise in the state process. Thus, the uniqueness of the mild solution of the Zakai equation and its regularity depends very much on the estimate we obtain in Theorem 2.1.
Let L = {L(t) : t ≥ 0} be a d-dimensional Lévy process. We consider the stochastic differential equations of the form where b : R d → R d and σ : R d → R d × R d are functions satisfying the following conditions.
Hypothesis 1 Let k ≥ d 2 . We assume that b ∈ C k (R d ; R d ) and σ ∈ C k (R d ; R d × R d ) with their derivatives are bounded. In particular, we suppose that σ is bounded from below and above.
Under Hypothesis 1, the existence and uniqueness of the solution to equation (1.1) is well established, see for e.g. [2,p. 367,Theorem 6.2.3]. In addition, X x has P-a.s. càdlàg trajectories.
Let (P t ) t≥0 be the Markovian semigroup associated to X defined by We have the following results.

Proposition 1.1 If σ and b satisfies Hypothesis 1, then
(1) for any t, s ≥ 0 we have P t • P s = P t+s , (2) the semigroup (P t ) t≥0 is Feller on C b (R d ) and C 0 (R d ).
Proof Since, by Hypothesis 1, σ and b are globally Lipschitz, the proof of item (1) is quite standard and can be found, for instance, in [2,Section 6.4.2]. Owing to Hypothesis 1 again, the C 0 -Feller Property of (P t ) t≥0 follows from [2,Theorem 6.7.2]. For the C b -Feller property we refer, for instance, to [2,Note 3].
The infinitesimal generator of (P t ) t≥0 is given by where the symbol ψ is defined by In case L is a d-dimensional Brownian motion and σ ∈ C ∞ b (R d ; R d × R d ) is bounded from below and above, A is a second order partial differential operator on L 2 (R d ) with domain D(A) = H 2 2 (R d ). Moreover, (P t ) t≥0 is an analytic semigroup on L 2 (R d ) and the following inequality holds for B = ∇ Let L be a pure jump Lévyprocess and B be a pseudo-differential operator induced by a symbol. The purpose of this article is to investigate under which additional conditions the estimate (1.2) holds.

Notation 1.1 For any nonnegative integers
For any ρ ≥ 0 we define the function · : R ξ → ξ ρ := (1 + |ξ | 2 ) ρ 2 ∈ R. The following inequality, also called the Peetre inequality is used in several places: for any s ∈ R there exists a constant c s > 0 such that Let U ⊂ R d be a non-empty set and f, g : U → [0, ∞). We set f (x) g(x), x ∈ U , iff there exists a constant C > 0 such that f (x) ≤ Cg(x) for all x ∈ U . Moreover, if f and g depend on a further variable z ∈ Z, the statement for all z ∈ Z, f (x, z) g(x, z), x ∈ U means that for every z ∈ Z there exists a real number C z > 0 such that f (x, z) ≤ C z g(x, z) Similarly as above,  one may handle the case if the functions depend on a further variable. Let S(R d ) be the Schwartz space of functions belong to C ∞ (R d ) where all derivatives decreases faster than any power of |x| as |x| approaches infinity. Let S (R d ) be the dual of S(R d ). For any pair of functions f, g ∈ S(R d ), we define (f, g) by Throughout this paper, we define the Fourier transform F as follows: Its inverse is F −1 and is defined by we denote the set of all complex valued continuous (resp. measurable and bounded) functions f : U → C. Furthermore, for m ∈ N we define Finally, we denote by N the set of positive integers and N 0 = N ∪ {0}.

Hoh's Symbols Associated to Lévy Processes
Throughout the remaining article, let L = {L(t) : t ≥ 0} be a family of d-dimensional Lévy processes and let us denote by Let A be the infinitesimal generator of (T t ) t≥0 defined by An alternative way of defining A makes use of the Lévy symbols. In particular, let ψ : Then, there exist a vector l ∈ R d , a positive definite matrix Q and a Lévy measure ν such that ψ can be written in the following form Here, ψ is called the Lévy symbol of the Lévy process L (see [23]). It can be shown that if L is a Lévy process with symbol ψ, then the infinitesimal generator defined by (2.1) can also be written as (see e.g. [2,4,17]) If ν has bounded second moments, the operator A is well defined on C 2 b (R d ), has values in B b (R d ) and satisfies the positive maximum principle (see e.g. [17,Theorem 4.5.13 ]). Therefore, A generates a Feller semigroup on C ∞ b (R d ) and a sub-Markovian semigroup on L 2 (R d ) (see e.g. [18, Theorem 2.6.9 and Theorem 2.6.10]).
A further important property of a Lévy symbol is, that it is negative definite.
If a negative definite function ψ is continuous, then Moreover, every real-valued continuous negative definite function ψ has a representation where c ≥ 0 is a constant, q ≥ 0 is a quadratic form and ν is a symmetric Borel measure on R d \ {0} called Lévy measure having the property that The Blumenthal-Getoor index of a Lévy process with Lévy measure ν is defined by the inf{p : |x|≤1 |x| p ν(dx) < ∞}.
The Blumenthal-Getoor index has been generalized in several directions, see e.g. [4, p. 124]. Here, in this article we modify also the Blumenthal-Getoor index, but similarly to the characterization of pseudo-differential operators.

Definition 2.2
Let L be a normalized Lévy process with symbol ψ and ψ ∈ C k (R d \ {0}) for some k ∈ N 0 . Then the Blumenthal-Getoor index of order k is defined by be the upper and be the lower Blumenthal-Getoor index s + of order k. Here α denotes a multi-index. If k = ∞ then Blumenthal-Getoor index of infinity order is defined by Remark 2.1 The Blumenthal-Getoor index of order infinity is defined for the sake of completeness. In this paper, we are interested in Lévy processes with Blumenthal-Getoor index of finite order k.
In order to define the resolvent of an operator A associated to a symbol ψ, we need to characterize the range of the symbol of ψ. Here, one has different possibilities at one's disposal, depending on whether one comes from analysis or probability theory. Thus, one can introduce the sector condition or the type of a symbol. Both definitions describe the range of a symbol. The sector condition reads as follows. [3, p. 116]) Let L be a Lévy process with symbol ψ. We say that the symbol ψ satisfies the sector condition, if there exists a κ > 0 such that For δ ∈ [0, π] we define δ := {z ∈ C \ {0} : | arg(z)| < δ}. Now, the type of a symbol is given as follows.

Definition 2.4
Let L be a Lévy process with symbol ψ. We say the symbol ψ is of type Remark 2.2 In general, one uses the essential range of a symbol ψ to characterize its type. But in our case ψ is continuous so we work with the closure of the range of ψ.

Remark 2.3
If a symbol ψ is of type (0, θ), then it satisfies the sector condition with κ = tan(θ ) and vice versa.
In addition, taking into account that the multiplication operator m(ξ ) = ξ s is an .
The generalized Blumenthal-Getoor index of order 0 and the type of a symbol can be calculated in many cases. Here, we give some examples.

Example 2.4
It is interesting to note that, by subordination, one can produce sectorial symbols given a non-sectorial one. The typical example ψ(ξ ) = iξ (non-sectorial) becomes sectorial if we use (iξ ) α , i.e. the Bernstein function f (s) = s α with α ∈ (0, 1). There are many of Bernstein functions having this property, for references see e.g. [12,16].
Let L = {L(t) : t ≥ 0} be a d-dimensional Lévy process without any Gaussian component. We consider the stochastic differential equations of the form t≥0 be the associated Markovian semigroup of X defined by Then, (P t ) t≥0 is a Feller semigroup and one can compute its infinitesimal generator. Again, one way of computing A is done by Hoh's symbols (see [15]). In particular, one has where the symbol a(., .) is defined by Alternatively, we can give an explicit form of a(·, ·) in term of the Lévy symbol of the driving noise L. In fact, if ψ is a Lévy symbol of the Lévy process L = {L(t) : t ≥ 0}, then it is shown in [25,Theorem 3.1], that the symbol a : Symbols also arise in the context of pseudo-differential operators, whereas the term symbol is defined in the following way (in Appendix B we give a short summary of some definition and theorems that we need for our proof).
• for any two multi-indices α and β there exists a positive constant C α,β > 0 depending only on α and β such that The symbolic calculus for pseudo-differential operators is well established, see [21,27,30]. However, in case one considers symbols associated to the solution to stochastic differential equations driven by Lévy processes, the derivatives of the symbol will not be necessarily continuous at the origin, i.e. at {0}. The behavior of ξ at the origin corresponds to the perturbation of the solution X of Eq. (2.3) by the large jumps of the Lévy process L. To illustrate this fact, let us assume that the symbol a(x, ξ ) = a(ξ ) is independent from x and positive definite. Then the symbol corresponds to a Lévy process. Now, let us assume that the Lévy process has a symmetric Lévy measure ν such that for all ≥ 2 the moments R d \{0} |y| ν(dy) are bounded. Then by [15, Proposition 2; p.793] the Lévy symbol a is infinitely often differentiable and one has which means that if all moments of the Lévy measure are bounded, i.e. the moments of the large jumps are bounded, then the Lévy symbol will be infinitely often differentiable at the origin. Now, let us assume that the Lévy process is symmetric and r-stable with r < 2. It is well known that the Lévy symbol is |ξ | r and the large jumps have only bounded moments up to order with < r. In case r < 1 the Lévy symbol is only once continuously differentiable at the origin and in case r > 1, twice continuous differentiable at the origin. Now, one may ask the question: does the non differentiability at the origin have any effect on the smoothing property of the corresponding Markovian semigroup (P t ) t≥0 . Again, let us assume that the Lévy process is symmetric and r-stable with r < 2. Then, the infinitesimal generator of the corresponding Markovian semigroup which implies that the discontinuity of the derivatives at the origin will not have an effect on the smoothing property of the corresponding semigroup (P t ) t≥0 . However, the discontinuity at the origin has to be taken into account and we will relax the definition of the symbols slightly and define a wider class of the Hoh symbols.

Remark 2.5
Let us assume that the Lévy symbol ψ has a generalized Blumenthal-Getoor index s of order k ≥ 1 and σ 1, We also make the following observation.
Remark 2.6 Hypothesis 1 can be reformulated as follows. We assume that b ∈ C k b (R d ) and Now, we can formulate the following Theorem.
If q and ψ are k-times differentiable on R d , then we do not impose any further condition on ψ and q. If ψ and q are only k-times differentiable on R d \{0}, we assume that ψ(0) = 0, and q(0) = 0, and there exist two constants γ ψ > 0 and γ q > 0 for which one has for all multi-indices α with |α| ≤ k.
If ψ is of type (ω, θ ) and has the generalized Blumenthal-Getoor index s 1 and q has the upper Blumenthal-Getoor index less or equal to s 2 with s 2 < s 1 , then for any ρ ∈ R there exists a constant C > 0 such that Remark 2.7 In the following we will write a(x, ξ ) to denote the symbol and a(x, D) to denote the corresponding pseudo-differential operator given by (2.7) In particular, if (2.7) is true, then an application of Proposition A.1 gives the assertion. Hence, we are going to show that under the conditions of Theorem 2.1 estimate (2.7) holds.
In the first step, we assume that ψ, q, σ and b are Schwartz functions defined on R d , i.e. ψ, q, σ, b ∈ S(R d ). In the second step we replace ψ and q by sequences {ψ n : n ∈ N} ⊂ S(R d ) and {q n : n ∈ N} ⊂ S(R d ) converging respectively to ψ and q. We will also replace σ and b by sequences {σ n : n ∈ N} ⊂ S(R d ) and {b n : n ∈ N} ⊂ S(R d ) converging respectively to σ and b, in an appropriate sense that will be made precise later.
Step 2: If ψ and q belong to C k (R d ), since S(R d ) is dense in C k (R d ), there exist sequences {ψ n : n ∈ N}, {σ n : n ∈ N}, {q n : n ∈ N} and {b n : n ∈ N} ⊂ S(R d ) converging to ψ, σ , q, and b, respectively. If ψ, q ∈ C k (R d \ {0}), then we show that there exist {ψ n : n ∈ N} and {q n : n ∈ N} ⊂ S(R d ) such that for all multi-indices α and β with |α| ≤ k and |β| ≤ k, the following limit holds Put ψ n =ψ n n and q n =q n n . Now, a straightforward calculations shows that for any multi-index α with |α| ≤ k one has for n → ∞. Furthermore, one has for n → ∞. Let π n be the symbol of the composition q n • r n where q n and r n are as above. Put .
Because of (B.1) and (B.2) one has Let us now replace in Step I ψ and q by the sequences {ψ n : n ∈ N} and {q n : n ∈ N} constructed above, and σ and b by sequences {σ n : n ∈ N} and {b n : n ∈ N} converging to σ and b in C k (R d ). The Lebesgue Dominated Convergence Theorem gives the assertion.
To tackle the case where ω = 0 it is sufficient to shift λ and use a similar argument to the above.
Acknowledgments Open access funding provided by Montanuniversität Leoben. This work was supported by the Austrian Science foundation (FWF), Project number P23591-N12. We would like to thank the anonymous referee for his insightful comments and remarks which improved the manuscript. We also thank him/her for providing us with Example 2.4.
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Definition A.2 Let U be a Banach space and let A be the generator of a degenerate analytic C 0 -semigroup on U . We say that A is of type (ω, θ, K), where ω ∈ R, θ ∈ (0, π 2 ) and K > 0, if ω + π 2 +θ ⊆ ρ(A) and The theorem below gives some characterizations of the analytic C 0 -semigroups that we will use later on.
Theorem A.1 Let (T (t)) t≥0 be a degenerate C 0 -semigroup of type (M, ω) for some M > 0 and ω ∈ R. Let A be the generator of T . Let ω > ω. The following statements are equivalent: (1) T is an analytic C 0 -semigroup on θ for some θ ∈ (0, π 2 ) and for every θ < θ there exists a constant C 1,θ such that e −ω z T (z) ≤ C 1,θ for all z ∈ θ .
(2) There exists a δ ∈ (0, π 2 ) such that: ω + π 2 +δ ⊂ ρ(A), and for every δ ∈ (0, δ) there exists a constant C 2,δ > 0 such that: T is differentiable for t > 0 and there exists a constant C 3 such that: The proof can be found in [22,Theorem 2.5.2] for exponentially stable analytic C 0 -semigroups with bounded and invertible generator, and can be transferred to arbitrary analytic C 0 -semigroup by the following observation. If T is a C 0 -semigroup of type (M, ω, K) and A is the generator of T , then for any ω > ω the C 0 -semigroup (e −ω t T (t)) t≥0 is exponentially stable and the generator of this semigroup, A − ω I U , is invertible.
For our purpose we need an estimate which is very similar to estimate (3) of Theorem A.1.

Appendix B. Symbol Classes and Pseudo-differential Operators
In this section we shortly introduce pseudo-differential operators and their symbols. In addition we introduce the definitions and theorems which are necessary to our purpose. For a detailed introduction on pseudo-differential operators and their symbols in the context of partial differential equations we recommend the books [21,27,30], in the context of Markov processes we recommend the books [17][18][19] or the survey [4].
In order to treat pseudo-differential operators different classes of symbols have been introduced. Here, we closely follow the definition of [27].
Definition B.1 Let U ⊂ R d , m ∈ R, and ρ, δ two real numbers such that 0 ≤ ρ ≤ 1 and • for any two multi-indices α and β and any compact set K ⊂ U there exists C K,α,β such that We call any function a(x, ξ ) belonging to ∪ m∈R S m 0,0 (R d , R d ) a symbol. For many estimates, one does not need that the function is infinitely often differentiable. For this reason, one introduces also the following classes. • and for any two multi-indices α and β with |α|+|β| ≤ k, there exists a positive constant C α,β > 0 depending only on α and β such that The product of two pseudo-differential operators is again a pseudo-differential operator and can be characterized as follows.  The existence of the formal adjoint is given by the following Theorem.
is the adjoint operator of a(x, D). In addition, a * (x, ξ ) has the following expansion In the next theorem we give sufficient condition for a pseudo-differential operator a(x, D) to be continuous. In fact, analyzing the proof of Theorem 9.7 [30, p. 79] one can see that the condition of the differentiability at the origin can be relaxed. Here, it is important to mention that the proof relies on the Theorem 2.5 [14, p. 120] (see also Theorem 4.23 [1]), from which one can clearly see the extension of the Theorem 9.7 of [30] to symbols, whose derivatives have a singularity at {0}. Moreover, analyzing line by line of the proof of Theorem 9.7, one can give an estimate of the norm of the operator. To investigate the inverse of a pseudo-differential operator one should introduce the set of elliptic and hypoelliptic symbols.
• For arbitrary multi-indices α and β and for any compact set K ⊂ U there exists a constants C K,α,β such that ∂ α ξ ∂ β x a(x, ξ, λ) a −1 (x, ξ, λ) ≤ C K,α,β |ξ | + |λ| One can classify the inverse of the each member of {a(x, ξ, λ) : λ ∈ }, but one has to introduce the set of properly supported operators. Let a(x, ξ ) be a symbol with kernel K a and let supp(K a ) denote the support of K a (the smallest closed subset Z ⊂ U × U such that K a | (U ×U)\Z = 0). Consider the canonical projections 1 , 2 : supp(K a ) → U , obtained by restricting the corresponding projections of the direct product U × U . Recall that a continuous map f : M → N between topological spaces M and N is called proper if for any compact K ⊂ N the inverse image f −1 (K) is a compact in M. A symbol a(x, ξ ) is called properly supported if both projections 1 , 2 : supp(K a ) → U are proper maps. For more details see [27, p. 18]. We will denote by HypL In order to deal with operators depending on a parameter, one can treat a family of symbols by refining the definition of symbol classes see for instance, [21, p. 19]. For this aim, we introduce some class of functions. A positive continuous function : Now, one can choose the target weight M in such a way that it depends on a parameter, say λ. Now the multiplication theorem for the composition operator can be restated as follows.