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

详细信息    查看全文

• 作者：Pani W. Fernando ; Erika Hausenblas ; Paul Razafimandimby
• 关键词：Hoh’s symbol ; Markovian semigroup ; Pseudo ; differential operator L’evy process ; Generalized Blumenthal ; Getoor index ; Sobolev ; Slobodeckii spaces
• 刊名：Potential Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：46
• 期：1
• 页码：1-21
• 全文大小：356KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Potential Theory; Probability Theory and Stochastic Processes; Geometry; Functional Analysis;
• 出版者：Springer Netherlands
• ISSN：1572-929X
• 卷排序：46

文摘

We consider the stochastic differential equation (SDE) of the form$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{rcl} dX^ x(t) &=& \sigma(X(t-)) dL(t) \\ X^ x(0)&=&x,\quad x\in{\mathbb{R}}^ d, \end{array}\right. \end{array} $$ where \(\sigma :{\mathbb {R}}^ d\to {\mathbb {R}}^ d\) 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 \((\mathcal {P}_{t})_{t\ge 0}\) be the Markovian semigroup associated to X defined by \(\left ({\mathcal {P}}_{t} f\right ) (x) := \mathbb {E} \left [ f(X^ x(t))\right ]\), t≥0, \(x\in {\mathbb {R}}^{d}\), \(f\in \mathcal {B}_{b}({\mathbb {R}}^{d})\). Let B be a pseudo–differential operator characterized by its symbol q. Fix \(\rho \in \mathbb {R}\). In this article we investigate under which conditions on σ, L and q there exist two constants γ>0 and C>0 such that$$\left| B {\mathcal{P}}_{t} u \right|_{H^{\rho}_{2}} \le C \, t^{-\gamma} \,\left| u \right|_{H^{\rho}_{2}}, \quad \forall u \in {H^{\rho}_{2}}(\mathbb{R}^{d} ),\, t>0. $$KeywordsHoh’s symbolMarkovian semigroupPseudo-differential operator L’evy processGeneralized Blumenthal-Getoor indexSobolev-Slobodeckii spacesThis work was supported by the Austrian Science foundation (FWF), Project number P23591-N12.