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

Abstract

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

$$\left| B {\mathcal{P}}_{t} u \right|_{H^{\rho}_{2}} \le C \, t^{-\gamma} \,\left| u \right|_{H^{\rho}_{2}}, \quad \forall u \in {H^{\rho}_{2}}(\mathbb{R}^{d} ),\, t>0. $$