Analytic properties of Markov semigroup generated by Stochastic Differential Equations driven by L\'evy processes

TL;DR

This paper studies the analytic properties of Markov semigroups generated by stochastic differential equations driven by Lévy processes, focusing on conditions for certain regularity estimates in Sobolev spaces.

Contribution

It establishes conditions under which the Markov semigroup exhibits specific smoothing properties in Sobolev spaces for SDEs driven by Lévy processes.

Findings

Derived bounds for the semigroup in Sobolev spaces

Identified conditions on σ, L, and q for regularity estimates

Provided a framework for analyzing pseudo-differential operators associated with the semigroup

Abstract

We consider the stochastic differential equations of the form \begin{equation*} \begin{cases} dX^ x(t) = \sigma(X(t-)) dL(t) \\ X^ x(0)=x,\quad x\in\mathbb{R}^ d, \end{cases} \end{equation*} where $\sigma:\mathbb{R}^ d\to \mathbb{R}^ d$ is Lipschitz continuous and $L=\{L(t):t\ge 0\}$ is a L\'evy process. Under this condition on $\sigma$ it is well known that the above problem has a unique solution $X$. Let $(\mathcal{P}_{t})_{t\ge0}$ be the Markovian semigroup associated to $X$ defined by $( \mathcal{P}_t f) (x) := \mathbb{E} [ f(X^ x(t))]$, $t\ge 0$, $x\in \mathbb{R}^d$, $f\in \mathcal{B}_b(\mathbb{R}^d)$. 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 $\sigma$, $L$ and $q$ there exist two constants $\gamma>0$ and $C>0$ such that $$ \lvert B \mathcal{P}_t u \rvert_{H^\rho_2} \le C \, t^{-\gamma} \,\lvert u \rvert_{H^\rho_2}, \quad \forall u \in {H^\rho_2}(\mathbb{R}^d ),\, t>0. $$