On Markovian semigroups of L\'evy driven SDEs, symbols and pseudo--differential operators

TL;DR

This paper investigates the analytic properties of nonlocal transition semigroups linked to Le9vy-driven SDEs, establishing conditions for analyticity, and explores applications including the strong Feller property and error estimates for approximation schemes.

Contribution

It provides new conditions under which the semigroup is analytic on Besov spaces and applies these results to prove the strong Feller property and error bounds for SDE approximations.

Findings

Semigroup analyticity on Besov spaces under specific conditions

Proof of the strong Feller property for the semigroup

Weak error estimates for SDE approximation schemes

Abstract

We analyse analytic properties of nonlocal transition semigroups associated with a class of stochastic differential equations (SDEs) in $\mathbb{R}^d$ driven by pure jump--type L\'evy processes. First, we will show under which conditions the semigroup will be analytic on the Besov space $B_{p,q}^ m(\mathbb{R}^d)$ with $1\le p, q<\infty$ and $m\in\mathbb{R}$. Secondly, we present some applications by proving the strong Feller property and give weak error estimates for approximating schemes of the SDEs over the Besov space $B_{\infty,\infty}^ m(\mathbb{R}^d)$.