On weak solution of SDE driven by inhomogeneous singular L\'evy noise

TL;DR

This paper establishes the existence, uniqueness, and regularity properties of weak solutions to a class of inhomogeneous SDEs driven by singular Lévy noise, using an adapted parametrix method.

Contribution

It introduces a novel approach to analyze inhomogeneous Lévy-driven SDEs with different scaling properties, proving key properties of the solutions.

Findings

Weak solutions are uniquely defined and Markov.

The solutions possess the strong Feller property.

The heat kernel is decomposed into principal and residual parts with negligible residual in short time.

Abstract

We study a time-inhomogeneous SDE in $\R^d$ driven by a cylindrical L\'evy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit `principal part' and a `residual part', subject to certain $L^\infty(dx)\otimes L^1(dy)$ and $L^\infty(dx)\otimes L^\infty(dy)$-estimates showing that this part is negligible in a short time, in a sense. The main tool of the construction is the analytic parametrix method, specially adapted to L\'evy-type generators with strong spatial inhomogeneities.