# Transition probabilities of L\'evy-type processes: Parametrix construction

## TL;DR

This paper establishes existence, uniqueness, and heat kernel estimates for Le9vy-type processes under weak regularity conditions, broadening the scope of applicable stochastic models.

## Contribution

It introduces a new parametrix construction method for Le9vy-type processes with minimal regularity assumptions on the symbol.

## Key findings

- Existence and uniqueness results for Le9vy-driven SDEs with Hf6lder continuous coefficients
- Heat kernel estimates for Le9vy and Le9vy-type processes
- Extensive list of processes satisfying the assumptions

## Abstract

We present an existence result for L\'evy-type processes which requires only weak regularity assumptions on the symbol $q(x,\xi)$ with respect to the space variable $x$. Applications range from existence and uniqueness results for L\'evy-driven SDEs with H\"older continuous coefficients to existence results for stable-like processes and L\'evy-type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of L\'evy and L\'evy-type processes. The paper includes an extensive list of L\'evy(-type) processes satisfying the assumptions of our results.