Non-symmetric L\'evy-type operators

TL;DR

This paper develops a general method for constructing solutions to non-symmetric Lévy-type operators, proving existence and uniqueness of weak fundamental solutions under broad conditions, including more general jump kernels.

Contribution

It introduces a parametrix-based approach to establish existence and uniqueness for non-symmetric Lévy-type operators with general coefficients.

Findings

Proved existence of weak fundamental solutions for a class of non-symmetric operators.

Established uniqueness of solutions under specified conditions.

Extended results to operators with more general jump kernels.

Abstract

We present a general approach to the parametrix construction. We apply it to prove the uniqueness and existence of a weak fundamental solution for the equation $\partial_t =\mathcal{L}$ with non-symmetric non-local operators $$ \mathcal{L}f(x):= b(x)\cdot \nabla f(x)+ \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\, dz\,, $$ under certain assumptions on $b$, $\kappa$ and $J$. The result allows more general coefficients even for $J(z)=|z|^{-d-1}$.