Non-symmetric Lévy-type operators

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 $$\mathcal{L}f(x):=b(x)\cdot \nabla f(x)+{\int }_{{\mathbb{R}}^d}(f(x+z)-f(x)-1_{|z|<1}⟨z,\nabla f(x)⟩)\kappa (x,z)J(z)dz,$$ under certain assumptions on $b$, $\kappa$ and $J$. The result allows more general coefficients even for $J(z)=|z{|}^{-d-1}$.