Nonsymmetric Lévy‐type operators

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 ∂tu=Lu$\partial _t u ={\mathcal {L}}u$ with nonsymmetric nonlocal operators Lf(x):=b(x)·∇f(x)+∫Rd(f(x+z)−f(x)−1|z|<1z,∇f(x))κ(x,z)J(z)dz,$$\begin{equation*} \mathcal {L}f(x):= b(x)\cdot \nabla f(x)+ \int {\mathbb {R}^d}(f(x+z)-f(x)- {\bf 1}{|z|<1} \left)\kappa (x,z)J(z), dz,, \end{equation*}$$under certain assumptions on b$b$, κ$\kappa$, and J$J$. The result allows more general coefficients even for J(z)=|z|−d−1$J(z)=|z|^{-d-1}$ .