We prove the uniqueness and the existence of the fundamental solution for the equation $\partial_t =\mathcal{L}^{\kappa}$ for non-symmetric non-local operators

$$\mathcal{L}^{\kappa}f(x):= b(x)\cdot \nabla f(x)+ \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\, dz\,,$$

under certain assumptions on $b$, $\kappa$ and $J$. In particular, $J\colon \mathbb{R}^d \to [0,\infty]$ is a Levy density, i.e.,

$$\int_{\mathbb{R}^d}(1\land |x|^2)J(x)dx<\infty\,,$$

the function $\kappa(x,z)$ satisfies $0<c^{-1}\leq \kappa(x,z)\leq c$, and $|\kappa(x,z)-\kappa(y,z)|\leq c|x-y|^{\epsilon_{\kappa}}$ for some $\epsilon_\kappa \in (0, 1]$.

The solution gives rise to a semigroup which we also study.

During the talk, based on [1, 2, 3], a general approach by the so-called parametrix method will be discussed.

References

1. Tomasz Grzywny and Karol Szczypkowski, Heat kernels of non-symmetric Lévy-type operators, J. Differential Equations,

267(10):6004–6064, 2019.

2. Jakub Minecki and Karol Szczypkowski, Non-symmetric non-local operators, preprint.

3. Karol Szczypkowski, Fundamental solution for super-critical non-symmetric Lévy-type operators, preprint.