Non-symmetric Lévy-type operators
A talk in the Nonlocal Equations: Analysis and Numerics series by
Karol Szczypkowski from Wrocław
| Abstract: | 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
$$
\mathcal{L}^{\kappa}f(x):= b(x)\cdot \nabla f(x)+ \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \langle z
,\nabla f(x)\rangle)\kappa(x,z)J(z)\, dz\,,
$$
under certain assumptions on $b$, $\kappa$ and $J$.
In particular, $J\colon \mathbb{R}^d \to [0,\infty]$
is a Lévy 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.
[GS] Tomasz Grzywny and Karol Szczypkowski, Heat kernels of non-symmetric Lévy-type operators,
J. Differential Equations, 267(10):6004--6064, 2019.
[MS] Jakub Minecki and Karol Szczypkowski, Non-symmetric
non-local operators, preprint.
[S] Karol Szczypkowski, Fundamental solution for super-critical non-symmetric
Lévy-type operators, preprint.
Within the CRC this talk is associated to the project(s): A7 |