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arXiv:1807.04257 [math.AP]AbstractReferencesReviewsResources

Fundamental solution for super-critical non-symmetric Lévy-type operators

Karol Szczypkowski

Published 2018-07-11Version 1

We construct the fundamental solution (the heat kernel) $p^{\kappa}$ to the equation $\partial_t =\mathcal{L}^{\kappa}$, where under certain assumptions the operator $\mathcal{L}^{\kappa}$ takes the form, $$ \mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\, dz\,. $$ We concentrate on the case when the order of the operator is positive and smaller or equal 1 (but without excluding higher orders up to 2). Our approach rests on imposing conditions on the expression $$ \int_{r\leq |z|<1} z \kappa(x,z)J(z)dz . $$ The result is new even for $1$-stable L{\'e}vy measure $J(z)=|z|^{-d-1}$.

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