## arXiv Analytics

### arXiv:1807.04257 [math.AP]AbstractReferencesReviewsResources

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

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 $J(z)=|z|^{-d-1}$.

Related articles: Most relevant | Search more
arXiv:1409.2289 [math.AP] (Published 2014-09-08)
Convergence and Divergence of Approximations in terms of the Derivatives of Heat Kernel
arXiv:1208.1808 [math.AP] (Published 2012-08-09, updated 2013-03-01)
The heat kernel on an asymptotically conic manifold
arXiv:1804.06444 [math.AP] (Published 2018-04-17)
The Fundamental Solution to the p-Laplacian in a class of Hörmander Vector Fields