{ "id": "1807.04257", "version": "v1", "published": "2018-07-11T17:36:52.000Z", "updated": "2018-07-11T17:36:52.000Z", "title": "Fundamental solution for super-critical non-symmetric Lévy-type operators", "authors": [ "Karol Szczypkowski" ], "categories": [ "math.AP" ], "abstract": "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, $$\\mathcal{L}^{\\kappa}f(x):= \\int_{\\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \\left)\\kappa(x,z)J(z)\\, dz\\,.$$ 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 $J(z)=|z|^{-d-1}$.", "revisions": [ { "version": "v1", "updated": "2018-07-11T17:36:52.000Z" } ], "analyses": { "keywords": [ "super-critical non-symmetric lévy-type operators", "fundamental solution", "heat kernel", "excluding higher orders", "approach rests" ], "note": { "typesetting": "TeX", "pages": 0, "language": "en", "license": "arXiv", "status": "editable" } } }