Heat kernels of non-symmetric Levy-type operators

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Grzywny, Tomasz¹  Szczypkowski, Karol¹ 

[1] Wroclaw Univ Technol, Wydzial Matemat, Wyb Wyspianskiego 27, PL-50370 Wroclaw, Poland

关键词: Heat kernel estimates;    Levy-type operator;    Non-symmetric non-local operator;    Non-symmetric Markov process;    Feller semigroup;    Levi's parametrix method;   

DOI  :  10.1016/j.jde.2019.06.013

来源: Elsevier

【 摘 要 】

We construct the fundamental solution (the heat kernel) p(kappa) to the equation partial derivative(t) = L-kappa, where under certain assumptions the operator L-kappa takes one of the following forms, L-k f(x) := integral(Rd)(f(x + z) -f (x) - 1(vertical bar z vertical bar<1) < z, del f (x)>)kappa(x, z)J(z)dz, L-kappa f(x) := integral(Rd)(f(x + z) - f (x))kappa(x, z)J(z)dz, L-kappa f(x) := 1/2 integral(Rd)(f(x + z) + f(x - z) - 2f(x))kappa(x, z)J(z)dz. In particular, J: R-d -> [0, infinity] is a Levy density, i.e., integral(Rd) (1 Lambda vertical bar x vertical bar(2)) J (x)dx < infinity. The function kappa(x, z) is assumed to be Borel measurable on R-d x R-d satisfying 0 < kappa(0) <= k(x,z) <= kappa(1), and vertical bar kappa(x, z) - kappa(y, z)vertical bar <= kappa(2)vertical bar x - y vertical bar(beta) for some beta is an element of (0, 1). We prove the uniqueness, estimates, regularity and other qualitative properties of p(kappa). (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free