Estimates of transition densities and their derivatives for jump Lévy processes

Abstract

We give upper and lower estimates of densities of convolution semigroups of probability measures under explicit assumptions on the corresponding Lévy measure and the Lévy-Khinchin exponent. We obtain also estimates of derivatives of densities.

Uniform pointwise asymptotics of solutions to quasi-geostrophic equation

“…where k 1 , k 2 ∈ N 2 . Hence, by Hölder inequality, (21), (10), (15) and (20), for p > 2 α−1 , we get…”

Heat kernels of non-symmetric jump processes with exponentially decaying jumping kernel

“…In this subsection, we will combine some results in [KR16,KS15,KS17] to prove Proposition 3.1. Recall that we have assumed that ν :…”

“…Now we are going to establish heat kernel estimates for the process Y obtained in Lemma 3.4, which will imply heat kernel estimate and gradient estimate of X as a consequence of (3.14). To do this, we will check conditions (E), (D), (P) and (C) (when β < 1) in [KS17] for the process X and Y , and apply [KS17,Theorem 4] and [KS15,Theorem 1].…”

“…In Section 3, we discuss gradient estimates for the heat kernel of isotropic unimodal Lévy process with jumping kernel J(|x|)dx, which follows from the results in [KS15,KS17]. We only use Proposition 3.2 in the proof of our main theorem, but Proposition 3.1 itself is of independent interest.…”

“…Proof. Define h(t) := 1 Ψ −1 (t −1 ) as in [KS15] and denote q t (|x|) = q t (x). Applying Lemmas 3.5, 3.6 and 3.8 to [KS15, Thoerem 1] for the process Y in Lemma 3.4, we have…”