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Regularity of fundamental solutions for Lévy-type operators

Szczypkowski Karol

Bulletin of the Polish Academy of Sciences. Mathematics

2023

Vol. 71, no 2

169--191

For a class of non-symmetric non-local Lévy-type operators Lκ, which include those of the form Lκf(x) := Rd(f(x + z) − f(x) − 1|z|<1⟨z, ∇f(x)⟩)κ(x, z)J(z) dz, e prove regularity of the fundamental solution pκ to the equation ∂t = Lκ.

On dual generators for non-local semi-Dirichlet forms

Uemura T.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 2

199--214

Let k(x, y) be a measurable function defined on E × E off the diagonal, where E is a locally compact separable metric space, and let m be a positive Radon measure on E with full support. In 2012, we showed that a quadratic form having k as a Lévy kernel becomes a lower bounded semi-Dirichlet form on L2(E, m) which is non-local and regular. Then there associates a Hunt process corresponding to the semi-Dirichlet form. In the case where E = Rd, we will show that the dual form of the semi-Dirichlet form also produces a Hunt process by taking a killing. As a byproduct, a precise description of the infinitesimal generator of the dual form is also given.