INTRINSIC COMPOUND KERNEL ESTIMATES FOR THE TRANSITION
PROBABILITY DENSITY OF LÉVY-TYPE PROCESSES AND THEIR
APPLICATIONS

Victoria Knopova

Aleksei Kulik

Abstract: Starting with an integro-differential operator , we prove that its
-closure is the generator of a Feller process , which admits a transition
probability density. To construct this transition probability density, we develop a version of
the parametrix method and a verification procedure, which proves that the constructed object
is the claimed one. As a part of the construction, we prove the intrinsic upper and lower
estimates on the density. As an application of the constructed estimates we state the necessary
and (separately) sufficient conditions under which a given Borel measure belongs to the Kato
and Dynkin classes with respect to the constructed transition probability density.

2010 AMS Mathematics Subject Classification: Primary: 60J35; Secondary: 60J75,
35S05, 35S10, 47G30.

Keywords and phrases: Transition probability density, Lévy-type processes,
pseudo-differential operator, generator, Levi’s parametrix method.