Feller Semigroups Obtained by Variable Order Subordination

Abstract

For certain classes of negative definite symbols $q(x,\xi)$ and state space dependent Bernstein function $f(x,s)$ we prove that $-p(x,D)$, the pseudo-differential operator with symbol $-p(x,\xi)=-f(x,q(x,\xi))$, extends to the generator of a Feller semigroup. Our result extends previously known results related to operators of variable (fractional) order of differentiation, or variable order fractional powers. New concrete examples are given.