Appl Math Optim 45:269–281 (2002)
2002 Springer-Verlag New York Inc.
Perturbations of Pseudodifferential Operators with Negative
Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld,
Postfach 100131, D-33501 Bielefeld, Germany
Communicated by M. R¨ockner
Abstract. Pseudodifferential operators with negative deﬁnite symbols appear as
generators of jump-type Markov processes. The purpose of this paper is to treat the
large jumps of the process by a perturbation approach for the generator. This is of
particular interest since in this way the generators are made accessible to a symbolic
calculus of pseudodifferential operators. The main auxiliary result consists of a
characterization of tightness of the jump measures in terms of the symbol.
Key Words. L´evy-type process, Generator, Symbolic calculus, Tightness.
AMS Classiﬁcation. 35S05, 60J75, 47D06.
A generator of a Feller semigroup on C
) satisﬁes the positive maximum principle
and therefore by a result of Courr`ege  it has a representation as a pseudodifferential
− p(x, D)ϕ(x) =
for ϕ ∈ C
) in the domain of the generator. Here ˆϕ denotes the Fourier transform
of ϕ, ·,· stands for the inner product in R
, and d¯ξ is deﬁned as d¯ξ = (2π)
The operators arising in this situation are characterized by the property that the symbols
→ C, which determine the operator, are for any ﬁxed x ∈ R