Appl Math Optim 45:237–250 (2002)
2002 Springer-Verlag New York Inc.
On the Log-Laplace Equation for Nonlocal Operators
Generating Sub-Markovian Semigroups
and V. Moroz
Department of Mathematics, University of Wales Swansea,
Singleton Park, Swansea SA2 8PP, Wales
Department of Mathematics, University of Bristol,
University Walk, Bristol BS8 1TW, England
Communicated by M. R¨ockner
Abstract. We consider the semilinear Cauchy problem for a class of pseudo-
lems with negative deﬁnite nonlinearity play an important role in constructing
branching measure-valuedprocesses. We establish local existence and uniqueness of
solutions in the contextof the Dirichlet space associated to the problem. Comparison
and global properties of solutions are also studied.
Key Words. Pseudo-differential operators, Sub-Markovian semigroups, Semi-
linear Cauchy problem, Dirichlet spaces, Branching processes.
AMS Classiﬁcation. Primary 35S10, Secondary 47D07, 60J80.
We consider a class of pseudo-differential operators
p(x, D)u(x) = (2π)
p(x,ξ)ˆu(ξ ) dξ,
V. Moroz has been supported by a one-year scholarship of the DAAD – German Academic Exchange