Positivity 5: 239–257, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
On a Class of Elliptic-Parabolic Equations on
FRANCESCO ALTOMARE and ELISABETTA M. MANGINO
Dipartimento Interuniversitario di Matematica, Campus Universitario, Università di Bari, Via
Edoardo Orabona 4, I-70125 Bari, Italy.
E-mail: email@example.com, firstname.lastname@example.org
(Received 23 July 1999; accepted 6 December 1999)
Abstract. We study a class of degenerate elliptic second order differential operators acting on some
polynomial weighted function spaces on [0, +∞[. We show that these operators are the generators
-semigroups of positive operators which, in turn, are the transition semigroups associated with
right-continuous normal Markov processes with state space [0, +∞]. Approximation and qualitative
properties of both the semigroups and the Markov processes are investigated as well. Most of the
results of the paper depend on a representation of the semigroups we give in terms of powers of
particular positive operators of discrete type we introduced and studied in a previous paper.
Mathematics Subject Classiﬁcation (2000): 47D06, 41A36, 60J35
Key words: diffusion equation, Feller semigroups, Markov process, positive linear operator
We consider the following degenerate elliptic-parabolic equation
(x, t) = α(x)
(x, t) (x > 0,t 0) (0.1)
with boundary conditions
(x, t) = lim
1 + x
(x, t) = 0 (t 0) (0.2)
and initial condition
u(x, 0) = u
(x) (x 0), (0.3)
where m is an integer number greater than 2, α is a continuous real-valued function
on [0, +∞[ which is differentiable at 0 and for which there exist α
x(1 + x)
(x > 0)