Linear additive functionals of
superdiffusions and related nonlinear P.D.E.
Authors:
E. B. Dynkin and S. E. Kuznetsov
Journal:
Trans. Amer. Math. Soc. 348 (1996), 1959-1987
MSC (1991):
Primary 60J60, 35J65; Secondary 60J80, 31C15, 60J25, 60J55, 31C45, 35J60
DOI:
https://doi.org/10.1090/S0002-9947-96-01602-9
MathSciNet review:
1348859
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a second order elliptic differential operator in a bounded smooth domain
in
and let
. We get necessary and sufficient conditions on measures
under which there exists a positive solution of the boundary value problem
The conditions are stated both analytically (in terms of capacities related to the Green's and Poisson kernels) and probabilistically (in terms of branching measure-valued processes called -superdiffusions).
We also investigate a closely related subject --- linear additive functionals of superdiffusions. For a superdiffusion in an arbitrary domain in
, we establish a 1-1 correspondence between a class of such functionals and a class of
-excessive functions
(which we describe in terms of their Martin integral representation). The Laplace transform of
satisfies an integral equation which can be considered as a substitute for (*).
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Additional Information
E. B. Dynkin
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853-7901
Email:
ebd1@cornell.edu
S. E. Kuznetsov
Affiliation:
Central Economics and Mathematical Institute, Russian Academy of Sciences, 117418, Moscow, Russia
Address at time of publication:
Department of Mathematics, Cornell University, Ithaca, New York 14853-7901
Email:
sk47@cornell.edu
DOI:
https://doi.org/10.1090/S0002-9947-96-01602-9
Received by editor(s):
March 29, 1995
Additional Notes:
Partially supported by National Science Foundation Grant DMS-9301315
Article copyright:
© Copyright 1996
American Mathematical Society