Appl Math Optim 37:151–188 (1998)
1998 Springer-Verlag New York Inc.
On Interacting Systems of Hilbert-Space-Valued Diffusions
A. G. Bhatt,
R. L. Karandikar,
and J. Xiong
Indian Statistical Institute, New Delhi, India
Center for Stochastic Processes, University of North Carolina,
Chapel Hill, NC 27599-3260, USA
Department of Mathematics, University of Tennessee,
Knoxville, TN 37996-1300, USA
Abstract. A nonlinear Hilbert-space-valuedstochasticdifferentialequation where
(L being the generator of the evolution semigroup) is not nuclear is investigated
in this paper. Under the assumption of nuclearity of L
, the existence of a unique
solution lying in the Hilbert space H has been shown by Dawson in an early paper.
is not nuclear, a solution in most cases lies not in H but in a larger
Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove
under suitable conditions that a unique strong solution can still be found to lie in
the space H itself. Uniqueness of the weak solution is proved without moment
assumptions on the initial random variable.
A second problem considered is the asymptotic behavior of the sequence of
empirical measures determined by the solutions of an interacting system of H-
valueddiffusions.Itis shownthat the sequence converges in probability to the unique
of the martingale problem posed by the corresponding McKean–Vlasov
Key Words. Martingale problem, Nuclear, Interacting Hilbert-space-valued dif-
fusions, McKean–Vlasov equation, Propagation of chaos.
AMS Classiﬁcation. Primary 60J60, Secondary 60B10.
This research was supported by the National Science Foundation and the Air Force Ofﬁce of Scientiﬁc
Research Grant No. F49620 92 J 0154 and the Army Research Ofﬁce Grant No. DAAL03 92 G 0008. The
ﬁrst author was supported by National Board for Higher Mathematics, Bombay, India.