On Interacting Systems of Hilbert-Space-Valued Diffusions View Full Text


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Article Info

DATE

1998-04

AUTHORS

A. G. Bhatt, G. Kallianpur, R. L. Karandikar, J. Xiong

ABSTRACT

A nonlinear Hilbert-space-valued stochastic differential equation where L-1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L-1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L-1 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 -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ0 of the martingale problem posed by the corresponding McKean—Vlasov equation. More... »

PAGES

151-188

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s002459900072

DOI

http://dx.doi.org/10.1007/s002459900072

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1025039530


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