On interacting systems of Hilbert-space-valued diffusions

Bhatt, A. G. ; Kallianpur, G. ; Karandikar, R. L. ; Xiong, J. (1998) On interacting systems of Hilbert-space-valued diffusions Applied Mathematics and Optimization, 37 (2). pp. 151-188. ISSN 0095-4616

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Official URL: http://www.springerlink.com/content/cleg3jarmfwcl2...

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

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.

Item Type:Article
Source:Copyright of this article belongs to Springer.
Keywords:Martingale Problem; Nuclear; Interacting Hilbert-space-valued Diffusions; Mckean-vlasov Equation; Propagation of Chaos; AMS Classification; Primary 60J60; Secondary 60B10
ID Code:73319
Deposited On:02 Dec 2011 08:38
Last Modified:02 Dec 2011 08:38

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