On Interacting Systems of Hilbert-Space-Valued Diffusions |
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Authors: | A G Bhatt G Kallianpur R L Karandikar J Xiong |
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Institution: | (1) Indian Statistical Institute, New Delhi, India , IN;(2) Center for Stochastic Processes, University of North Carolina, Chapel Hill, NC 27599-3260, USA , US;(3) Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA, US |
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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 Λ
0
of the martingale problem posed by the corresponding McKean—Vlasov equation.
Accepted 4 April 1996 |
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Keywords: | , Martingale problem, Nuclear, Interacting Hilbert-space-valued diffusions, McKean—,Vlasov equation, Propagation of,,,,,chaos, AMS Classification, Primary 60J60, Secondary 60B10, |
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