On Interacting Systems of Hilbert-Space-Valued Diffusions

Journal Article · Sun Mar 15 04:00:00 EST 1998 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S002459900072· OSTI ID:21064299

Bhatt, A G ^[1]; Kallianpur, G ^[2]; Karandikar, R L ^[1]; Xiong, J ^[2]

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1. Indian Statistical Institute, New Delhi (India)
2. Center for Stochastic Processes, University of North Carolina, Chapel Hill, NC 27599-3260 (United States)

A nonlinear Hilbert-space-valued stochastic differential equation where L{sup -1} (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L{sup -1} , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L{sup -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 {lambda}{sub 0} of the martingale problem posed by the corresponding McKean-Vlasov equation.

• OSTI ID: 21064299
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 2 Vol. 37; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ASYMPTOTIC SOLUTIONS
BOLTZMANN-VLASOV EQUATION
DIFFUSION
GROUP THEORY
HILBERT SPACE
MATHEMATICAL EVOLUTION
NONLINEAR PROBLEMS
PROBABILITY
RANDOMNESS
STOCHASTIC PROCESSES