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A Segal-Langevin Type Stochastic Differential Equation on a Space Of Generalized Functionals

Published online by Cambridge University Press:  20 November 2018

Gopinath Kallianpur
Affiliation:
Department of Statistics University of North Carolina Chapel Hill, NC 27599-3260 U.S.A.
Itaru Mitoma
Affiliation:
Department of Mathematics Saga University Saga 840, Japan
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Abstract

Let E′ be the dual of a nuclear Fréchet space E and L*(t) the adjoint operator of a diffusion operator L(t) of infinitely many variables, which has a formal expression:

A weak form of the stochastic differential equation

dX(t) = dW(t) + L*(t)X(t)dt

is introduced and the existence of a unique solution is established. The solution process is a random linear functional (in the sense of I. E. Segal) on a space of generalized functionals on E′. The above is an appropriate model for the central limit theorem for an interacting system of spatially extended neurons. Applications to the latter problem are discussed.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1992

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