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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  and
Itaru Mitoma
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 Saga840, Japan
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Abstract

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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.

Keywords

46F2560F0560H1535K22Weak solutionSDEFréchet derivativegeneralized functional spacecentral limit theoremsystem of neurons
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 3 , 01 June 1992 , pp. 524 - 552
Copyright
Copyright © Canadian Mathematical Society 1992

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