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Gaussian and Non–Gaussian Distribution–Valued Ornstein–Uhlenbeck Processes

Published online by Cambridge University Press:  20 November 2018

Tomasz Bojdecki  and
Luis G. Gorostiza
Tomasz Bojdecki
Affiliation:
Institute of Mathematics, University of Warsaw, Poland
Luis G. Gorostiza
Affiliation:
Centro de Investigación y de Estudios Avanzados, Mexico, D.F., Mexico
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Generalized (distribution-valued) Ornstein-Uhlenbeck processes, which by definition are solutions of generalized Langevin equations, arise in many investigations on fluctuation limits of particle systems (eg. Bojdecki and Gorostiza [1], Dawson, Fleischmann and Gorostiza [5], Fernández [7], Gorostiza [8,9], Holley and Stroock [10], Itô [12], Kallianpur and Pérez-Abreu [16], Kallianpur and Wolpert [14], Kotelenez [17], Martin-Löf [19], Mitoma [22], Uchiyama [25]). The state space for such a process is the strong dual Φ′ of a nuclear space Φ. A generalized Langevin equation for a Φ′-valued process X ≡ {Xt} is a stochastic evolution equation of the form

where {At} is a family of linear operators on Φ and Z ≡ {Zt} is a Φ'-valued semimartingale (in some sense) with independent increments. Equations of the type (1.1) where Z does not have independent increments also arise in applications (eg. [9,14,20]) but here we are interested precisely in the case when Z has independent increments (we restrict the term generalized Langevin equation to this case in accordance with the classical Langevin equation).

Keywords

60G2060H1560H05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 6 , 01 December 1991 , pp. 1136 - 1149
Copyright
Copyright © Canadian Mathematical Society 1991

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