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

Bojdecki, Tomasz ; Gorostiza, Luis G.

Canadian journal of mathematics, Tome 43 (1991) no. 6, pp. 1136-1149

Résumé

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).

Détail

DOI : 10.4153/CJM-1991-066-6

Mots-clés : 60G20, 60H15, 60H05

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