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

Abstract

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 formwhere {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).

Stochastic Fubini Theorem for Semimartingales in Hilbert Space

“…It is often necessary to have sufficient conditions for the process Y l to be a version of the process Y 2 (e.g. [1], proof of Theorem 2.11). In the real case, i.e.…”

Stochastic Evolution Equations with Lévy Noise in the Dual of a Nuclear Space

“…Properties of solutions to stochastic evolution equations of the form (1.2) have been studied by many authors (e.g. [6,7,8,9,11,13,20,24,25,34,35,39]). In most of these works the authors assumed that Φ and Ψ are nuclear Fréchet spaces, and the driving noise L is a Wiener process or a square integrable martingale, although the case where L is assumed to be a semimartingale with independent increments has also been considered (e.g.…”

“…In most of these works the authors assumed that Φ and Ψ are nuclear Fréchet spaces, and the driving noise L is a Wiener process or a square integrable martingale, although the case where L is assumed to be a semimartingale with independent increments has also been considered (e.g. see [6]).…”

“…Here it is worth to mention that unlike other works that establish existence and uniqueness of solutions to equations of the form (1.2) (e.g. [6,24]), we do not require for the family of generators (A(t) : t ≥ 0) that each A(t) be a continuous and linear operator on Ψ.…”

Fubini theorem for anticipating stochastic integrals in Hilbert space

“…The stochastic Fubini theorem, which has been a useful tool for the analysis of some classes of stochastic differential equations (see, e.g., [2], [3], and [11]), has been studied by several authors. In the case where integrands are "adapted" to a given filtration, Le6n [11] established a result which generalizes those of [3], [6], [7], and [12], and in the case where the integrands are real-valued and "anticipating" with respect to the given filtration, Berger [1] used the homogeneous chaos expansion of the integrands to prove a similar result to that of [11].…”