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Gaussian and non-Gaussian distribution-valued Ornstein-Uhlenbeck processes. (English) Zbl 0753.60037

Let \(\Phi\) be a nuclear Fréchet space and let \(\{A_ t:0\leq t\leq T\}\) be a family of continuous operators from \(\Phi\) into \(\Phi\), generating an evolution system \(\{U_{s,t}:0\leq s\leq t\leq T\}\) with some minimal regularity conditions. The aim of the paper is to determine when a given \(\Phi'\)-valued process \(X\) is a generalized Ornstein-Uhlenbeck process, i.e., satisfies the equation \(dX_ t=A^*_ tX_ tdt+dZ_ t\), where \(Z\) is some \(\Phi'\)-valued process with independent increments. The most general sufficient condition is: \[ E[e^{i\langle X_ t,\varphi\rangle}\mid\langle X_ r,\psi\rangle,\;r\leq s,\;\psi\in\Phi]=e^{i\langle X_ s,U_{s,t}\varphi\rangle}H(s,t;\varphi),\;\varphi\in\Phi,\;0\leq s\leq t\leq T, \] for some deterministic complex-valued \(H(s,t;\varphi)\). In most cases, this condition is proved to be necessary as well. Moreover, this condition can be quite frequently given somewhat more explicit form which permits to obtain the laws of the increments of the process \(Z\). The paper generalizes earlier results, obtained for Gaussian processes [authors, Probab. Theory Relat. Fields 73, 227-244 (1986; Zbl 0581.60094)].

MSC:

60G20 Generalized stochastic processes
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
60J99 Markov processes
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