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Some results on stochastic convolutions arising in Volterra equations perturbed by noise

Philippe Clément, Giuseppe Da Prato (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Regularity of stochastic convolutions corresponding to a Volterra equation, perturbed by a white noise, is studied. Under suitable assumptions, hölderianity of the corresponding trajectories is proved.

Some thoughts about Segal's ergodic theorem

Daniel W. Stroock (2010)

Colloquium Mathematicae

Over fifty years ago, Irving Segal proved a theorem which leads to a characterization of those orthogonal transformations on a Hilbert space which induce ergodic transformations. Because Segal did not present his result in a way which made it readily accessible to specialists in ergodic theory, it was difficult for them to appreciate what he had done. The purpose of this note is to state and prove Segal's result in a way which, I hope, will win it the recognition which it deserves.

Stochastic continuity and approximation

Leon Brown, Bertram Schreiber (1996)

Studia Mathematica

This work is concerned with the study of stochastic processes which are continuous in probability, over various parameter spaces, from the point of view of approximation and extension. A stochastic version of the classical theorem of Mergelyan on polynomial approximation is shown to be valid for subsets of the plane whose boundaries are sets of rational approximation. In a similar vein, one can obtain a version in the context of continuity in probability of the theorem of Arakelyan on the uniform...

Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions

Naresh C. Jain, Michael B. Marcus (1974)

Annales de l'institut Fourier

Let { X ( t ) , t ∈ [ 0 , 1 ] n } be a stochastically continuous, separable, Gaussian process with E [ X ( t + h ) - X ( t ) ] 2 = σ 2 ( | h | ) . A sufficient condition, in terms of the monotone rearrangement of σ , is obtained for X ( t ) to have continuous sample paths almost surely. This result is applied to a wide class of random series of functions, in particular, to random Fourier series.

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