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Stochastic evolution equations driven by Liouville fractional Brownian motion

Zdzisław Brzeźniak, Jan van Neerven, Donna Salopek (2012)

Czechoslovak Mathematical Journal

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ ( H , E ) -valued functions with respect to H -cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1 . For 0 < β < 1 2 we show that a function Φ : ( 0 , T ) → ℒ ( H , E ) is stochastically integrable with respect to an H -cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H -cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations...

The falling apart of the tagged fragment and the asymptotic disintegration of the brownian height fragmentation

Gerónimo Uribe Bravo (2009)

Annales de l'I.H.P. Probabilités et statistiques

We present a further analysis of the fragmentation at heights of the normalized brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the brownian height fragmentation when it is...

Uniqueness of Brownian motion on Sierpiński carpets

Martin Barlow, Richard F. Bass, Takashi Kumagai, Alexander Teplyaev (2010)

Journal of the European Mathematical Society

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpi´nski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

Universality of the asymptotics of the one-sided exit problem for integrated processes

Frank Aurzada, Steffen Dereich (2013)

Annales de l'I.H.P. Probabilités et statistiques

We consider the one-sided exit problem – also called one-sided barrier problem – for ( α -fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function α ↦ θ ( α ) such that the probability that any α -fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time T behaves as T - θ ( α ) + o ( 1 ) for large T . We also investigate when the fixed level can be replaced by a different barrier...

Wavelet estimation of the long memory parameter for Hermite polynomial of gaussian processes

M. Clausel, F. Roueff, M. S. Taqqu, C. Tudor (2014)

ESAIM: Probability and Statistics

We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long–memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener–Itô integral of order 2. This happens even if the original...

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