Semilinear stochastic equations with bilinear fractional noise

María J.  Garrido-Atienza; Bohdan  Maslowski; Jana  Šnupárková

doi:10.3934/dcdsb.2016088

Article Contents

2016, Volume 21, Issue 9: 3075-3094. Doi: 10.3934/dcdsb.2016088

This issue Previous Article Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions Next Article Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes

Semilinear stochastic equations with bilinear fractional noise

1.
Dpto. Ecuaciones Diferenciales y Análisis Numérico , Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2.
Charles University in Prague, Faculty of Mathematics and Physics, Sokolovska 83, Prague 8

Received: February 2016

Revised: March 2016

Published: November 2016

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Abstract

In the paper, we study existence and uniqueness of solutions to semilinear stochastic evolution systems, driven by a fractional Brownian motion with bilinear noise term, and the long time behavior of solutions to such equations. For this purpose, we study at first the random evolution operator defined by the corresponding bilinear equation which is later used to define the mild solution of the semilinear equation. The mild solution is also shown to be weak in the PDE sense. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. We show that the solution generates a random dynamical system that, under appropriate stability and compactness conditions, possesses a random attractor.

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Mathematics Subject Classification: Primary: 60H15; Secondary: 60G22, 37L55.

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