Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces

Zdzisław  Brzeźniak; Paul André Razafimandimby

doi:10.3934/dcdsb.2016.21.1051

Article Contents

2016, Volume 21, Issue 4: 1051-1077. Doi: 10.3934/dcdsb.2016.21.1051

This issue Previous Article Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization Next Article On the spectral stability of standing waves of the one-dimensional $M^5$-model

Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces

Zdzisław Brzeźniak^1, and
Paul André Razafimandimby^2,

1.
Department of Mathematics, University of York, Heslington Road, York YO10 5DD
2.
Department of Mathematics and Information Technology, Montanuniversität Leoben, Franz Josef Straße 18, 8700 Leoben

Received: December 31, 2014

Revised: September 30, 2015

Published: May 2016

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Abstract

The main goal of this paper is to generalize to Banach spaces the well-known results for diffusions on Hilbert spaces obtained in [Peszat, S. and Zabczyk, J. (1995). Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23(1): 157-172.]. More precisely, we are aiming to prove the strong Feller property and irreducibility of the solutions to a stochastic evolution equations (SEEs) in Banach spaces. We give sufficient conditions on the path space and the coefficients of the SEEs for these aforementioned properties to hold. We apply our result to investigate the long-time behavior of a stochastic nonlinear heat equations on $L^p$-space with $p>4$. Our result implies the uniqueness of the invariant measure, if it exists, for the stochastic nonlinear heat equations on $L^p$-space with $p>4$.

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Mathematics Subject Classification: Primary: 60H15, 37L40; Secondary: 35R60.

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