# American Institute of Mathematical Sciences

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June  2016, 21(4): 1051-1077. doi: 10.3934/dcdsb.2016.21.1051

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

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

Received  January 2015 Revised  October 2015 Published  March 2016

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 $p>4$.
Citation: Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051
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