DCDS-B
Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises
Jiahui Zhu Zdzisław Brzeźniak
Discrete & Continuous Dynamical Systems - B 2016, 21(9): 3269-3299 doi: 10.3934/dcdsb.2016097
We study a class of abstract nonlinear stochastic equations of hyperbolic type driven by jump noises, which covers both beam equations with nonlocal, nonlinear terms and nonlinear wave equations. We derive an Itô formula for the local mild solution which plays an important role in the proof of our main results. Under appropriate conditions, we prove the non-explosion and the asymptotic stability of the mild solution.
keywords: Itô formula. local and global mild solution Stochastic nonlinear beam equation Lyapunov function Poisson random measure
DCDS-B
Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces
Zdzisław Brzeźniak Paul André Razafimandimby
Discrete & Continuous Dynamical Systems - B 2016, 21(4): 1051-1077 doi: 10.3934/dcdsb.2016.21.1051
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 $p>4$.
keywords: M-type 2 Banach spaces. stochastic evolution equations Markov semigroup strong Feller Bismut-Elworthy-Li formula invariant measure irreducible

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