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Ergodic behaviour of stochastic parabolic equations. (English) Zbl 0935.60041
The main goal of the paper is to investigate ergodic properties of Markov processes induced by semilinear stochastic equations of the form \(dX=(AX+F(X)) dt+\Sigma (X) dW\) in a Hilbert space \(H\), where \(A\) is an unbounded operator on \(H\) generating a strongly continuous semigroup, \(W\) is an \(H\)-valued Wiener process and \(F\) and \(\Sigma \) are nonlinear, \(H\)-valued and \(\mathcal L(H)\)-valued, respectively, functions on \(H\). Most of the results of the paper are however applicable to general strong Feller irreducible Markov processes taking values in Polish spaces. Criteria for existence and uniqueness of finite and \(\sigma \)-finite invariant measures are found and the relation between recurrence of the process and existence of invariant measures is studied in detail. An important ratio (or Hopf type) ergodic theorem is proven and a number of its consequences is given; in particular, convergence of transition probability kernels to the invariant measure in the norm of total variation is established. The results are applied to stochastic parabolic semilinear equations.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents
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