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On ergodicity of some Markov processes. (English) Zbl 1214.60035
The article proves some ergodicity results for Feller processes on general (i.e., not necessarily locally compact) Polish state spaces. These are then applied to stochastic evolution equations with additive noise in Hilbert spaces.
\((\chi, \rho)\) is supposed to be a metric Polish space. \(C_b(\chi)\) denotes the space of bounded continuous functions on \(\chi\), and Lip\(_b(\chi)\) denotes bounded Lipschitz continuous functions. The ball of size \(\delta\) with center \(x\) is denoted by \(B(x,\delta)\). \(((Z^x_t)_{t \geq 0}, x \in \chi)\) is supposed to be a stochastically continuous Markov family with transition semigroup \((P_t)_{t \geq 0}\). \((P_t)\) is supposed to be a Feller semigroup, i.e., \(P_t(C_b(\chi)) \subseteq C_b(\chi)\).
The authors introduce the “e-property”: The semigroup \((P_t)_{t\geq 0}\) has the e-property if for any \(\psi \in\) Lip\(_b(\chi)\), for any \(x \in \chi\), and for any \(\varepsilon > 0\) there exists \(\delta > 0\) such that for all \(z\) with \(d(x,z) < \delta\) and for all \(t \geq 0\): \[ |P_t \psi(x) - P_t \psi(z)| < \varepsilon \] i.e., if the family of functions \((P_t\psi)_{t \geq 0}\) is equicontinuous at every \(x \in \chi\).
They also define for every \(T > 0\) and any probability measure \(\mu\) on \(\chi\) \[ Q^T \mu = \frac{1}{T} \int_0^T P^*_s \mu ds \] where \(P^*_s \mu\) is the usual dual of \(P_s\) acting on the space of finite positive measures: \[ P^*_s \mu(B) = \int_\chi P_s 1_B(x) \mu(dx) \] The set \(\tau\) is defined as the set of all \(x \in \chi\) for which \((Q^T \delta_x)_{T > 0}\) is a tight family of probability measures.
The first main theorem states that if \((P_t)_{t \geq 0}\) has the e-property, and if there exists \(z \in \chi\) such that for every \(\delta > 0\) and for every \(x \in \chi\) \[ \liminf_{T\rightarrow \infty} (Q^T\delta_x)(B(z,\delta)) > 0, \] then there exists a unique invariant probability measure \(\mu^*\) for \((P_t)_{t \geq 0}\). Further \(Q^T\nu\) converges weakly to \(\mu^*\) for every probability measure \(\nu\) whose support is included in \(\tau\).
The authors proceed by giving an example where the set \(\tau\) does not equal \(\chi\). This serves as a motivation for the next result. A semigroup \((P_t)_{t \geq 0}\) is called weak-\(^*\) mean ergodic if there exists a measure \(\mu^*\) on \(\chi\) such that for every probability measure \(\nu\) on \(\chi\), \(Q^T\nu\) converges weakly to \(\mu^*\) as \(T\) goes to infinity.
Theorem 2 gives a refinement of Theorem 1 in the case where the lower bound is satisfied locally uniformly: Let \((P_t)_{t \geq 0}\) again be a Feller semigroup satisfying the e-property. Now assume that there exists \(z \in \chi\) such that for every bounded set \(A\) and every \(\delta > 0\) \[ \inf_{x\in A} \liminf_{T\rightarrow \infty} (Q^T \delta_x)(B(z,\delta)) > 0 \] Further assume that for every \(\varepsilon > 0\) and every \(x \in \chi\) there exists a bounded Borel set \(D \subset \chi\) such that \[ \liminf_{T\rightarrow \infty} (Q^T\delta_x)(D) > 1 - \varepsilon \] Then there exists a unique invariant probability measure \(\mu^*\), the semigroup \((P_t)_{t \geq 0}\) is weak-\(^*\) mean ergodic, and for every \(\psi \in \) Lip\(_b(\chi)\) and every probability measure \(\nu\) on \(\chi\) the weak law of large numbers holds: \[ \lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T \psi(Z_s) ds = \int_\chi \psi(x) \mu^*(dx) \] where the convergence holds in \(P_\nu\)-probability.
These results are then applied to stochastic evolution equations in Hilbert spaces.

MSC:
60J25 Continuous-time Markov processes on general state spaces
37A30 Ergodic theorems, spectral theory, Markov operators
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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